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What is Hypothesis Testing? Types and Methods

• Soumyaa Rawat
• Jul 23, 2021

Hypothesis Testing  

Hypothesis testing is the act of testing a hypothesis or a supposition in relation to a statistical parameter. Analysts implement hypothesis testing in order to test if a hypothesis is plausible or not. 

In data science and statistics , hypothesis testing is an important step as it involves the verification of an assumption that could help develop a statistical parameter. For instance, a researcher establishes a hypothesis assuming that the average of all odd numbers is an even number. 

In order to find the plausibility of this hypothesis, the researcher will have to test the hypothesis using hypothesis testing methods. Unlike a hypothesis that is ‘supposed’ to stand true on the basis of little or no evidence, hypothesis testing is required to have plausible evidence in order to establish that a statistical hypothesis is true. 

Perhaps this is where statistics play an important role. A number of components are involved in this process. But before understanding the process involved in hypothesis testing in research methodology, we shall first understand the types of hypotheses that are involved in the process. Let us get started! 

Types of Hypotheses

In data sampling, different types of hypothesis are involved in finding whether the tested samples test positive for a hypothesis or not. In this segment, we shall discover the different types of hypotheses and understand the role they play in hypothesis testing.

Alternative Hypothesis

Alternative Hypothesis (H1) or the research hypothesis states that there is a relationship between two variables (where one variable affects the other). The alternative hypothesis is the main driving force for hypothesis testing. 

It implies that the two variables are related to each other and the relationship that exists between them is not due to chance or coincidence. 

When the process of hypothesis testing is carried out, the alternative hypothesis is the main subject of the testing process. The analyst intends to test the alternative hypothesis and verifies its plausibility.

Null Hypothesis

The Null Hypothesis (H0) aims to nullify the alternative hypothesis by implying that there exists no relation between two variables in statistics. It states that the effect of one variable on the other is solely due to chance and no empirical cause lies behind it. 

The null hypothesis is established alongside the alternative hypothesis and is recognized as important as the latter. In hypothesis testing, the null hypothesis has a major role to play as it influences the testing against the alternative hypothesis. 

(Must read: What is ANOVA test? )

Non-Directional Hypothesis

The Non-directional hypothesis states that the relation between two variables has no direction. 

Simply put, it asserts that there exists a relation between two variables, but does not recognize the direction of effect, whether variable A affects variable B or vice versa. 

Directional Hypothesis

The Directional hypothesis, on the other hand, asserts the direction of effect of the relationship that exists between two variables. 

Herein, the hypothesis clearly states that variable A affects variable B, or vice versa. 

Statistical Hypothesis

A statistical hypothesis is a hypothesis that can be verified to be plausible on the basis of statistics. 

By using data sampling and statistical knowledge, one can determine the plausibility of a statistical hypothesis and find out if it stands true or not. 

(Related blog: z-test vs t-test )

Performing Hypothesis Testing  

Now that we have understood the types of hypotheses and the role they play in hypothesis testing, let us now move on to understand the process in a better manner. 

In hypothesis testing, a researcher is first required to establish two hypotheses - alternative hypothesis and null hypothesis in order to begin with the procedure. 

To establish these two hypotheses, one is required to study data samples, find a plausible pattern among the samples, and pen down a statistical hypothesis that they wish to test. 

A random population of samples can be drawn, to begin with hypothesis testing. Among the two hypotheses, alternative and null, only one can be verified to be true. Perhaps the presence of both hypotheses is required to make the process successful. 

At the end of the hypothesis testing procedure, either of the hypotheses will be rejected and the other one will be supported. Even though one of the two hypotheses turns out to be true, no hypothesis can ever be verified 100%. 

(Read also: Types of data sampling techniques )

Therefore, a hypothesis can only be supported based on the statistical samples and verified data. Here is a step-by-step guide for hypothesis testing.

Establish the hypotheses

First things first, one is required to establish two hypotheses - alternative and null, that will set the foundation for hypothesis testing. 

These hypotheses initiate the testing process that involves the researcher working on data samples in order to either support the alternative hypothesis or the null hypothesis. 

Generate a testing plan

Once the hypotheses have been formulated, it is now time to generate a testing plan. A testing plan or an analysis plan involves the accumulation of data samples, determining which statistic is to be considered and laying out the sample size. 

All these factors are very important while one is working on hypothesis testing.

Analyze data samples

As soon as a testing plan is ready, it is time to move on to the analysis part. Analysis of data samples involves configuring statistical values of samples, drawing them together, and deriving a pattern out of these samples. 

While analyzing the data samples, a researcher needs to determine a set of things -

Significance Level - The level of significance in hypothesis testing indicates if a statistical result could have significance if the null hypothesis stands to be true.

Testing Method - The testing method involves a type of sampling-distribution and a test statistic that leads to hypothesis testing. There are a number of testing methods that can assist in the analysis of data samples. 

Test statistic - Test statistic is a numerical summary of a data set that can be used to perform hypothesis testing.

P-value - The P-value interpretation is the probability of finding a sample statistic to be as extreme as the test statistic, indicating the plausibility of the null hypothesis. 

Infer the results

The analysis of data samples leads to the inference of results that establishes whether the alternative hypothesis stands true or not. When the P-value is less than the significance level, the null hypothesis is rejected and the alternative hypothesis turns out to be plausible. 

Methods of Hypothesis Testing

As we have already looked into different aspects of hypothesis testing, we shall now look into the different methods of hypothesis testing. All in all, there are 2 most common types of hypothesis testing methods. They are as follows -

Frequentist Hypothesis Testing

The frequentist hypothesis or the traditional approach to hypothesis testing is a hypothesis testing method that aims on making assumptions by considering current data. 

The supposed truths and assumptions are based on the current data and a set of 2 hypotheses are formulated. A very popular subtype of the frequentist approach is the Null Hypothesis Significance Testing (NHST). 

The NHST approach (involving the null and alternative hypothesis) has been one of the most sought-after methods of hypothesis testing in the field of statistics ever since its inception in the mid-1950s. 

Bayesian Hypothesis Testing

A much unconventional and modern method of hypothesis testing, the Bayesian Hypothesis Testing claims to test a particular hypothesis in accordance with the past data samples, known as prior probability, and current data that lead to the plausibility of a hypothesis. 

The result obtained indicates the posterior probability of the hypothesis. In this method, the researcher relies on ‘prior probability and posterior probability’ to conduct hypothesis testing on hand. 

On the basis of this prior probability, the Bayesian approach tests a hypothesis to be true or false. The Bayes factor, a major component of this method, indicates the likelihood ratio among the null hypothesis and the alternative hypothesis. 

The Bayes factor is the indicator of the plausibility of either of the two hypotheses that are established for hypothesis testing.  

(Also read - Introduction to Bayesian Statistics ) 

To conclude, hypothesis testing, a way to verify the plausibility of a supposed assumption can be done through different methods - the Bayesian approach or the Frequentist approach. 

Although the Bayesian approach relies on the prior probability of data samples, the frequentist approach assumes without a probability. A number of elements involved in hypothesis testing are - significance level, p-level, test statistic, and method of hypothesis testing. 

(Also read: Introduction to probability distributions )

A significant way to determine whether a hypothesis stands true or not is to verify the data samples and identify the plausible hypothesis among the null hypothesis and alternative hypothesis. 

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Hypothesis Testing – A Deep Dive into Hypothesis Testing, The Backbone of Statistical Inference

• September 21, 2023

Explore the intricacies of hypothesis testing, a cornerstone of statistical analysis. Dive into methods, interpretations, and applications for making data-driven decisions.

In this Blog post we will learn:

• What is Hypothesis Testing?
• Steps in Hypothesis Testing 2.1. Set up Hypotheses: Null and Alternative 2.2. Choose a Significance Level (α) 2.3. Calculate a test statistic and P-Value 2.4. Make a Decision
• Example : Testing a new drug.
• Example in python

1. What is Hypothesis Testing?

In simple terms, hypothesis testing is a method used to make decisions or inferences about population parameters based on sample data. Imagine being handed a dice and asked if it’s biased. By rolling it a few times and analyzing the outcomes, you’d be engaging in the essence of hypothesis testing.

Think of hypothesis testing as the scientific method of the statistics world. Suppose you hear claims like “This new drug works wonders!” or “Our new website design boosts sales.” How do you know if these statements hold water? Enter hypothesis testing.

2. Steps in Hypothesis Testing

• Set up Hypotheses : Begin with a null hypothesis (H0) and an alternative hypothesis (Ha).
• Choose a Significance Level (α) : Typically 0.05, this is the probability of rejecting the null hypothesis when it’s actually true. Think of it as the chance of accusing an innocent person.
• Calculate Test statistic and P-Value : Gather evidence (data) and calculate a test statistic.
• p-value : This is the probability of observing the data, given that the null hypothesis is true. A small p-value (typically ≤ 0.05) suggests the data is inconsistent with the null hypothesis.
• Decision Rule : If the p-value is less than or equal to α, you reject the null hypothesis in favor of the alternative.

2.1. Set up Hypotheses: Null and Alternative

Before diving into testing, we must formulate hypotheses. The null hypothesis (H0) represents the default assumption, while the alternative hypothesis (H1) challenges it.

For instance, in drug testing, H0 : “The new drug is no better than the existing one,” H1 : “The new drug is superior .”

2.2. Choose a Significance Level (α)

When You collect and analyze data to test H0 and H1 hypotheses. Based on your analysis, you decide whether to reject the null hypothesis in favor of the alternative, or fail to reject / Accept the null hypothesis.

The significance level, often denoted by $α$, represents the probability of rejecting the null hypothesis when it is actually true.

In other words, it’s the risk you’re willing to take of making a Type I error (false positive).

Type I Error (False Positive) :

• Symbolized by the Greek letter alpha (α).
• Occurs when you incorrectly reject a true null hypothesis . In other words, you conclude that there is an effect or difference when, in reality, there isn’t.
• The probability of making a Type I error is denoted by the significance level of a test. Commonly, tests are conducted at the 0.05 significance level , which means there’s a 5% chance of making a Type I error .
• Commonly used significance levels are 0.01, 0.05, and 0.10, but the choice depends on the context of the study and the level of risk one is willing to accept.

Example : If a drug is not effective (truth), but a clinical trial incorrectly concludes that it is effective (based on the sample data), then a Type I error has occurred.

Type II Error (False Negative) :

• Symbolized by the Greek letter beta (β).
• Occurs when you accept a false null hypothesis . This means you conclude there is no effect or difference when, in reality, there is.
• The probability of making a Type II error is denoted by β. The power of a test (1 – β) represents the probability of correctly rejecting a false null hypothesis.

Example : If a drug is effective (truth), but a clinical trial incorrectly concludes that it is not effective (based on the sample data), then a Type II error has occurred.

Balancing the Errors :

In practice, there’s a trade-off between Type I and Type II errors. Reducing the risk of one typically increases the risk of the other. For example, if you want to decrease the probability of a Type I error (by setting a lower significance level), you might increase the probability of a Type II error unless you compensate by collecting more data or making other adjustments.

It’s essential to understand the consequences of both types of errors in any given context. In some situations, a Type I error might be more severe, while in others, a Type II error might be of greater concern. This understanding guides researchers in designing their experiments and choosing appropriate significance levels.

2.3. Calculate a test statistic and P-Value

Test statistic : A test statistic is a single number that helps us understand how far our sample data is from what we’d expect under a null hypothesis (a basic assumption we’re trying to test against). Generally, the larger the test statistic, the more evidence we have against our null hypothesis. It helps us decide whether the differences we observe in our data are due to random chance or if there’s an actual effect.

P-value : The P-value tells us how likely we would get our observed results (or something more extreme) if the null hypothesis were true. It’s a value between 0 and 1. – A smaller P-value (typically below 0.05) means that the observation is rare under the null hypothesis, so we might reject the null hypothesis. – A larger P-value suggests that what we observed could easily happen by random chance, so we might not reject the null hypothesis.

2.4. Make a Decision

Relationship between $α$ and P-Value

When conducting a hypothesis test:

We then calculate the p-value from our sample data and the test statistic.

Finally, we compare the p-value to our chosen $α$:

• If $p−value≤α$: We reject the null hypothesis in favor of the alternative hypothesis. The result is said to be statistically significant.
• If $p−value>α$: We fail to reject the null hypothesis. There isn’t enough statistical evidence to support the alternative hypothesis.

3. Example : Testing a new drug.

Imagine we are investigating whether a new drug is effective at treating headaches faster than drug B.

Setting Up the Experiment : You gather 100 people who suffer from headaches. Half of them (50 people) are given the new drug (let’s call this the ‘Drug Group’), and the other half are given a sugar pill, which doesn’t contain any medication.

• Set up Hypotheses : Before starting, you make a prediction:
• Null Hypothesis (H0): The new drug has no effect. Any difference in healing time between the two groups is just due to random chance.
• Alternative Hypothesis (H1): The new drug does have an effect. The difference in healing time between the two groups is significant and not just by chance.

Calculate Test statistic and P-Value : After the experiment, you analyze the data. The “test statistic” is a number that helps you understand the difference between the two groups in terms of standard units.

For instance, let’s say:

• The average healing time in the Drug Group is 2 hours.
• The average healing time in the Placebo Group is 3 hours.

The test statistic helps you understand how significant this 1-hour difference is. If the groups are large and the spread of healing times in each group is small, then this difference might be significant. But if there’s a huge variation in healing times, the 1-hour difference might not be so special.

Imagine the P-value as answering this question: “If the new drug had NO real effect, what’s the probability that I’d see a difference as extreme (or more extreme) as the one I found, just by random chance?”

For instance:

• P-value of 0.01 means there’s a 1% chance that the observed difference (or a more extreme difference) would occur if the drug had no effect. That’s pretty rare, so we might consider the drug effective.
• P-value of 0.5 means there’s a 50% chance you’d see this difference just by chance. That’s pretty high, so we might not be convinced the drug is doing much.
• If the P-value is less than ($α$) 0.05: the results are “statistically significant,” and they might reject the null hypothesis , believing the new drug has an effect.
• If the P-value is greater than ($α$) 0.05: the results are not statistically significant, and they don’t reject the null hypothesis , remaining unsure if the drug has a genuine effect.

4. Example in python

For simplicity, let’s say we’re using a t-test (common for comparing means). Let’s dive into Python:

Making a Decision : “The results are statistically significant! p-value < 0.05 , The drug seems to have an effect!” If not, we’d say, “Looks like the drug isn’t as miraculous as we thought.”

5. Conclusion

Hypothesis testing is an indispensable tool in data science, allowing us to make data-driven decisions with confidence. By understanding its principles, conducting tests properly, and considering real-world applications, you can harness the power of hypothesis testing to unlock valuable insights from your data.

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In today’s data-driven world , decisions are based on data all the time. Hypothesis plays a crucial role in that process, whether it may be making business decisions, in the health sector, academia, or in quality improvement. Without hypothesis & hypothesis tests, you risk drawing the wrong conclusions and making bad decisions. In this tutorial, you will look at Hypothesis Testing in Statistics.

What Is Hypothesis Testing in Statistics?

Hypothesis Testing is a type of statistical analysis in which you put your assumptions about a population parameter to the test. It is used to estimate the relationship between 2 statistical variables.

Let's discuss few examples of statistical hypothesis from real-life - 

• A teacher assumes that 60% of his college's students come from lower-middle-class families.
• A doctor believes that 3D (Diet, Dose, and Discipline) is 90% effective for diabetic patients.

Now that you know about hypothesis testing, look at the two types of hypothesis testing in statistics.

Hypothesis Testing Formula

Z = ( x̅ – μ0 ) / (σ /√n)

• Here, x̅ is the sample mean,
• μ0 is the population mean,
• σ is the standard deviation,
• n is the sample size.

How Hypothesis Testing Works?

An analyst performs hypothesis testing on a statistical sample to present evidence of the plausibility of the null hypothesis. Measurements and analyses are conducted on a random sample of the population to test a theory. Analysts use a random population sample to test two hypotheses: the null and alternative hypotheses.

The null hypothesis is typically an equality hypothesis between population parameters; for example, a null hypothesis may claim that the population means return equals zero. The alternate hypothesis is essentially the inverse of the null hypothesis (e.g., the population means the return is not equal to zero). As a result, they are mutually exclusive, and only one can be correct. One of the two possibilities, however, will always be correct.

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Null Hypothesis and Alternate Hypothesis

The Null Hypothesis is the assumption that the event will not occur. A null hypothesis has no bearing on the study's outcome unless it is rejected.

H0 is the symbol for it, and it is pronounced H-naught.

The Alternate Hypothesis is the logical opposite of the null hypothesis. The acceptance of the alternative hypothesis follows the rejection of the null hypothesis. H1 is the symbol for it.

Let's understand this with an example.

A sanitizer manufacturer claims that its product kills 95 percent of germs on average. 

To put this company's claim to the test, create a null and alternate hypothesis.

H0 (Null Hypothesis): Average = 95%.

Alternative Hypothesis (H1): The average is less than 95%.

Another straightforward example to understand this concept is determining whether or not a coin is fair and balanced. The null hypothesis states that the probability of a show of heads is equal to the likelihood of a show of tails. In contrast, the alternate theory states that the probability of a show of heads and tails would be very different.

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Hypothesis Testing Calculation With Examples

Let's consider a hypothesis test for the average height of women in the United States. Suppose our null hypothesis is that the average height is 5'4". We gather a sample of 100 women and determine that their average height is 5'5". The standard deviation of population is 2.

To calculate the z-score, we would use the following formula:

z = ( x̅ – μ0 ) / (σ /√n)

z = (5'5" - 5'4") / (2" / √100)

z = 0.5 / (0.045)

 We will reject the null hypothesis as the z-score of 11.11 is very large and conclude that there is evidence to suggest that the average height of women in the US is greater than 5'4".

Steps of Hypothesis Testing

Step 1: specify your null and alternate hypotheses.

It is critical to rephrase your original research hypothesis (the prediction that you wish to study) as a null (Ho) and alternative (Ha) hypothesis so that you can test it quantitatively. Your first hypothesis, which predicts a link between variables, is generally your alternate hypothesis. The null hypothesis predicts no link between the variables of interest.

Step 2: Gather Data

For a statistical test to be legitimate, sampling and data collection must be done in a way that is meant to test your hypothesis. You cannot draw statistical conclusions about the population you are interested in if your data is not representative.

Step 3: Conduct a Statistical Test

Other statistical tests are available, but they all compare within-group variance (how to spread out the data inside a category) against between-group variance (how different the categories are from one another). If the between-group variation is big enough that there is little or no overlap between groups, your statistical test will display a low p-value to represent this. This suggests that the disparities between these groups are unlikely to have occurred by accident. Alternatively, if there is a large within-group variance and a low between-group variance, your statistical test will show a high p-value. Any difference you find across groups is most likely attributable to chance. The variety of variables and the level of measurement of your obtained data will influence your statistical test selection.

Step 4: Determine Rejection Of Your Null Hypothesis

Your statistical test results must determine whether your null hypothesis should be rejected or not. In most circumstances, you will base your judgment on the p-value provided by the statistical test. In most circumstances, your preset level of significance for rejecting the null hypothesis will be 0.05 - that is, when there is less than a 5% likelihood that these data would be seen if the null hypothesis were true. In other circumstances, researchers use a lower level of significance, such as 0.01 (1%). This reduces the possibility of wrongly rejecting the null hypothesis.

Step 5: Present Your Results 

The findings of hypothesis testing will be discussed in the results and discussion portions of your research paper, dissertation, or thesis. You should include a concise overview of the data and a summary of the findings of your statistical test in the results section. You can talk about whether your results confirmed your initial hypothesis or not in the conversation. Rejecting or failing to reject the null hypothesis is a formal term used in hypothesis testing. This is likely a must for your statistics assignments.

Types of Hypothesis Testing

To determine whether a discovery or relationship is statistically significant, hypothesis testing uses a z-test. It usually checks to see if two means are the same (the null hypothesis). Only when the population standard deviation is known and the sample size is 30 data points or more, can a z-test be applied.

A statistical test called a t-test is employed to compare the means of two groups. To determine whether two groups differ or if a procedure or treatment affects the population of interest, it is frequently used in hypothesis testing.

Chi-Square 

You utilize a Chi-square test for hypothesis testing concerning whether your data is as predicted. To determine if the expected and observed results are well-fitted, the Chi-square test analyzes the differences between categorical variables from a random sample. The test's fundamental premise is that the observed values in your data should be compared to the predicted values that would be present if the null hypothesis were true.

Hypothesis Testing and Confidence Intervals

Both confidence intervals and hypothesis tests are inferential techniques that depend on approximating the sample distribution. Data from a sample is used to estimate a population parameter using confidence intervals. Data from a sample is used in hypothesis testing to examine a given hypothesis. We must have a postulated parameter to conduct hypothesis testing.

Bootstrap distributions and randomization distributions are created using comparable simulation techniques. The observed sample statistic is the focal point of a bootstrap distribution, whereas the null hypothesis value is the focal point of a randomization distribution.

A variety of feasible population parameter estimates are included in confidence ranges. In this lesson, we created just two-tailed confidence intervals. There is a direct connection between these two-tail confidence intervals and these two-tail hypothesis tests. The results of a two-tailed hypothesis test and two-tailed confidence intervals typically provide the same results. In other words, a hypothesis test at the 0.05 level will virtually always fail to reject the null hypothesis if the 95% confidence interval contains the predicted value. A hypothesis test at the 0.05 level will nearly certainly reject the null hypothesis if the 95% confidence interval does not include the hypothesized parameter.

Simple and Composite Hypothesis Testing

Depending on the population distribution, you can classify the statistical hypothesis into two types.

Simple Hypothesis: A simple hypothesis specifies an exact value for the parameter.

Composite Hypothesis: A composite hypothesis specifies a range of values.

A company is claiming that their average sales for this quarter are 1000 units. This is an example of a simple hypothesis.

Suppose the company claims that the sales are in the range of 900 to 1000 units. Then this is a case of a composite hypothesis.

One-Tailed and Two-Tailed Hypothesis Testing

The One-Tailed test, also called a directional test, considers a critical region of data that would result in the null hypothesis being rejected if the test sample falls into it, inevitably meaning the acceptance of the alternate hypothesis.

In a one-tailed test, the critical distribution area is one-sided, meaning the test sample is either greater or lesser than a specific value.

In two tails, the test sample is checked to be greater or less than a range of values in a Two-Tailed test, implying that the critical distribution area is two-sided.

If the sample falls within this range, the alternate hypothesis will be accepted, and the null hypothesis will be rejected.

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Right Tailed Hypothesis Testing

If the larger than (>) sign appears in your hypothesis statement, you are using a right-tailed test, also known as an upper test. Or, to put it another way, the disparity is to the right. For instance, you can contrast the battery life before and after a change in production. Your hypothesis statements can be the following if you want to know if the battery life is longer than the original (let's say 90 hours):

• The null hypothesis is (H0 <= 90) or less change.
• A possibility is that battery life has risen (H1) > 90.

The crucial point in this situation is that the alternate hypothesis (H1), not the null hypothesis, decides whether you get a right-tailed test.

Left Tailed Hypothesis Testing

Alternative hypotheses that assert the true value of a parameter is lower than the null hypothesis are tested with a left-tailed test; they are indicated by the asterisk "<".

Suppose H0: mean = 50 and H1: mean not equal to 50

According to the H1, the mean can be greater than or less than 50. This is an example of a Two-tailed test.

In a similar manner, if H0: mean >=50, then H1: mean <50

Here the mean is less than 50. It is called a One-tailed test.

Type 1 and Type 2 Error

A hypothesis test can result in two types of errors.

Type 1 Error: A Type-I error occurs when sample results reject the null hypothesis despite being true.

Type 2 Error: A Type-II error occurs when the null hypothesis is not rejected when it is false, unlike a Type-I error.

Suppose a teacher evaluates the examination paper to decide whether a student passes or fails.

H0: Student has passed

H1: Student has failed

Type I error will be the teacher failing the student [rejects H0] although the student scored the passing marks [H0 was true]. 

Type II error will be the case where the teacher passes the student [do not reject H0] although the student did not score the passing marks [H1 is true].

Level of Significance

The alpha value is a criterion for determining whether a test statistic is statistically significant. In a statistical test, Alpha represents an acceptable probability of a Type I error. Because alpha is a probability, it can be anywhere between 0 and 1. In practice, the most commonly used alpha values are 0.01, 0.05, and 0.1, which represent a 1%, 5%, and 10% chance of a Type I error, respectively (i.e. rejecting the null hypothesis when it is in fact correct).

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A p-value is a metric that expresses the likelihood that an observed difference could have occurred by chance. As the p-value decreases the statistical significance of the observed difference increases. If the p-value is too low, you reject the null hypothesis.

Here you have taken an example in which you are trying to test whether the new advertising campaign has increased the product's sales. The p-value is the likelihood that the null hypothesis, which states that there is no change in the sales due to the new advertising campaign, is true. If the p-value is .30, then there is a 30% chance that there is no increase or decrease in the product's sales.  If the p-value is 0.03, then there is a 3% probability that there is no increase or decrease in the sales value due to the new advertising campaign. As you can see, the lower the p-value, the chances of the alternate hypothesis being true increases, which means that the new advertising campaign causes an increase or decrease in sales.

Why is Hypothesis Testing Important in Research Methodology?

Hypothesis testing is crucial in research methodology for several reasons:

• Provides evidence-based conclusions: It allows researchers to make objective conclusions based on empirical data, providing evidence to support or refute their research hypotheses.
• Supports decision-making: It helps make informed decisions, such as accepting or rejecting a new treatment, implementing policy changes, or adopting new practices.
• Adds rigor and validity: It adds scientific rigor to research using statistical methods to analyze data, ensuring that conclusions are based on sound statistical evidence.
• Contributes to the advancement of knowledge: By testing hypotheses, researchers contribute to the growth of knowledge in their respective fields by confirming existing theories or discovering new patterns and relationships.

Limitations of Hypothesis Testing

Hypothesis testing has some limitations that researchers should be aware of:

• It cannot prove or establish the truth: Hypothesis testing provides evidence to support or reject a hypothesis, but it cannot confirm the absolute truth of the research question.
• Results are sample-specific: Hypothesis testing is based on analyzing a sample from a population, and the conclusions drawn are specific to that particular sample.
• Possible errors: During hypothesis testing, there is a chance of committing type I error (rejecting a true null hypothesis) or type II error (failing to reject a false null hypothesis).
• Assumptions and requirements: Different tests have specific assumptions and requirements that must be met to accurately interpret results.

After reading this tutorial, you would have a much better understanding of hypothesis testing, one of the most important concepts in the field of Data Science . The majority of hypotheses are based on speculation about observed behavior, natural phenomena, or established theories.

If you are interested in statistics of data science and skills needed for such a career, you ought to explore Simplilearn’s Post Graduate Program in Data Science.

If you have any questions regarding this ‘Hypothesis Testing In Statistics’ tutorial, do share them in the comment section. Our subject matter expert will respond to your queries. Happy learning!

1. What is hypothesis testing in statistics with example?

Hypothesis testing is a statistical method used to determine if there is enough evidence in a sample data to draw conclusions about a population. It involves formulating two competing hypotheses, the null hypothesis (H0) and the alternative hypothesis (Ha), and then collecting data to assess the evidence. An example: testing if a new drug improves patient recovery (Ha) compared to the standard treatment (H0) based on collected patient data.

2. What is hypothesis testing and its types?

Hypothesis testing is a statistical method used to make inferences about a population based on sample data. It involves formulating two hypotheses: the null hypothesis (H0), which represents the default assumption, and the alternative hypothesis (Ha), which contradicts H0. The goal is to assess the evidence and determine whether there is enough statistical significance to reject the null hypothesis in favor of the alternative hypothesis.

Types of hypothesis testing:

• One-sample test: Used to compare a sample to a known value or a hypothesized value.
• Two-sample test: Compares two independent samples to assess if there is a significant difference between their means or distributions.
• Paired-sample test: Compares two related samples, such as pre-test and post-test data, to evaluate changes within the same subjects over time or under different conditions.
• Chi-square test: Used to analyze categorical data and determine if there is a significant association between variables.
• ANOVA (Analysis of Variance): Compares means across multiple groups to check if there is a significant difference between them.

3. What are the steps of hypothesis testing?

The steps of hypothesis testing are as follows:

• Formulate the hypotheses: State the null hypothesis (H0) and the alternative hypothesis (Ha) based on the research question.
• Set the significance level: Determine the acceptable level of error (alpha) for making a decision.
• Collect and analyze data: Gather and process the sample data.
• Compute test statistic: Calculate the appropriate statistical test to assess the evidence.
• Make a decision: Compare the test statistic with critical values or p-values and determine whether to reject H0 in favor of Ha or not.
• Draw conclusions: Interpret the results and communicate the findings in the context of the research question.

4. What are the 2 types of hypothesis testing?

• One-tailed (or one-sided) test: Tests for the significance of an effect in only one direction, either positive or negative.
• Two-tailed (or two-sided) test: Tests for the significance of an effect in both directions, allowing for the possibility of a positive or negative effect.

The choice between one-tailed and two-tailed tests depends on the specific research question and the directionality of the expected effect.

5. What are the 3 major types of hypothesis?

The three major types of hypotheses are:

• Null Hypothesis (H0): Represents the default assumption, stating that there is no significant effect or relationship in the data.
• Alternative Hypothesis (Ha): Contradicts the null hypothesis and proposes a specific effect or relationship that researchers want to investigate.
• Nondirectional Hypothesis: An alternative hypothesis that doesn't specify the direction of the effect, leaving it open for both positive and negative possibilities.

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6a.2 - steps for hypothesis tests, the logic of hypothesis testing section  .

A hypothesis, in statistics, is a statement about a population parameter, where this statement typically is represented by some specific numerical value. In testing a hypothesis, we use a method where we gather data in an effort to gather evidence about the hypothesis.

How do we decide whether to reject the null hypothesis?

• If the sample data are consistent with the null hypothesis, then we do not reject it.
• If the sample data are inconsistent with the null hypothesis, but consistent with the alternative, then we reject the null hypothesis and conclude that the alternative hypothesis is true.

Six Steps for Hypothesis Tests Section  

In hypothesis testing, there are certain steps one must follow. Below these are summarized into six such steps to conducting a test of a hypothesis.

• Set up the hypotheses and check conditions : Each hypothesis test includes two hypotheses about the population. One is the null hypothesis, notated as \(H_0 \), which is a statement of a particular parameter value. This hypothesis is assumed to be true until there is evidence to suggest otherwise. The second hypothesis is called the alternative, or research hypothesis, notated as \(H_a \). The alternative hypothesis is a statement of a range of alternative values in which the parameter may fall. One must also check that any conditions (assumptions) needed to run the test have been satisfied e.g. normality of data, independence, and number of success and failure outcomes.
• Decide on the significance level, \(\alpha \): This value is used as a probability cutoff for making decisions about the null hypothesis. This alpha value represents the probability we are willing to place on our test for making an incorrect decision in regards to rejecting the null hypothesis. The most common \(\alpha \) value is 0.05 or 5%. Other popular choices are 0.01 (1%) and 0.1 (10%).
• Calculate the test statistic: Gather sample data and calculate a test statistic where the sample statistic is compared to the parameter value. The test statistic is calculated under the assumption the null hypothesis is true and incorporates a measure of standard error and assumptions (conditions) related to the sampling distribution.
• Calculate probability value (p-value), or find the rejection region: A p-value is found by using the test statistic to calculate the probability of the sample data producing such a test statistic or one more extreme. The rejection region is found by using alpha to find a critical value; the rejection region is the area that is more extreme than the critical value. We discuss the p-value and rejection region in more detail in the next section.
• Make a decision about the null hypothesis: In this step, we decide to either reject the null hypothesis or decide to fail to reject the null hypothesis. Notice we do not make a decision where we will accept the null hypothesis.
• State an overall conclusion : Once we have found the p-value or rejection region, and made a statistical decision about the null hypothesis (i.e. we will reject the null or fail to reject the null), we then want to summarize our results into an overall conclusion for our test.

We will follow these six steps for the remainder of this Lesson. In the future Lessons, the steps will be followed but may not be explained explicitly.

Step 1 is a very important step to set up correctly. If your hypotheses are incorrect, your conclusion will be incorrect. In this next section, we practice with Step 1 for the one sample situations.

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What Is Hypothesis Testing?

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Hypothesis Testing: 4 Steps and Example

Hypothesis testing, sometimes called significance testing, is an act in statistics whereby an analyst tests an assumption regarding a population parameter. The methodology employed by the analyst depends on the nature of the data used and the reason for the analysis.

Hypothesis testing is used to assess the plausibility of a hypothesis by using sample data. Such data may come from a larger population or a data-generating process. The word "population" will be used for both of these cases in the following descriptions.

Key Takeaways

• Hypothesis testing is used to assess the plausibility of a hypothesis by using sample data.
• The test provides evidence concerning the plausibility of the hypothesis, given the data.
• Statistical analysts test a hypothesis by measuring and examining a random sample of the population being analyzed.
• The four steps of hypothesis testing include stating the hypotheses, formulating an analysis plan, analyzing the sample data, and analyzing the result.

How Hypothesis Testing Works

In hypothesis testing, an  analyst  tests a statistical sample, intending to provide evidence on the plausibility of the null hypothesis. Statistical analysts measure and examine a random sample of the population being analyzed. All analysts use a random population sample to test two different hypotheses: the null hypothesis and the alternative hypothesis.

The null hypothesis is usually a hypothesis of equality between population parameters; e.g., a null hypothesis may state that the population mean return is equal to zero. The alternative hypothesis is effectively the opposite of a null hypothesis. Thus, they are mutually exclusive , and only one can be true. However, one of the two hypotheses will always be true.

The null hypothesis is a statement about a population parameter, such as the population mean, that is assumed to be true.

• State the hypotheses.
• Formulate an analysis plan, which outlines how the data will be evaluated.
• Carry out the plan and analyze the sample data.
• Analyze the results and either reject the null hypothesis, or state that the null hypothesis is plausible, given the data.

Example of Hypothesis Testing

If an individual wants to test that a penny has exactly a 50% chance of landing on heads, the null hypothesis would be that 50% is correct, and the alternative hypothesis would be that 50% is not correct. Mathematically, the null hypothesis is represented as Ho: P = 0.5. The alternative hypothesis is shown as "Ha" and is identical to the null hypothesis, except with the equal sign struck-through, meaning that it does not equal 50%.

A random sample of 100 coin flips is taken, and the null hypothesis is tested. If it is found that the 100 coin flips were distributed as 40 heads and 60 tails, the analyst would assume that a penny does not have a 50% chance of landing on heads and would reject the null hypothesis and accept the alternative hypothesis.

If there were 48 heads and 52 tails, then it is plausible that the coin could be fair and still produce such a result. In cases such as this where the null hypothesis is "accepted," the analyst states that the difference between the expected results (50 heads and 50 tails) and the observed results (48 heads and 52 tails) is "explainable by chance alone."

When Did Hypothesis Testing Begin?

Some statisticians attribute the first hypothesis tests to satirical writer John Arbuthnot in 1710, who studied male and female births in England after observing that in nearly every year, male births exceeded female births by a slight proportion. Arbuthnot calculated that the probability of this happening by chance was small, and therefore it was due to “divine providence.”

What are the Benefits of Hypothesis Testing?

Hypothesis testing helps assess the accuracy of new ideas or theories by testing them against data. This allows researchers to determine whether the evidence supports their hypothesis, helping to avoid false claims and conclusions. Hypothesis testing also provides a framework for decision-making based on data rather than personal opinions or biases. By relying on statistical analysis, hypothesis testing helps to reduce the effects of chance and confounding variables, providing a robust framework for making informed conclusions.

What are the Limitations of Hypothesis Testing?

Hypothesis testing relies exclusively on data and doesn’t provide a comprehensive understanding of the subject being studied. Additionally, the accuracy of the results depends on the quality of the available data and the statistical methods used. Inaccurate data or inappropriate hypothesis formulation may lead to incorrect conclusions or failed tests. Hypothesis testing can also lead to errors, such as analysts either accepting or rejecting a null hypothesis when they shouldn’t have. These errors may result in false conclusions or missed opportunities to identify significant patterns or relationships in the data.

Hypothesis testing refers to a statistical process that helps researchers determine the reliability of a study. By using a well-formulated hypothesis and set of statistical tests, individuals or businesses can make inferences about the population that they are studying and draw conclusions based on the data presented. All hypothesis testing methods have the same four-step process, which includes stating the hypotheses, formulating an analysis plan, analyzing the sample data, and analyzing the result.

Sage. " Introduction to Hypothesis Testing ," Page 4.

Elder Research. " Who Invented the Null Hypothesis? "

Formplus. " Hypothesis Testing: Definition, Uses, Limitations and Examples ."

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A Beginner’s Guide to Hypothesis Testing in Business

• 30 Mar 2021

Becoming a more data-driven decision-maker can bring several benefits to your organization, enabling you to identify new opportunities to pursue and threats to abate. Rather than allowing subjective thinking to guide your business strategy, backing your decisions with data can empower your company to become more innovative and, ultimately, profitable.

If you’re new to data-driven decision-making, you might be wondering how data translates into business strategy. The answer lies in generating a hypothesis and verifying or rejecting it based on what various forms of data tell you.

Below is a look at hypothesis testing and the role it plays in helping businesses become more data-driven.

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What Is Hypothesis Testing?

To understand what hypothesis testing is, it’s important first to understand what a hypothesis is.

A hypothesis or hypothesis statement seeks to explain why something has happened, or what might happen, under certain conditions. It can also be used to understand how different variables relate to each other. Hypotheses are often written as if-then statements; for example, “If this happens, then this will happen.”

Hypothesis testing , then, is a statistical means of testing an assumption stated in a hypothesis. While the specific methodology leveraged depends on the nature of the hypothesis and data available, hypothesis testing typically uses sample data to extrapolate insights about a larger population.

Hypothesis Testing in Business

When it comes to data-driven decision-making, there’s a certain amount of risk that can mislead a professional. This could be due to flawed thinking or observations, incomplete or inaccurate data , or the presence of unknown variables. The danger in this is that, if major strategic decisions are made based on flawed insights, it can lead to wasted resources, missed opportunities, and catastrophic outcomes.

The real value of hypothesis testing in business is that it allows professionals to test their theories and assumptions before putting them into action. This essentially allows an organization to verify its analysis is correct before committing resources to implement a broader strategy.

As one example, consider a company that wishes to launch a new marketing campaign to revitalize sales during a slow period. Doing so could be an incredibly expensive endeavor, depending on the campaign’s size and complexity. The company, therefore, may wish to test the campaign on a smaller scale to understand how it will perform.

In this example, the hypothesis that’s being tested would fall along the lines of: “If the company launches a new marketing campaign, then it will translate into an increase in sales.” It may even be possible to quantify how much of a lift in sales the company expects to see from the effort. Pending the results of the pilot campaign, the business would then know whether it makes sense to roll it out more broadly.

Related: 9 Fundamental Data Science Skills for Business Professionals

Key Considerations for Hypothesis Testing

1. alternative hypothesis and null hypothesis.

In hypothesis testing, the hypothesis that’s being tested is known as the alternative hypothesis . Often, it’s expressed as a correlation or statistical relationship between variables. The null hypothesis , on the other hand, is a statement that’s meant to show there’s no statistical relationship between the variables being tested. It’s typically the exact opposite of whatever is stated in the alternative hypothesis.

For example, consider a company’s leadership team that historically and reliably sees $12 million in monthly revenue. They want to understand if reducing the price of their services will attract more customers and, in turn, increase revenue.

In this case, the alternative hypothesis may take the form of a statement such as: “If we reduce the price of our flagship service by five percent, then we’ll see an increase in sales and realize revenues greater than $12 million in the next month.”

The null hypothesis, on the other hand, would indicate that revenues wouldn’t increase from the base of $12 million, or might even decrease.

Check out the video below about the difference between an alternative and a null hypothesis, and subscribe to our YouTube channel for more explainer content.

2. Significance Level and P-Value

Statistically speaking, if you were to run the same scenario 100 times, you’d likely receive somewhat different results each time. If you were to plot these results in a distribution plot, you’d see the most likely outcome is at the tallest point in the graph, with less likely outcomes falling to the right and left of that point.

With this in mind, imagine you’ve completed your hypothesis test and have your results, which indicate there may be a correlation between the variables you were testing. To understand your results' significance, you’ll need to identify a p-value for the test, which helps note how confident you are in the test results.

In statistics, the p-value depicts the probability that, assuming the null hypothesis is correct, you might still observe results that are at least as extreme as the results of your hypothesis test. The smaller the p-value, the more likely the alternative hypothesis is correct, and the greater the significance of your results.

3. One-Sided vs. Two-Sided Testing

When it’s time to test your hypothesis, it’s important to leverage the correct testing method. The two most common hypothesis testing methods are one-sided and two-sided tests , or one-tailed and two-tailed tests, respectively.

Typically, you’d leverage a one-sided test when you have a strong conviction about the direction of change you expect to see due to your hypothesis test. You’d leverage a two-sided test when you’re less confident in the direction of change.

4. Sampling

To perform hypothesis testing in the first place, you need to collect a sample of data to be analyzed. Depending on the question you’re seeking to answer or investigate, you might collect samples through surveys, observational studies, or experiments.

A survey involves asking a series of questions to a random population sample and recording self-reported responses.

Observational studies involve a researcher observing a sample population and collecting data as it occurs naturally, without intervention.

Finally, an experiment involves dividing a sample into multiple groups, one of which acts as the control group. For each non-control group, the variable being studied is manipulated to determine how the data collected differs from that of the control group.

Learn How to Perform Hypothesis Testing

Hypothesis testing is a complex process involving different moving pieces that can allow an organization to effectively leverage its data and inform strategic decisions.

If you’re interested in better understanding hypothesis testing and the role it can play within your organization, one option is to complete a course that focuses on the process. Doing so can lay the statistical and analytical foundation you need to succeed.

Do you want to learn more about hypothesis testing? Explore Business Analytics —one of our online business essentials courses —and download our Beginner’s Guide to Data & Analytics .

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Hypothesis Testing

Hypothesis testing is a tool for making statistical inferences about the population data. It is an analysis tool that tests assumptions and determines how likely something is within a given standard of accuracy. Hypothesis testing provides a way to verify whether the results of an experiment are valid.

A null hypothesis and an alternative hypothesis are set up before performing the hypothesis testing. This helps to arrive at a conclusion regarding the sample obtained from the population. In this article, we will learn more about hypothesis testing, its types, steps to perform the testing, and associated examples.

What is Hypothesis Testing in Statistics?

Hypothesis testing uses sample data from the population to draw useful conclusions regarding the population probability distribution . It tests an assumption made about the data using different types of hypothesis testing methodologies. The hypothesis testing results in either rejecting or not rejecting the null hypothesis.

Hypothesis Testing Definition

Hypothesis testing can be defined as a statistical tool that is used to identify if the results of an experiment are meaningful or not. It involves setting up a null hypothesis and an alternative hypothesis. These two hypotheses will always be mutually exclusive. This means that if the null hypothesis is true then the alternative hypothesis is false and vice versa. An example of hypothesis testing is setting up a test to check if a new medicine works on a disease in a more efficient manner.

Null Hypothesis

The null hypothesis is a concise mathematical statement that is used to indicate that there is no difference between two possibilities. In other words, there is no difference between certain characteristics of data. This hypothesis assumes that the outcomes of an experiment are based on chance alone. It is denoted as \(H_{0}\). Hypothesis testing is used to conclude if the null hypothesis can be rejected or not. Suppose an experiment is conducted to check if girls are shorter than boys at the age of 5. The null hypothesis will say that they are the same height.

Alternative Hypothesis

The alternative hypothesis is an alternative to the null hypothesis. It is used to show that the observations of an experiment are due to some real effect. It indicates that there is a statistical significance between two possible outcomes and can be denoted as \(H_{1}\) or \(H_{a}\). For the above-mentioned example, the alternative hypothesis would be that girls are shorter than boys at the age of 5.

Hypothesis Testing P Value

In hypothesis testing, the p value is used to indicate whether the results obtained after conducting a test are statistically significant or not. It also indicates the probability of making an error in rejecting or not rejecting the null hypothesis.This value is always a number between 0 and 1. The p value is compared to an alpha level, \(\alpha\) or significance level. The alpha level can be defined as the acceptable risk of incorrectly rejecting the null hypothesis. The alpha level is usually chosen between 1% to 5%.

Hypothesis Testing Critical region

All sets of values that lead to rejecting the null hypothesis lie in the critical region. Furthermore, the value that separates the critical region from the non-critical region is known as the critical value.

Hypothesis Testing Formula

Depending upon the type of data available and the size, different types of hypothesis testing are used to determine whether the null hypothesis can be rejected or not. The hypothesis testing formula for some important test statistics are given below:

• z = \(\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\). \(\overline{x}\) is the sample mean, \(\mu\) is the population mean, \(\sigma\) is the population standard deviation and n is the size of the sample.
• t = \(\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}\). s is the sample standard deviation.
• \(\chi ^{2} = \sum \frac{(O_{i}-E_{i})^{2}}{E_{i}}\). \(O_{i}\) is the observed value and \(E_{i}\) is the expected value.

We will learn more about these test statistics in the upcoming section.

Types of Hypothesis Testing

Selecting the correct test for performing hypothesis testing can be confusing. These tests are used to determine a test statistic on the basis of which the null hypothesis can either be rejected or not rejected. Some of the important tests used for hypothesis testing are given below.

Hypothesis Testing Z Test

A z test is a way of hypothesis testing that is used for a large sample size (n ≥ 30). It is used to determine whether there is a difference between the population mean and the sample mean when the population standard deviation is known. It can also be used to compare the mean of two samples. It is used to compute the z test statistic. The formulas are given as follows:

• One sample: z = \(\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\).
• Two samples: z = \(\frac{(\overline{x_{1}}-\overline{x_{2}})-(\mu_{1}-\mu_{2})}{\sqrt{\frac{\sigma_{1}^{2}}{n_{1}}+\frac{\sigma_{2}^{2}}{n_{2}}}}\).

Hypothesis Testing t Test

The t test is another method of hypothesis testing that is used for a small sample size (n < 30). It is also used to compare the sample mean and population mean. However, the population standard deviation is not known. Instead, the sample standard deviation is known. The mean of two samples can also be compared using the t test.

• One sample: t = \(\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}\).
• Two samples: t = \(\frac{(\overline{x_{1}}-\overline{x_{2}})-(\mu_{1}-\mu_{2})}{\sqrt{\frac{s_{1}^{2}}{n_{1}}+\frac{s_{2}^{2}}{n_{2}}}}\).

Hypothesis Testing Chi Square

The Chi square test is a hypothesis testing method that is used to check whether the variables in a population are independent or not. It is used when the test statistic is chi-squared distributed.

One Tailed Hypothesis Testing

One tailed hypothesis testing is done when the rejection region is only in one direction. It can also be known as directional hypothesis testing because the effects can be tested in one direction only. This type of testing is further classified into the right tailed test and left tailed test.

Right Tailed Hypothesis Testing

The right tail test is also known as the upper tail test. This test is used to check whether the population parameter is greater than some value. The null and alternative hypotheses for this test are given as follows:

\(H_{0}\): The population parameter is ≤ some value

\(H_{1}\): The population parameter is > some value.

If the test statistic has a greater value than the critical value then the null hypothesis is rejected

Left Tailed Hypothesis Testing

The left tail test is also known as the lower tail test. It is used to check whether the population parameter is less than some value. The hypotheses for this hypothesis testing can be written as follows:

\(H_{0}\): The population parameter is ≥ some value

\(H_{1}\): The population parameter is < some value.

The null hypothesis is rejected if the test statistic has a value lesser than the critical value.

Two Tailed Hypothesis Testing

In this hypothesis testing method, the critical region lies on both sides of the sampling distribution. It is also known as a non - directional hypothesis testing method. The two-tailed test is used when it needs to be determined if the population parameter is assumed to be different than some value. The hypotheses can be set up as follows:

\(H_{0}\): the population parameter = some value

\(H_{1}\): the population parameter ≠ some value

The null hypothesis is rejected if the test statistic has a value that is not equal to the critical value.

Hypothesis Testing Steps

Hypothesis testing can be easily performed in five simple steps. The most important step is to correctly set up the hypotheses and identify the right method for hypothesis testing. The basic steps to perform hypothesis testing are as follows:

• Step 1: Set up the null hypothesis by correctly identifying whether it is the left-tailed, right-tailed, or two-tailed hypothesis testing.
• Step 2: Set up the alternative hypothesis.
• Step 3: Choose the correct significance level, \(\alpha\), and find the critical value.
• Step 4: Calculate the correct test statistic (z, t or \(\chi\)) and p-value.
• Step 5: Compare the test statistic with the critical value or compare the p-value with \(\alpha\) to arrive at a conclusion. In other words, decide if the null hypothesis is to be rejected or not.

Hypothesis Testing Example

The best way to solve a problem on hypothesis testing is by applying the 5 steps mentioned in the previous section. Suppose a researcher claims that the mean average weight of men is greater than 100kgs with a standard deviation of 15kgs. 30 men are chosen with an average weight of 112.5 Kgs. Using hypothesis testing, check if there is enough evidence to support the researcher's claim. The confidence interval is given as 95%.

Step 1: This is an example of a right-tailed test. Set up the null hypothesis as \(H_{0}\): \(\mu\) = 100.

Step 2: The alternative hypothesis is given by \(H_{1}\): \(\mu\) > 100.

Step 3: As this is a one-tailed test, \(\alpha\) = 100% - 95% = 5%. This can be used to determine the critical value.

1 - \(\alpha\) = 1 - 0.05 = 0.95

0.95 gives the required area under the curve. Now using a normal distribution table, the area 0.95 is at z = 1.645. A similar process can be followed for a t-test. The only additional requirement is to calculate the degrees of freedom given by n - 1.

Step 4: Calculate the z test statistic. This is because the sample size is 30. Furthermore, the sample and population means are known along with the standard deviation.

z = \(\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\).

\(\mu\) = 100, \(\overline{x}\) = 112.5, n = 30, \(\sigma\) = 15

z = \(\frac{112.5-100}{\frac{15}{\sqrt{30}}}\) = 4.56

Step 5: Conclusion. As 4.56 > 1.645 thus, the null hypothesis can be rejected.

Hypothesis Testing and Confidence Intervals

Confidence intervals form an important part of hypothesis testing. This is because the alpha level can be determined from a given confidence interval. Suppose a confidence interval is given as 95%. Subtract the confidence interval from 100%. This gives 100 - 95 = 5% or 0.05. This is the alpha value of a one-tailed hypothesis testing. To obtain the alpha value for a two-tailed hypothesis testing, divide this value by 2. This gives 0.05 / 2 = 0.025.

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Important Notes on Hypothesis Testing

• Hypothesis testing is a technique that is used to verify whether the results of an experiment are statistically significant.
• It involves the setting up of a null hypothesis and an alternate hypothesis.
• There are three types of tests that can be conducted under hypothesis testing - z test, t test, and chi square test.
• Hypothesis testing can be classified as right tail, left tail, and two tail tests.

Examples on Hypothesis Testing

• Example 1: The average weight of a dumbbell in a gym is 90lbs. However, a physical trainer believes that the average weight might be higher. A random sample of 5 dumbbells with an average weight of 110lbs and a standard deviation of 18lbs. Using hypothesis testing check if the physical trainer's claim can be supported for a 95% confidence level. Solution: As the sample size is lesser than 30, the t-test is used. \(H_{0}\): \(\mu\) = 90, \(H_{1}\): \(\mu\) > 90 \(\overline{x}\) = 110, \(\mu\) = 90, n = 5, s = 18. \(\alpha\) = 0.05 Using the t-distribution table, the critical value is 2.132 t = \(\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}\) t = 2.484 As 2.484 > 2.132, the null hypothesis is rejected. Answer: The average weight of the dumbbells may be greater than 90lbs
• Example 2: The average score on a test is 80 with a standard deviation of 10. With a new teaching curriculum introduced it is believed that this score will change. On random testing, the score of 38 students, the mean was found to be 88. With a 0.05 significance level, is there any evidence to support this claim? Solution: This is an example of two-tail hypothesis testing. The z test will be used. \(H_{0}\): \(\mu\) = 80, \(H_{1}\): \(\mu\) ≠ 80 \(\overline{x}\) = 88, \(\mu\) = 80, n = 36, \(\sigma\) = 10. \(\alpha\) = 0.05 / 2 = 0.025 The critical value using the normal distribution table is 1.96 z = \(\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\) z = \(\frac{88-80}{\frac{10}{\sqrt{36}}}\) = 4.8 As 4.8 > 1.96, the null hypothesis is rejected. Answer: There is a difference in the scores after the new curriculum was introduced.
• Example 3: The average score of a class is 90. However, a teacher believes that the average score might be lower. The scores of 6 students were randomly measured. The mean was 82 with a standard deviation of 18. With a 0.05 significance level use hypothesis testing to check if this claim is true. Solution: The t test will be used. \(H_{0}\): \(\mu\) = 90, \(H_{1}\): \(\mu\) < 90 \(\overline{x}\) = 110, \(\mu\) = 90, n = 6, s = 18 The critical value from the t table is -2.015 t = \(\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}\) t = \(\frac{82-90}{\frac{18}{\sqrt{6}}}\) t = -1.088 As -1.088 > -2.015, we fail to reject the null hypothesis. Answer: There is not enough evidence to support the claim.

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FAQs on Hypothesis Testing

What is hypothesis testing.

Hypothesis testing in statistics is a tool that is used to make inferences about the population data. It is also used to check if the results of an experiment are valid.

What is the z Test in Hypothesis Testing?

The z test in hypothesis testing is used to find the z test statistic for normally distributed data . The z test is used when the standard deviation of the population is known and the sample size is greater than or equal to 30.

What is the t Test in Hypothesis Testing?

The t test in hypothesis testing is used when the data follows a student t distribution . It is used when the sample size is less than 30 and standard deviation of the population is not known.

What is the formula for z test in Hypothesis Testing?

The formula for a one sample z test in hypothesis testing is z = \(\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\) and for two samples is z = \(\frac{(\overline{x_{1}}-\overline{x_{2}})-(\mu_{1}-\mu_{2})}{\sqrt{\frac{\sigma_{1}^{2}}{n_{1}}+\frac{\sigma_{2}^{2}}{n_{2}}}}\).

What is the p Value in Hypothesis Testing?

The p value helps to determine if the test results are statistically significant or not. In hypothesis testing, the null hypothesis can either be rejected or not rejected based on the comparison between the p value and the alpha level.

What is One Tail Hypothesis Testing?

When the rejection region is only on one side of the distribution curve then it is known as one tail hypothesis testing. The right tail test and the left tail test are two types of directional hypothesis testing.

What is the Alpha Level in Two Tail Hypothesis Testing?

To get the alpha level in a two tail hypothesis testing divide \(\alpha\) by 2. This is done as there are two rejection regions in the curve.

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Hypothesis testing involves formulating assumptions about population parameters based on sample statistics and rigorously evaluating these assumptions against empirical evidence. This article sheds light on the significance of hypothesis testing and the critical steps involved in the process.

What is Hypothesis Testing?

Hypothesis testing is a statistical method that is used to make a statistical decision using experimental data. Hypothesis testing is basically an assumption that we make about a population parameter. It evaluates two mutually exclusive statements about a population to determine which statement is best supported by the sample data. 

Example: You say an average height in the class is 30 or a boy is taller than a girl. All of these is an assumption that we are assuming, and we need some statistical way to prove these. We need some mathematical conclusion whatever we are assuming is true.

Defining Hypotheses

Key Terms of Hypothesis Testing

• P-value: The P value , or calculated probability, is the probability of finding the observed/extreme results when the null hypothesis(H0) of a study-given problem is true. If your P-value is less than the chosen significance level then you reject the null hypothesis i.e. accept that your sample claims to support the alternative hypothesis.
• Test Statistic: The test statistic is a numerical value calculated from sample data during a hypothesis test, used to determine whether to reject the null hypothesis. It is compared to a critical value or p-value to make decisions about the statistical significance of the observed results.
• Critical value : The critical value in statistics is a threshold or cutoff point used to determine whether to reject the null hypothesis in a hypothesis test.
• Degrees of freedom: Degrees of freedom are associated with the variability or freedom one has in estimating a parameter. The degrees of freedom are related to the sample size and determine the shape.

Why do we use Hypothesis Testing?

Hypothesis testing is an important procedure in statistics. Hypothesis testing evaluates two mutually exclusive population statements to determine which statement is most supported by sample data. When we say that the findings are statistically significant, thanks to hypothesis testing. 

One-Tailed and Two-Tailed Test

One tailed test focuses on one direction, either greater than or less than a specified value. We use a one-tailed test when there is a clear directional expectation based on prior knowledge or theory. The critical region is located on only one side of the distribution curve. If the sample falls into this critical region, the null hypothesis is rejected in favor of the alternative hypothesis.

One-Tailed Test

There are two types of one-tailed test:

Two-Tailed Test

A two-tailed test considers both directions, greater than and less than a specified value.We use a two-tailed test when there is no specific directional expectation, and want to detect any significant difference.

What are Type 1 and Type 2 errors in Hypothesis Testing?

In hypothesis testing, Type I and Type II errors are two possible errors that researchers can make when drawing conclusions about a population based on a sample of data. These errors are associated with the decisions made regarding the null hypothesis and the alternative hypothesis.

How does Hypothesis Testing work?

Step 1: define null and alternative hypothesis.

We first identify the problem about which we want to make an assumption keeping in mind that our assumption should be contradictory to one another, assuming Normally distributed data.

Step 2 – Choose significance level

Step 3 – Collect and Analyze data.

Gather relevant data through observation or experimentation. Analyze the data using appropriate statistical methods to obtain a test statistic.

Step 4-Calculate Test Statistic

The data for the tests are evaluated in this step we look for various scores based on the characteristics of data. The choice of the test statistic depends on the type of hypothesis test being conducted.

There are various hypothesis tests, each appropriate for various goal to calculate our test. This could be a Z-test , Chi-square , T-test , and so on.

• Z-test : If population means and standard deviations are known. Z-statistic is commonly used.
• t-test : If population standard deviations are unknown. and sample size is small than t-test statistic is more appropriate.
• Chi-square test : Chi-square test is used for categorical data or for testing independence in contingency tables
• F-test : F-test is often used in analysis of variance (ANOVA) to compare variances or test the equality of means across multiple groups.

We have a smaller dataset, So, T-test is more appropriate to test our hypothesis.

T-statistic is a measure of the difference between the means of two groups relative to the variability within each group. It is calculated as the difference between the sample means divided by the standard error of the difference. It is also known as the t-value or t-score.

Step 5 – Comparing Test Statistic:

In this stage, we decide where we should accept the null hypothesis or reject the null hypothesis. There are two ways to decide where we should accept or reject the null hypothesis.

Method A: Using Crtical values

Comparing the test statistic and tabulated critical value we have,

• If Test Statistic>Critical Value: Reject the null hypothesis.
• If Test Statistic≤Critical Value: Fail to reject the null hypothesis.

Note: Critical values are predetermined threshold values that are used to make a decision in hypothesis testing. To determine critical values for hypothesis testing, we typically refer to a statistical distribution table , such as the normal distribution or t-distribution tables based on.

Method B: Using P-values

We can also come to an conclusion using the p-value,

Note : The p-value is the probability of obtaining a test statistic as extreme as, or more extreme than, the one observed in the sample, assuming the null hypothesis is true. To determine p-value for hypothesis testing, we typically refer to a statistical distribution table , such as the normal distribution or t-distribution tables based on.

Step 7- Interpret the Results

At last, we can conclude our experiment using method A or B.

Calculating test statistic

To validate our hypothesis about a population parameter we use statistical functions . We use the z-score, p-value, and level of significance(alpha) to make evidence for our hypothesis for normally distributed data .

1. Z-statistics:

When population means and standard deviations are known.

• μ represents the population mean, 
• σ is the standard deviation
• and n is the size of the sample.

2. T-Statistics

T test is used when n<30,

t-statistic calculation is given by:

• t = t-score,
• x̄ = sample mean
• μ = population mean,
• s = standard deviation of the sample,
• n = sample size

3. Chi-Square Test

Chi-Square Test for Independence categorical Data (Non-normally distributed) using:

• i,j are the rows and columns index respectively.

Real life Hypothesis Testing example

Let’s examine hypothesis testing using two real life situations,

Case A: D oes a New Drug Affect Blood Pressure?

Imagine a pharmaceutical company has developed a new drug that they believe can effectively lower blood pressure in patients with hypertension. Before bringing the drug to market, they need to conduct a study to assess its impact on blood pressure.

• Before Treatment: 120, 122, 118, 130, 125, 128, 115, 121, 123, 119
• After Treatment: 115, 120, 112, 128, 122, 125, 110, 117, 119, 114

Step 1 : Define the Hypothesis

• Null Hypothesis : (H 0 )The new drug has no effect on blood pressure.
• Alternate Hypothesis : (H 1 )The new drug has an effect on blood pressure.

Step 2: Define the Significance level

Let’s consider the Significance level at 0.05, indicating rejection of the null hypothesis.

If the evidence suggests less than a 5% chance of observing the results due to random variation.

Step 3 : Compute the test statistic

Using paired T-test analyze the data to obtain a test statistic and a p-value.

The test statistic (e.g., T-statistic) is calculated based on the differences between blood pressure measurements before and after treatment.

t = m/(s/√n)

• m  = mean of the difference i.e X after, X before
• s  = standard deviation of the difference (d) i.e d i ​= X after, i ​− X before,
• n  = sample size,

then, m= -3.9, s= 1.8 and n= 10

we, calculate the , T-statistic = -9 based on the formula for paired t test

Step 4: Find the p-value

The calculated t-statistic is -9 and degrees of freedom df = 9, you can find the p-value using statistical software or a t-distribution table.

thus, p-value = 8.538051223166285e-06

Step 5: Result

• If the p-value is less than or equal to 0.05, the researchers reject the null hypothesis.
• If the p-value is greater than 0.05, they fail to reject the null hypothesis.

Conclusion: Since the p-value (8.538051223166285e-06) is less than the significance level (0.05), the researchers reject the null hypothesis. There is statistically significant evidence that the average blood pressure before and after treatment with the new drug is different.

Python Implementation of Hypothesis Testing

Let’s create hypothesis testing with python, where we are testing whether a new drug affects blood pressure. For this example, we will use a paired T-test. We’ll use the scipy.stats library for the T-test.

Scipy is a mathematical library in Python that is mostly used for mathematical equations and computations.

We will implement our first real life problem via python,

In the above example, given the T-statistic of approximately -9 and an extremely small p-value, the results indicate a strong case to reject the null hypothesis at a significance level of 0.05. 

• The results suggest that the new drug, treatment, or intervention has a significant effect on lowering blood pressure.
• The negative T-statistic indicates that the mean blood pressure after treatment is significantly lower than the assumed population mean before treatment.

Case B : Cholesterol level in a population

Data: A sample of 25 individuals is taken, and their cholesterol levels are measured.

Cholesterol Levels (mg/dL): 205, 198, 210, 190, 215, 205, 200, 192, 198, 205, 198, 202, 208, 200, 205, 198, 205, 210, 192, 205, 198, 205, 210, 192, 205.

Populations Mean = 200

Population Standard Deviation (σ): 5 mg/dL(given for this problem)

Step 1: Define the Hypothesis

• Null Hypothesis (H 0 ): The average cholesterol level in a population is 200 mg/dL.
• Alternate Hypothesis (H 1 ): The average cholesterol level in a population is different from 200 mg/dL.

As the direction of deviation is not given , we assume a two-tailed test, and based on a normal distribution table, the critical values for a significance level of 0.05 (two-tailed) can be calculated through the z-table and are approximately -1.96 and 1.96.

Step 4: Result

Since the absolute value of the test statistic (2.04) is greater than the critical value (1.96), we reject the null hypothesis. And conclude that, there is statistically significant evidence that the average cholesterol level in the population is different from 200 mg/dL

Limitations of Hypothesis Testing

• Although a useful technique, hypothesis testing does not offer a comprehensive grasp of the topic being studied. Without fully reflecting the intricacy or whole context of the phenomena, it concentrates on certain hypotheses and statistical significance.
• The accuracy of hypothesis testing results is contingent on the quality of available data and the appropriateness of statistical methods used. Inaccurate data or poorly formulated hypotheses can lead to incorrect conclusions.
• Relying solely on hypothesis testing may cause analysts to overlook significant patterns or relationships in the data that are not captured by the specific hypotheses being tested. This limitation underscores the importance of complimenting hypothesis testing with other analytical approaches.

Hypothesis testing stands as a cornerstone in statistical analysis, enabling data scientists to navigate uncertainties and draw credible inferences from sample data. By systematically defining null and alternative hypotheses, choosing significance levels, and leveraging statistical tests, researchers can assess the validity of their assumptions. The article also elucidates the critical distinction between Type I and Type II errors, providing a comprehensive understanding of the nuanced decision-making process inherent in hypothesis testing. The real-life example of testing a new drug’s effect on blood pressure using a paired T-test showcases the practical application of these principles, underscoring the importance of statistical rigor in data-driven decision-making.

Frequently Asked Questions (FAQs)

1. what are the 3 types of hypothesis test.

There are three types of hypothesis tests: right-tailed, left-tailed, and two-tailed. Right-tailed tests assess if a parameter is greater, left-tailed if lesser. Two-tailed tests check for non-directional differences, greater or lesser.

2.What are the 4 components of hypothesis testing?

Null Hypothesis ( ): No effect or difference exists. Alternative Hypothesis ( ): An effect or difference exists. Significance Level ( ): Risk of rejecting null hypothesis when it’s true (Type I error). Test Statistic: Numerical value representing observed evidence against null hypothesis.

3.What is hypothesis testing in ML?

Statistical method to evaluate the performance and validity of machine learning models. Tests specific hypotheses about model behavior, like whether features influence predictions or if a model generalizes well to unseen data.

4.What is the difference between Pytest and hypothesis in Python?

Pytest purposes general testing framework for Python code while Hypothesis is a Property-based testing framework for Python, focusing on generating test cases based on specified properties of the code.

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1.2: The 7-Step Process of Statistical Hypothesis Testing

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• Penn State's Department of Statistics
• The Pennsylvania State University

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We will cover the seven steps one by one.

Step 1: State the Null Hypothesis

The null hypothesis can be thought of as the opposite of the "guess" the researchers made: in this example, the biologist thinks the plant height will be different for the fertilizers. So the null would be that there will be no difference among the groups of plants. Specifically, in more statistical language the null for an ANOVA is that the means are the same. We state the null hypothesis as: \[H_{0}: \ \mu_{1} = \mu_{2} = \ldots = \mu_{T}\] for \(T\) levels of an experimental treatment.

Why do we do this? Why not simply test the working hypothesis directly? The answer lies in the Popperian Principle of Falsification. Karl Popper (a philosopher) discovered that we can't conclusively confirm a hypothesis, but we can conclusively negate one. So we set up a null hypothesis which is effectively the opposite of the working hypothesis. The hope is that based on the strength of the data, we will be able to negate or reject the null hypothesis and accept an alternative hypothesis. In other words, we usually see the working hypothesis in \(H_{A}\).

Step 2: State the Alternative Hypothesis

\[H_{A}: \ \text{treatment level means not all equal}\]

The reason we state the alternative hypothesis this way is that if the null is rejected, there are many possibilities.

For example, \(\mu_{1} \neq \mu_{2} = \ldots = \mu_{T}\) is one possibility, as is \(\mu_{1} = \mu_{2} \neq \mu_{3} = \ldots = \mu_{T}\). Many people make the mistake of stating the alternative hypothesis as \(mu_{1} \neq mu_{2} \neq \ldots \neq \mu_{T}\), which says that every mean differs from every other mean. This is a possibility, but only one of many possibilities. To cover all alternative outcomes, we resort to a verbal statement of "not all equal" and then follow up with mean comparisons to find out where differences among means exist. In our example, this means that fertilizer 1 may result in plants that are really tall, but fertilizers 2, 3, and the plants with no fertilizers don't differ from one another. A simpler way of thinking about this is that at least one mean is different from all others.

Step 3: Set \(\alpha\)

If we look at what can happen in a hypothesis test, we can construct the following contingency table:

You should be familiar with type I and type II errors from your introductory course. It is important to note that we want to set \(\alpha\) before the experiment ( a priori ) because the Type I error is the more grievous error to make. The typical value of \(\alpha\) is 0.05, establishing a 95% confidence level. For this course, we will assume \(\alpha\) =0.05, unless stated otherwise.

Step 4: Collect Data

Remember the importance of recognizing whether data is collected through an experimental design or observational study.

Step 5: Calculate a test statistic

For categorical treatment level means, we use an \(F\) statistic, named after R.A. Fisher. We will explore the mechanics of computing the \(F\) statistic beginning in Chapter 2. The \(F\) value we get from the data is labeled \(F_{\text{calculated}}\).

Step 6: Construct Acceptance / Rejection regions

As with all other test statistics, a threshold (critical) value of \(F\) is established. This \(F\) value can be obtained from statistical tables or software and is referred to as \(F_{\text{critical}}\) or \(F_{\alpha}\). As a reminder, this critical value is the minimum value for the test statistic (in this case the F test) for us to be able to reject the null.

The \(F\) distribution, \(F_{\alpha}\), and the location of acceptance and rejection regions are shown in the graph below:

Step 7: Based on steps 5 and 6, draw a conclusion about H0

If the \(F_{\text{\calculated}}\) from the data is larger than the \(F_{\alpha}\), then you are in the rejection region and you can reject the null hypothesis with \((1 - \alpha)\) level of confidence.

Note that modern statistical software condenses steps 6 and 7 by providing a \(p\)-value. The \(p\)-value here is the probability of getting an \(F_{\text{calculated}}\) even greater than what you observe assuming the null hypothesis is true. If by chance, the \(F_{\text{calculated}} = F_{\alpha}\), then the \(p\)-value would exactly equal \(\alpha\). With larger \(F_{\text{calculated}}\) values, we move further into the rejection region and the \(p\) - value becomes less than \(\alpha\). So the decision rule is as follows:

If the \(p\) - value obtained from the ANOVA is less than \(\alpha\), then reject \(H_{0}\) and accept \(H_{A}\).

If you are not familiar with this material, we suggest that you review course materials from your basic statistics course.

Hypothesis Testing - Analysis of Variance (ANOVA)

Lisa Sullivan, PhD

Professor of Biostatistics

Boston University School of Public Health

Introduction

This module will continue the discussion of hypothesis testing, where a specific statement or hypothesis is generated about a population parameter, and sample statistics are used to assess the likelihood that the hypothesis is true. The hypothesis is based on available information and the investigator's belief about the population parameters. The specific test considered here is called analysis of variance (ANOVA) and is a test of hypothesis that is appropriate to compare means of a continuous variable in two or more independent comparison groups. For example, in some clinical trials there are more than two comparison groups. In a clinical trial to evaluate a new medication for asthma, investigators might compare an experimental medication to a placebo and to a standard treatment (i.e., a medication currently being used). In an observational study such as the Framingham Heart Study, it might be of interest to compare mean blood pressure or mean cholesterol levels in persons who are underweight, normal weight, overweight and obese.  

The technique to test for a difference in more than two independent means is an extension of the two independent samples procedure discussed previously which applies when there are exactly two independent comparison groups. The ANOVA technique applies when there are two or more than two independent groups. The ANOVA procedure is used to compare the means of the comparison groups and is conducted using the same five step approach used in the scenarios discussed in previous sections. Because there are more than two groups, however, the computation of the test statistic is more involved. The test statistic must take into account the sample sizes, sample means and sample standard deviations in each of the comparison groups.

If one is examining the means observed among, say three groups, it might be tempting to perform three separate group to group comparisons, but this approach is incorrect because each of these comparisons fails to take into account the total data, and it increases the likelihood of incorrectly concluding that there are statistically significate differences, since each comparison adds to the probability of a type I error. Analysis of variance avoids these problemss by asking a more global question, i.e., whether there are significant differences among the groups, without addressing differences between any two groups in particular (although there are additional tests that can do this if the analysis of variance indicates that there are differences among the groups).

The fundamental strategy of ANOVA is to systematically examine variability within groups being compared and also examine variability among the groups being compared.

Learning Objectives

After completing this module, the student will be able to:

• Perform analysis of variance by hand
• Appropriately interpret results of analysis of variance tests
• Distinguish between one and two factor analysis of variance tests
• Identify the appropriate hypothesis testing procedure based on type of outcome variable and number of samples

The ANOVA Approach

Consider an example with four independent groups and a continuous outcome measure. The independent groups might be defined by a particular characteristic of the participants such as BMI (e.g., underweight, normal weight, overweight, obese) or by the investigator (e.g., randomizing participants to one of four competing treatments, call them A, B, C and D). Suppose that the outcome is systolic blood pressure, and we wish to test whether there is a statistically significant difference in mean systolic blood pressures among the four groups. The sample data are organized as follows:

The hypotheses of interest in an ANOVA are as follows:

• H 0 : μ 1 = μ 2 = μ 3 ... = μ k
• H 1 : Means are not all equal.

where k = the number of independent comparison groups.

In this example, the hypotheses are:

• H 0 : μ 1 = μ 2 = μ 3 = μ 4
• H 1 : The means are not all equal.

The null hypothesis in ANOVA is always that there is no difference in means. The research or alternative hypothesis is always that the means are not all equal and is usually written in words rather than in mathematical symbols. The research hypothesis captures any difference in means and includes, for example, the situation where all four means are unequal, where one is different from the other three, where two are different, and so on. The alternative hypothesis, as shown above, capture all possible situations other than equality of all means specified in the null hypothesis.

Test Statistic for ANOVA

The test statistic for testing H 0 : μ 1 = μ 2 = ... =   μ k is:

and the critical value is found in a table of probability values for the F distribution with (degrees of freedom) df 1 = k-1, df 2 =N-k. The table can be found in "Other Resources" on the left side of the pages.

NOTE: The test statistic F assumes equal variability in the k populations (i.e., the population variances are equal, or s 1 2 = s 2 2 = ... = s k 2 ). This means that the outcome is equally variable in each of the comparison populations. This assumption is the same as that assumed for appropriate use of the test statistic to test equality of two independent means. It is possible to assess the likelihood that the assumption of equal variances is true and the test can be conducted in most statistical computing packages. If the variability in the k comparison groups is not similar, then alternative techniques must be used.

The F statistic is computed by taking the ratio of what is called the "between treatment" variability to the "residual or error" variability. This is where the name of the procedure originates. In analysis of variance we are testing for a difference in means (H 0 : means are all equal versus H 1 : means are not all equal) by evaluating variability in the data. The numerator captures between treatment variability (i.e., differences among the sample means) and the denominator contains an estimate of the variability in the outcome. The test statistic is a measure that allows us to assess whether the differences among the sample means (numerator) are more than would be expected by chance if the null hypothesis is true. Recall in the two independent sample test, the test statistic was computed by taking the ratio of the difference in sample means (numerator) to the variability in the outcome (estimated by Sp).  

The decision rule for the F test in ANOVA is set up in a similar way to decision rules we established for t tests. The decision rule again depends on the level of significance and the degrees of freedom. The F statistic has two degrees of freedom. These are denoted df 1 and df 2 , and called the numerator and denominator degrees of freedom, respectively. The degrees of freedom are defined as follows:

df 1 = k-1 and df 2 =N-k,

where k is the number of comparison groups and N is the total number of observations in the analysis.   If the null hypothesis is true, the between treatment variation (numerator) will not exceed the residual or error variation (denominator) and the F statistic will small. If the null hypothesis is false, then the F statistic will be large. The rejection region for the F test is always in the upper (right-hand) tail of the distribution as shown below.

Rejection Region for F   Test with a =0.05, df 1 =3 and df 2 =36 (k=4, N=40)

For the scenario depicted here, the decision rule is: Reject H 0 if F > 2.87.

The ANOVA Procedure

We will next illustrate the ANOVA procedure using the five step approach. Because the computation of the test statistic is involved, the computations are often organized in an ANOVA table. The ANOVA table breaks down the components of variation in the data into variation between treatments and error or residual variation. Statistical computing packages also produce ANOVA tables as part of their standard output for ANOVA, and the ANOVA table is set up as follows: 

where  

• X = individual observation,
• k = the number of treatments or independent comparison groups, and
• N = total number of observations or total sample size.

The ANOVA table above is organized as follows.

• The first column is entitled "Source of Variation" and delineates the between treatment and error or residual variation. The total variation is the sum of the between treatment and error variation.
• The second column is entitled "Sums of Squares (SS)" . The between treatment sums of squares is

and is computed by summing the squared differences between each treatment (or group) mean and the overall mean. The squared differences are weighted by the sample sizes per group (n j ). The error sums of squares is:

and is computed by summing the squared differences between each observation and its group mean (i.e., the squared differences between each observation in group 1 and the group 1 mean, the squared differences between each observation in group 2 and the group 2 mean, and so on). The double summation ( SS ) indicates summation of the squared differences within each treatment and then summation of these totals across treatments to produce a single value. (This will be illustrated in the following examples). The total sums of squares is:

and is computed by summing the squared differences between each observation and the overall sample mean. In an ANOVA, data are organized by comparison or treatment groups. If all of the data were pooled into a single sample, SST would reflect the numerator of the sample variance computed on the pooled or total sample. SST does not figure into the F statistic directly. However, SST = SSB + SSE, thus if two sums of squares are known, the third can be computed from the other two.

• The third column contains degrees of freedom . The between treatment degrees of freedom is df 1 = k-1. The error degrees of freedom is df 2 = N - k. The total degrees of freedom is N-1 (and it is also true that (k-1) + (N-k) = N-1).
• The fourth column contains "Mean Squares (MS)" which are computed by dividing sums of squares (SS) by degrees of freedom (df), row by row. Specifically, MSB=SSB/(k-1) and MSE=SSE/(N-k). Dividing SST/(N-1) produces the variance of the total sample. The F statistic is in the rightmost column of the ANOVA table and is computed by taking the ratio of MSB/MSE.  

A clinical trial is run to compare weight loss programs and participants are randomly assigned to one of the comparison programs and are counseled on the details of the assigned program. Participants follow the assigned program for 8 weeks. The outcome of interest is weight loss, defined as the difference in weight measured at the start of the study (baseline) and weight measured at the end of the study (8 weeks), measured in pounds.  

Three popular weight loss programs are considered. The first is a low calorie diet. The second is a low fat diet and the third is a low carbohydrate diet. For comparison purposes, a fourth group is considered as a control group. Participants in the fourth group are told that they are participating in a study of healthy behaviors with weight loss only one component of interest. The control group is included here to assess the placebo effect (i.e., weight loss due to simply participating in the study). A total of twenty patients agree to participate in the study and are randomly assigned to one of the four diet groups. Weights are measured at baseline and patients are counseled on the proper implementation of the assigned diet (with the exception of the control group). After 8 weeks, each patient's weight is again measured and the difference in weights is computed by subtracting the 8 week weight from the baseline weight. Positive differences indicate weight losses and negative differences indicate weight gains. For interpretation purposes, we refer to the differences in weights as weight losses and the observed weight losses are shown below.

Is there a statistically significant difference in the mean weight loss among the four diets?  We will run the ANOVA using the five-step approach.

• Step 1. Set up hypotheses and determine level of significance

H 0 : μ 1 = μ 2 = μ 3 = μ 4 H 1 : Means are not all equal              α=0.05

• Step 2. Select the appropriate test statistic.  

The test statistic is the F statistic for ANOVA, F=MSB/MSE.

• Step 3. Set up decision rule.  

The appropriate critical value can be found in a table of probabilities for the F distribution(see "Other Resources"). In order to determine the critical value of F we need degrees of freedom, df 1 =k-1 and df 2 =N-k. In this example, df 1 =k-1=4-1=3 and df 2 =N-k=20-4=16. The critical value is 3.24 and the decision rule is as follows: Reject H 0 if F > 3.24.

• Step 4. Compute the test statistic.  

To organize our computations we complete the ANOVA table. In order to compute the sums of squares we must first compute the sample means for each group and the overall mean based on the total sample.  

We can now compute

So, in this case:

Next we compute,

SSE requires computing the squared differences between each observation and its group mean. We will compute SSE in parts. For the participants in the low calorie diet:  

For the participants in the low fat diet:  

For the participants in the low carbohydrate diet:  

For the participants in the control group:

We can now construct the ANOVA table .

• Step 5. Conclusion.  

We reject H 0 because 8.43 > 3.24. We have statistically significant evidence at α=0.05 to show that there is a difference in mean weight loss among the four diets.    

ANOVA is a test that provides a global assessment of a statistical difference in more than two independent means. In this example, we find that there is a statistically significant difference in mean weight loss among the four diets considered. In addition to reporting the results of the statistical test of hypothesis (i.e., that there is a statistically significant difference in mean weight losses at α=0.05), investigators should also report the observed sample means to facilitate interpretation of the results. In this example, participants in the low calorie diet lost an average of 6.6 pounds over 8 weeks, as compared to 3.0 and 3.4 pounds in the low fat and low carbohydrate groups, respectively. Participants in the control group lost an average of 1.2 pounds which could be called the placebo effect because these participants were not participating in an active arm of the trial specifically targeted for weight loss. Are the observed weight losses clinically meaningful?

Another ANOVA Example

Calcium is an essential mineral that regulates the heart, is important for blood clotting and for building healthy bones. The National Osteoporosis Foundation recommends a daily calcium intake of 1000-1200 mg/day for adult men and women. While calcium is contained in some foods, most adults do not get enough calcium in their diets and take supplements. Unfortunately some of the supplements have side effects such as gastric distress, making them difficult for some patients to take on a regular basis.  

 A study is designed to test whether there is a difference in mean daily calcium intake in adults with normal bone density, adults with osteopenia (a low bone density which may lead to osteoporosis) and adults with osteoporosis. Adults 60 years of age with normal bone density, osteopenia and osteoporosis are selected at random from hospital records and invited to participate in the study. Each participant's daily calcium intake is measured based on reported food intake and supplements. The data are shown below.   

Is there a statistically significant difference in mean calcium intake in patients with normal bone density as compared to patients with osteopenia and osteoporosis? We will run the ANOVA using the five-step approach.

H 0 : μ 1 = μ 2 = μ 3 H 1 : Means are not all equal                            α=0.05

In order to determine the critical value of F we need degrees of freedom, df 1 =k-1 and df 2 =N-k.   In this example, df 1 =k-1=3-1=2 and df 2 =N-k=18-3=15. The critical value is 3.68 and the decision rule is as follows: Reject H 0 if F > 3.68.

To organize our computations we will complete the ANOVA table. In order to compute the sums of squares we must first compute the sample means for each group and the overall mean.  

 If we pool all N=18 observations, the overall mean is 817.8.

We can now compute:

Substituting:

SSE requires computing the squared differences between each observation and its group mean. We will compute SSE in parts. For the participants with normal bone density:

For participants with osteopenia:

For participants with osteoporosis:

We do not reject H 0 because 1.395 < 3.68. We do not have statistically significant evidence at a =0.05 to show that there is a difference in mean calcium intake in patients with normal bone density as compared to osteopenia and osterporosis. Are the differences in mean calcium intake clinically meaningful? If so, what might account for the lack of statistical significance?

One-Way ANOVA in R

The video below by Mike Marin demonstrates how to perform analysis of variance in R. It also covers some other statistical issues, but the initial part of the video will be useful to you.

Two-Factor ANOVA

The ANOVA tests described above are called one-factor ANOVAs. There is one treatment or grouping factor with k > 2 levels and we wish to compare the means across the different categories of this factor. The factor might represent different diets, different classifications of risk for disease (e.g., osteoporosis), different medical treatments, different age groups, or different racial/ethnic groups. There are situations where it may be of interest to compare means of a continuous outcome across two or more factors. For example, suppose a clinical trial is designed to compare five different treatments for joint pain in patients with osteoarthritis. Investigators might also hypothesize that there are differences in the outcome by sex. This is an example of a two-factor ANOVA where the factors are treatment (with 5 levels) and sex (with 2 levels). In the two-factor ANOVA, investigators can assess whether there are differences in means due to the treatment, by sex or whether there is a difference in outcomes by the combination or interaction of treatment and sex. Higher order ANOVAs are conducted in the same way as one-factor ANOVAs presented here and the computations are again organized in ANOVA tables with more rows to distinguish the different sources of variation (e.g., between treatments, between men and women). The following example illustrates the approach.

Consider the clinical trial outlined above in which three competing treatments for joint pain are compared in terms of their mean time to pain relief in patients with osteoarthritis. Because investigators hypothesize that there may be a difference in time to pain relief in men versus women, they randomly assign 15 participating men to one of the three competing treatments and randomly assign 15 participating women to one of the three competing treatments (i.e., stratified randomization). Participating men and women do not know to which treatment they are assigned. They are instructed to take the assigned medication when they experience joint pain and to record the time, in minutes, until the pain subsides. The data (times to pain relief) are shown below and are organized by the assigned treatment and sex of the participant.

Table of Time to Pain Relief by Treatment and Sex

The analysis in two-factor ANOVA is similar to that illustrated above for one-factor ANOVA. The computations are again organized in an ANOVA table, but the total variation is partitioned into that due to the main effect of treatment, the main effect of sex and the interaction effect. The results of the analysis are shown below (and were generated with a statistical computing package - here we focus on interpretation). 

 ANOVA Table for Two-Factor ANOVA

There are 4 statistical tests in the ANOVA table above. The first test is an overall test to assess whether there is a difference among the 6 cell means (cells are defined by treatment and sex). The F statistic is 20.7 and is highly statistically significant with p=0.0001. When the overall test is significant, focus then turns to the factors that may be driving the significance (in this example, treatment, sex or the interaction between the two). The next three statistical tests assess the significance of the main effect of treatment, the main effect of sex and the interaction effect. In this example, there is a highly significant main effect of treatment (p=0.0001) and a highly significant main effect of sex (p=0.0001). The interaction between the two does not reach statistical significance (p=0.91). The table below contains the mean times to pain relief in each of the treatments for men and women (Note that each sample mean is computed on the 5 observations measured under that experimental condition).  

Mean Time to Pain Relief by Treatment and Gender

Treatment A appears to be the most efficacious treatment for both men and women. The mean times to relief are lower in Treatment A for both men and women and highest in Treatment C for both men and women. Across all treatments, women report longer times to pain relief (See below).  

Notice that there is the same pattern of time to pain relief across treatments in both men and women (treatment effect). There is also a sex effect - specifically, time to pain relief is longer in women in every treatment.  

Suppose that the same clinical trial is replicated in a second clinical site and the following data are observed.

Table - Time to Pain Relief by Treatment and Sex - Clinical Site 2

The ANOVA table for the data measured in clinical site 2 is shown below.

Table - Summary of Two-Factor ANOVA - Clinical Site 2

Notice that the overall test is significant (F=19.4, p=0.0001), there is a significant treatment effect, sex effect and a highly significant interaction effect. The table below contains the mean times to relief in each of the treatments for men and women.  

Table - Mean Time to Pain Relief by Treatment and Gender - Clinical Site 2

Notice that now the differences in mean time to pain relief among the treatments depend on sex. Among men, the mean time to pain relief is highest in Treatment A and lowest in Treatment C. Among women, the reverse is true. This is an interaction effect (see below).  

Notice above that the treatment effect varies depending on sex. Thus, we cannot summarize an overall treatment effect (in men, treatment C is best, in women, treatment A is best).    

When interaction effects are present, some investigators do not examine main effects (i.e., do not test for treatment effect because the effect of treatment depends on sex). This issue is complex and is discussed in more detail in a later module. 

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Hypothesis Testing Steps & Examples

Table of Contents

What is a Hypothesis testing?

As per the definition from Oxford languages, a hypothesis is a supposition or proposed explanation made on the basis of limited evidence as a starting point for further investigation. As per the Dictionary page on Hypothesis , Hypothesis means a proposition or set of propositions, set forth as an explanation for the occurrence of some specified group of phenomena, either asserted merely as a provisional conjecture to guide investigation (working hypothesis) or accepted as highly probable in the light of established facts.

The hypothesis can be defined as the claim that can either be related to the truth about something that exists in the world, or, truth about something that’s needs to be established a fresh . In simple words, another word for the hypothesis is the “claim” . Until the claim is proven to be true, it is called the hypothesis. Once the claim is proved, it becomes the new truth or new knowledge about the thing. For example , let’s say that a claim is made that students studying for more than 6 hours a day gets more than 90% of marks in their examination. Now, this is just a claim or a hypothesis and not the truth in the real world. However, in order for the claim to become the truth for widespread adoption, it needs to be proved using pieces of evidence, e.g., data.  In order to reject this claim or otherwise, one needs to do some empirical analysis by gathering data samples and evaluating the claim. The process of gathering data and evaluating the claims or hypotheses with the goal to reject or otherwise (failing to reject) can be called as hypothesis testing . Note the wordings – “failing to reject”. It means that we don’t have enough evidence to reject the claim. Thus, until the time that new evidence comes up, the claim can be considered the truth. There are different techniques to test the hypothesis in order to reach the conclusion of whether the hypothesis can be used to represent the truth of the world.

One must note that the hypothesis testing never constitutes a proof that the hypothesis is absolute truth based on the observations. It only provides added support to consider the hypothesis as truth until the time that new evidences can against the hypotheses can be gathered. We can never be 100% sure about truth related to those hypotheses based on the hypothesis testing.

Simply speaking, hypothesis testing is a framework that can be used to assert whether the claim or the hypothesis made about a real-world/real-life event can be seen as the truth or otherwise based on the given data (evidences).

Hypothesis Testing Examples

Before we get ahead and start understanding more details about hypothesis and hypothesis testing steps, lets take a look at some  real-world examples  of how to think about hypothesis and hypothesis testing when dealing with real-world problems :

• Customers are churning because they ain’t getting response to their complaints or issues
• Customers are churning because there are other competitive services in the market which are providing these services at lower cost.
• Customers are churning because there are other competitive services which are providing more services at the same cost.
• It is claimed that a 500 gm sugar packet for a particular brand, say XYZA, contains sugar of less than 500 gm, say around 480gm.  Can this claim be taken as truth? How do we know that this claim is true? This is a hypothesis until proved.
• A group of doctors claims that quitting smoking increases lifespan. Can this claim be taken as new truth? The hypothesis is that quitting smoking results in an increase in lifespan.
• It is claimed that brisk walking for half an hour every day reverses diabetes. In order to accept this in your lifestyle, you may need evidence that supports this claim or hypothesis.
• It is claimed that doing Pranayama yoga for 30 minutes a day can help in easing stress by 50%. This can be termed as hypothesis and would require testing / validation for it to be established as a truth and recommended for widespread adoption.
• One common real-life example of hypothesis testing is election polling. In order to predict the outcome of an election, pollsters take a sample of the population and ask them who they plan to vote for. They then use hypothesis testing to assess whether their sample is representative of the population as a whole. If the results of the hypothesis test are significant, it means that the sample is representative and that the poll can be used to predict the outcome of the election. However, if the results are not significant, it means that the sample is not representative and that the poll should not be used to make predictions.
• Machine learning models make predictions based on the input data. Each of the machine learning model representing a function approximation can be taken as a hypothesis. All different models constitute what is called as hypothesis space .
• As part of a linear regression machine learning model , it is claimed that there is a relationship between the response variables and predictor variables? Can this hypothesis or claim be taken as truth? Let’s say, the hypothesis is that the housing price depends upon the average income of people already staying in the locality. How true is this hypothesis or claim? The relationship between response variable and each of the predictor variables can be evaluated using T-test and T-statistics .
• For linear regression model , one of the hypothesis is that there is no relationship between the response variable and any of the predictor variables. Thus, if b1, b2, b3 are three parameters, all of them is equal to 0. b1 = b2 = b3 = 0. This is where one performs F-test and use F-statistics to test this hypothesis.

You may note different hypotheses which are listed above. The next step would be validate some of these hypotheses. This is where data scientists will come into picture. One or more data scientists may be asked to work on different hypotheses. This would result in these data scientists looking for appropriate data related to the hypothesis they are working. This section will be detailed out in near future.

State the Hypothesis to begin Hypothesis Testing

The first step to hypothesis testing is defining or stating a hypothesis. Before the hypothesis can be tested, we need to formulate the hypothesis in terms of mathematical expressions. There are two important aspects to pay attention to, prior to the formulation of the hypothesis. The following represents different types of hypothesis that could be put to hypothesis testing:

• Claim made against the well-established fact : The case in which a fact is well-established, or accepted as truth or “knowledge” and a new claim is made about this well-established fact. For example , when you buy a packet of 500 gm of sugar, you assume that the packet does contain at the minimum 500 gm of sugar and not any less, based on the label of 500 gm on the packet. In this case, the fact is given or assumed to be the truth. A new claim can be made that the 500 gm sugar contains sugar weighing less than 500 gm. This claim needs to be tested before it is accepted as truth. Such cases could be considered for hypothesis testing if this is claimed that the assumption or the default state of being is not true. The claim to be established as new truth can be stated as “alternate hypothesis”. The opposite state can be stated as “null hypothesis”. Here the claim that the 500 gm packet consists of sugar less than 500 grams would be stated as alternate hypothesis. The opposite state which is the sugar packet consists 500 gm is null hypothesis.
• Claim to establish the new truth : The case in which there is some claim made about the reality that exists in the world (fact). For example , the fact that the housing price depends upon the average income of people already staying in the locality can be considered as a claim and not assumed to be true. Another example could be the claim that running 5 miles a day would result in a reduction of 10 kg of weight within a month. There could be varied such claims which when required to be proved as true have to go through hypothesis testing. The claim to be established as new truth can be stated as “alternate hypothesis”. The opposite state can be stated as “null hypothesis”. Running 5 miles a day would result in reduction of 10 kg within a month would be stated as alternate hypothesis.

Based on the above considerations, the following hypothesis can be stated for doing hypothesis testing.

• The packet of 500 gm of sugar contains sugar of weight less than 500 gm. (Claim made against the established fact). This is a new knowledge which requires hypothesis testing to get established and acted upon.
• The housing price depends upon the average income of the people staying in the locality. This is a new knowledge which requires hypothesis testing to get established and acted upon.
• Running 5 miles a day results in a reduction of 10 kg of weight within a month. This is a new knowledge which requires hypothesis testing to get established for widespread adoption.

Formulate Null & Alternate Hypothesis as Next Step

Once the hypothesis is defined or stated, the next step is to formulate the null and alternate hypothesis in order to begin hypothesis testing as described above.

What is a null hypothesis?

In the case where the given statement is a well-established fact or default state of being in the real world, one can call it a null hypothesis (in the simpler word, nothing new). Well-established facts don’t need any hypothesis testing and hence can be called the null hypothesis. In cases, when there are any new claims made which is not well established in the real world, the null hypothesis can be thought of as the default state or opposite state of that claim. For example , in the previous section, the claim or hypothesis is made that the students studying for more than 6 hours a day gets more than 90% of marks in their examination. The null hypothesis, in this case, will be that the claim is not true or real. The null hypothesis can be stated that there is no relationship or association between the students reading more than 6 hours a day and they getting 90% of the marks. Any occurrence is only a chance occurrence. Another example of hypothesis is when somebody is alleged that they have performed a crime.

Null hypothesis is denoted by letter H with 0, e.g., [latex]H_0[/latex]

What is an alternate hypothesis?

When the given statement is a claim (unexpected event in the real world) and not yet proven, one can call/formulate it as an alternate hypothesis and accordingly define a null hypothesis which is the opposite state of the hypothesis. The alternate hypothesis is a new knowledge or truth that needs to be established. In simple words, the hypothesis or claim that needs to be tested against reality in the real world can be termed the alternate hypothesis. In order to reach a conclusion that the claim (alternate hypothesis) can be considered the new knowledge or truth (based on the available evidence), it would be important to reject the null hypothesis. It should be noted that null and alternate hypotheses are mutually exclusive and at the same time asymmetric. In the example given in the previous section, the claim that the students studying for more than 6 hours get more than 90% of marks can be termed as the alternate hypothesis.

Alternate hypothesis is denoted with H subscript a, e.g., [latex]H_a[/latex]

Once the hypothesis is formulated as null([latex]H_0[/latex]) and alternate hypothesis ([latex]H_a[/latex]), there are two possible outcomes that can happen from hypothesis testing. These outcomes are the following:

• Reject the null hypothesis : There is enough evidence based on which one can reject the null hypothesis. Let’s understand this with the help of an example provided earlier in this section. The null hypothesis is that there is no relationship between the students studying more than 6 hours a day and getting more than 90% marks. In a sample of 30 students studying more than 6 hours a day, it was found that they scored 91% marks. Given that the null hypothesis is true, this kind of hypothesis testing result will be highly unlikely. This kind of result can’t happen by chance. That would mean that the claim can be taken as the new truth or new knowledge in the real world. One can go and take further samples of 30 students to perform some more testing to validate the hypothesis. If similar results show up with other tests, it can be said with very high confidence that there is enough evidence to reject the null hypothesis that there is no relationship between the students studying more than 6 hours a day and getting more than 90% marks. In such cases, one can go to accept the claim as new truth that the students studying more than 6 hours a day get more than 90% marks. The hypothesis can be considered the new truth until the time that new tests provide evidence against this claim.
• Fail to reject the null hypothesis : There is not enough evidence-based on which one can reject the null hypothesis (well-established fact or reality). Thus, one would fail to reject the null hypothesis. In a sample of 30 students studying more than 6 hours a day, the students were found to score 75%. Given that the null hypothesis is true, this kind of result is fairly likely or expected. With the given sample, one can’t reject the null hypothesis that there is no relationship between the students studying more than 6 hours a day and getting more than 90% marks.

Examples of formulating the null and alternate hypothesis

The following are some examples of the null and alternate hypothesis.

Hypothesis Testing Steps

Here is the diagram which represents the workflow of Hypothesis Testing.

Figure 1. Hypothesis Testing Steps

Based on the above, the following are some of the  steps to be taken when doing hypothesis testing:

• State the hypothesis : First and foremost, the hypothesis needs to be stated. The hypothesis could either be the statement that is assumed to be true or the claim which is made to be true.
• Formulate the hypothesis : This step requires one to identify the Null and Alternate hypotheses or in simple words, formulate the hypothesis. Take an example of the canned sauce weighing 500 gm as the Null Hypothesis.
• Set the criteria for a decision : Identify test statistics that could be used to assess the Null Hypothesis. The test statistics with the above example would be the average weight of the sugar packet, and t-statistics would be used to determine the P-value. For different kinds of problems, different kinds of statistics including Z-statistics, T-statistics, F-statistics, etc can be used.
• Identify the level of significance (alpha) : Before starting the hypothesis testing, one would be required to set the significance level (also called as  alpha ) which represents the value for which a P-value less than or equal to  alpha  is considered statistically significant. Typical values of  alpha  are 0.1, 0.05, and 0.01. In case the P-value is evaluated as statistically significant, the null hypothesis is rejected. In case, the P-value is more than the  alpha  value, the null hypothesis is failed to be rejected.
• Compute the test statistics : Next step is to calculate the test statistics (z-test, t-test, f-test, etc) to determine the P-value. If the sample size is more than 30, it is recommended to use z-statistics. Otherwise, t-statistics could be used. In the current example where 20 packets of canned sauce is selected for hypothesis testing, t-statistics will be calculated for the mean value of 505 gm (sample mean). The t-statistics would then be calculated as the difference of 505 gm (sample mean) and the population means (500 gm) divided by the sample standard deviation divided by the square root of sample size (20).
• Calculate the P-value of the test statistics : Once the test statistics have been calculated, find the P-value using either of t-table or a z-table. P-value is the probability of obtaining a test statistic (t-score or z-score) equal to or more extreme than the result obtained from the sample data, given that the null hypothesis H0 is true.
• Compare P-value with the level of significance : The significance level is set as the allowable range within which if the value appears, one will be failed to reject the Null Hypothesis. This region is also called as Non-rejection region . The value of alpha is compared with the p-value. If the p-value is less than the significance level, the test is statistically significant and hence, the null hypothesis will be rejected.

P-Value: Key to Statistical Hypothesis Testing

Once you formulate the hypotheses, there is the need to test those hypotheses. Meaning, say that the null hypothesis is stated as the statement that housing price does not depend upon the average income of people staying in the locality, it would be required to be tested by taking samples of housing prices and, based on the test results, this Null hypothesis could either be rejected or failed to be rejected . In hypothesis testing, the following two are the outcomes:

• Reject the Null hypothesis
• Fail to Reject the Null hypothesis

Take the above example of the sugar packet weighing 500 gm. The Null hypothesis is set as the statement that the sugar packet weighs 500 gm. After taking a sample of 20 sugar packets and testing/taking its weight, it was found that the average weight of the sugar packets came to 495 gm. The test statistics (t-statistics) were calculated for this sample and the P-value was determined. Let’s say the P-value was found to be 15%. Assuming that the level of significance is selected to be 5%, the test statistic is not statistically significant (P-value > 5%) and thus, the null hypothesis fails to get rejected. Thus, one could safely conclude that the sugar packet does weigh 500 gm. However, if the average weight of canned sauce would have found to be 465 gm, this is way beyond/away from the mean value of 500 gm and one could have ended up rejecting the Null Hypothesis based on the P-value .

Hypothesis Testing for Problem Analysis & Solution Implementation

Hypothesis testing can be applied in both problem analysis and solution implementation. The following represents method on how you can apply hypothesis testing technique for both problem and solution space:

• Problem Analysis : Hypothesis testing is a systematic way to validate assumptions or educated guesses during problem analysis. It allows for a structured investigation into the nature of a problem and its potential root causes. In this process, a null hypothesis and an alternative hypothesis are usually defined. The null hypothesis generally asserts that no significant change or effect exists, while the alternative hypothesis posits the opposite. Through controlled experiments, data collection, or statistical analysis, these hypotheses are then tested to determine their validity. For example, if a software company notices a sudden increase in user churn rate, they might hypothesize that the recent update to their application is the root cause. The null hypothesis could be that the update has no effect on churn rate, while the alternative hypothesis would assert that the update significantly impacts the churn rate. By analyzing user behavior and feedback before and after the update, and perhaps running A/B tests where one user group has the update and another doesn’t, the company can test these hypotheses. If the alternative hypothesis is confirmed, the company can then focus on identifying specific issues in the update that may be causing the increased churn, thereby moving closer to a solution.
• Solution Implementation : Hypothesis testing can also be a valuable tool during the solution implementation phase, serving as a method to evaluate the effectiveness of proposed remedies. By setting up a specific hypothesis about the expected outcome of a solution, organizations can create targeted metrics and KPIs to measure success. For example, if a retail business is facing low customer retention rates, they might implement a loyalty program as a solution. The hypothesis could be that introducing a loyalty program will increase customer retention by at least 15% within six months. The null hypothesis would state that the loyalty program has no significant effect on retention rates. To test this, the company can compare retention metrics from before and after the program’s implementation, possibly even setting up control groups for more robust analysis. By applying statistical tests to this data, the company can determine whether their hypothesis is confirmed or refuted, thereby gauging the effectiveness of their solution and making data-driven decisions for future actions.
• Tests of Significance
• Hypothesis testing for the Mean
• z-statistics vs t-statistics (Khan Academy)

Hypothesis testing quiz

The claim that needs to be established is set as ____________, the outcome of hypothesis testing is _________.

Please select 2 correct answers

P-value is defined as the probability of obtaining the result as extreme given the null hypothesis is true

There is a claim that doing pranayama yoga results in reversing diabetes. which of the following is true about null hypothesis.

In this post, you learned about hypothesis testing and related nuances such as the null and alternate hypothesis formulation techniques, ways to go about doing hypothesis testing etc. In data science, one of the reasons why one needs to understand the concepts of hypothesis testing is the need to verify the relationship between the dependent (response) and independent (predictor) variables. One would, thus, need to understand the related concepts such as hypothesis formulation into null and alternate hypothesis, level of significance, test statistics calculation, P-value, etc. Given that the relationship between dependent and independent variables is a sort of hypothesis or claim , the null hypothesis could be set as the scenario where there is no relationship between dependent and independent variables.

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Highlights: Closing arguments wrap in Trump hush money trial

What to know about the hush money trial.

• Prosecutors finished delivering their closing statements in the trial shortly before 8 p.m. Former President Donald Trump's lawyers presented their arguments this morning .
• Prosecutor Joshua Steinglass, who spoke for more than four hours, argued that Trump falsified business records to cover up what was essentially an illegal campaign contribution meant to help him get elected in 2016.
• Trump is charged with 34 counts of falsifying business records in connection with a hush money payment to adult film actor Stormy Daniels to buy her silence about an alleged affair with Trump. He has pleaded not guilty to the charges.
• Court adjourned for the day at 8 p.m. and will resume at 10 a.m., when the judge will give instructions to the jury before it begins deliberations.

Judge lays out timeline for the rest of the week

Gary Grumbach

Zoë Richards

Tomorrow's trial proceedings are expected to get underway at 10 a.m., instead of the regular 9:30 a.m., with Judge Juan Merchan saying he expects jury instructions to last about an hour.

After that, the case will be in the hands of the jury.

Merchan said tomorrow's proceedings will conclude at 4:30 p.m., but he left the door open for the rest of the week, noting that if proceedings are needed on Thursday and Friday, the timing will be determined by how deliberations are progressing.

Trump makes no comments after leaving courtroom

Katherine Koretski

Trump did not make any comments as he left the Manhattan courtroom after the prosecution delivered closing arguments that went until just before 8 p.m.

Trump, who has often spoken outside the courtroom, instead raised his fist as he left.

Closing arguments are done; court to resume at 10 a.m. tomorrow

Prosecutor Joshua Steinglass has finished his closing argument, which began shortly after 2 p.m.

Judge Juan Merchan told jurors they will start tomorrow at 10 a.m.

Merchan told jurors that jury instructions will take around an hour before deliberations begin. He said the plan is to go until 4:30 p.m. for the day.

Prosecutor gets fired up during end of closing argument

Phil Helsel

Prosecutor Joshua Steinglass began accelerating and emphasizing his delivery to jurors during closing arguments with minutes to go before an 8 p.m. deadline.

Steinglass reiterated to the jurors that it is a crime to willfully create inaccurate tax forms and that Trump’s intent to defraud in this case is clear. He argued that why else would Stormy Daniels be paid in what he described as an elaborate scheme, instead of all at once.

Steinglass argued that that and other steps show Trump wanted the issue to be kept quiet until after the election.

“The name of the game was concealment,” he said.

Defense objects to prosecutor's remarks about Trump and Fifth Avenue

Jillian Frankel

Prosecutor Joshua Steinglass urged the jury to hold Trump accountable, suggesting by way of analogy that he can’t shoot someone on Fifth Avenue during rush hour and get away with it.

Trump's defense team objected to the comment, which Judge Juan Merchan sustained.

Mixed level of visible engagement among jurors at this late hour

Laura Jarrett

At least one juror appears to be visibly engaged in prosecutor Joshua Steinglass’ presentation — offering an affirming smile.

Others, however, appear considerably less focused and can be seen twisting their hair and rubbing their faces.

The jury is approaching an 11-hour day at the courthouse.

Prosecutor talks about difference between reasonable doubt and certainty

Prosecutor Joshua Steinglass told the jury that it does not need to evaluate each piece of evidence alone and in a vacuum but as part of a whole that he argues proves Trump’s guilt.

“You will see that the people have proven this case beyond a reasonable doubt,” he said.

During his remarks, the defense objected. Judge Juan Merchan sustained the objection.

“I’ll instruct them on the law and the evidence,” Merchan said.

Prosecutor launches into rapid-fire recap of Trump’s involvement in Daniels and McDougal stories

Prosecutor Joshua Steinglass is recapping all of the evidence intended to show Trump’s direct involvement in the settlements with Karen McDougal and Stormy Daniels, beginning with an August 2015 Trump Tower meeting.

A screen the prosecution displayed during closing arguments read “Mr. Trump involved every step of the way” as Steinglass went through a timeline of events.

Joshua Steinglass passes 4-hour mark in his closing arguments

Prosecutor Joshua Steinglass has passed the four-hour mark since he began giving the prosecution’s closing argument in Trump’s trial.

Steinglass began giving the prosecution’s closing arguments at around 2:07 p.m., but there have been several breaks since then.

Today's trial proceedings to continue until 8 p.m., judge says

After he returned to the bench, Judge Juan Merchan indicated to the attorneys that the court will push forward until 8 p.m. but will need to wrap up after that.

That would make an 11-hour day for the jury.

Last recess of the day

Judge Juan Merchan announced at 6:52 p.m. what he said will be the last recess of the day.

It's expected to last just a few minutes.

Merchan earlier said that the plan was to go until at least 7 p.m. and “finish this out if we can.”

'A bold-faced lie': Prosecutor revisits Robert Costello's testimony

Given the hour, it was initially unclear why prosecutor Joshua Steinglass began revisiting the testimony of Robert Costello , a Trump ally and lawyer who has clashed with Michael Cohen.

But the prosecution's display of an email exchange between Costello and Cohen hinted that the DA's office aims to portray Trump’s attitude toward Cohen changing only after his former attorney's compliance was in doubt, not because of anything else Cohen did.

Recounting Costello's testimony, Steinglass argued that Costello's assertion that he was acting in Cohen’s best interest and that he didn’t care at all about the defendant’s interest "was a bold-faced lie.”

‘You guys good to go a little bit longer?’ prosecutor asks, as 7 p.m. draws near

Prosecutor Joshua Steinglass asked jurors, “You guys good to go a little bit longer?” and said “Alright!” after a bench meeting to discuss scheduling at around 6:30 p.m.

Judge Juan Merchan earlier today said the plan was to go until at least 7 p.m. and “finish this out if we can.”

Prosecutor refers to 'devastating' testimony by Hope Hicks

Given the largely chronological order of the prosecution's closing arguments, prosecutor Joshua Steinglass could be nearing the end of his remarks.

He discussed what he called Hope Hicks’ “devastating” testimony earlier in the trial, adding that she burst into tears because she realized the impact of what she had told the court.

Defense attorney Todd Blanche objected to that characterization, but Judge Juan Merchan allowed it.

Prosecutor argues Trump wanted to be 'involved in everything'

Prosecutor Joshua Steinglass mocked former Trump aide Madeleine Westerhout’s testimony in which she said Trump was often so busy that sometimes he absent-mindedly signed presidential proclamations.

Steinglass, who dismissed Westerhout's remarks as a narrative Trump’s team encouraged, said that overall she gave the opposite impression — that the former president remained very attentive to outlays of his personal expenses, and that his most frequent contacts included his former attorney Michael Cohen and a former top executive of his company, Allen Weisselberg. Westerhout's testimony also conveyed that Trump continued to be the sole signatory on his own accounts, even though he easily could have added other signatories, Steinglass argued.

Trump wanted to maintain control — and “he insists on signing his own checks," Steinglass said, adding that Trump boasted about his frugality and micromanagement in his books, which Steinglass read excerpts from.

Steinglass also rejected the defense's argument that Trump was too busy to be involved in certain financial transactions.

“He’s in charge of a company for 40 years. The defendant’s entire business philosophy was to be involved in everything,” Steinglass said.

Prosecutor: Cohen's time being cross-examined exceeded his legal work for Trump in 2017

Prosecutor Joshua Steinglass said that Michael Cohen did very few hours of legal work for Trump on 2017, and that “these payments had nothing to do with the retainer agreement and nothing to do with services rendered in 2017.”

“Cohen spent more time being cross-examined in this trial than he did doing legal work for Donald Trump in 2017,” Steinglass said. He also told the jury that none of the Trump invoices went through the Trump Organization’s legal department because they weren’t for legal services rendered.

Steinglass also commented on how Cohen was paid pretty well, and had the title of personal attorney for the president.

“He was making way more money than any government job would ever pay, and don’t I know that,” Steinglass joked.

Some jurors cracked smiles and small laughs when Steinglass joked about government salaries compared to what Cohen was making.

Prosecutor says ‘these documents are so damning that you almost have to laugh’ at defense's argument

Prosecutor Joshua Steinglass told the jury that “these documents are so damning that you almost have to laugh” at an argument presented by Trump’s defense.

Steinglass was referring to a comments by defense attorney Todd Blanche that the records were not false because, if they were false, they would have been destroyed.

Steinglass also argued that the 1099s forms on which Trump reported payments to Michael Cohen of $105,000 and $315,000 were another “unlawful means” through which the conspiracy was acted upon.

EXCLUSIVE: Elise Stefanik requests probe into Merchan's selection as judge

Rep. Elise Stefanik, R-N.Y., issued a complaint letter today to the New York State Commission on Judicial Conduct and an inspector general for the New York State Unified Court System, requesting an investigation into Judge Juan Merchan’s selection to preside over Trump’s hush money case.

Stefanik pointed to Merchan’s role as presiding judge for a pair of other cases related to Trump and his allies, saying, “The probability of three specific criminal cases being assigned to the same justice is infinitesimally small.”

“One cannot help but suspect that the ‘random selection’ at work in the assignment of Acting Justice Merchan, a Democrat Party donor, to these cases involving prominent Republicans, is in fact not random at all,” Stefanik wrote. “The simple answer to why Acting Justice Merchan has been assigned to these cases would seem to be that whoever made the assignment intentionally selected Acting Justice Merchan to handle them to increase the chance that Donald Trump, the Trump Organization, and Steven Bannon would ultimately be convicted.”

The letter marks a continued effort by Trump allies to attack people involved with the case by filing complaints. The board overseeing the judges has made clear that Merchan didn’t need to recuse himself over issues that some of his critics have called a conflict of interest.

Trump posts on Truth Social during break in courtroom action

Vaughn Hillyard

During the court's roughly 20-minute break, Trump on his Truth Social platform disparaged the proceedings as "boring" and a " filibuster ."

Trump's Truth Social account has been active today with posts referring to his criminal trial and the closing arguments, which have continued as the prosecution continues its argument into this evening.

Judge says closing arguments to continue into the evening

Adam Edelman

Judge Juan Merchan announced a short courtroom break and said the plan is to go until at least 7 p.m. and "finish this out if we can."

“I was watching the jurors, they look pretty alert to me. I don’t think we’re losing anyone. So I think right now we’re going to try to finish this out if we can," he told the attorneys.

“Let’s see what we can do," Merchan continued, adding that they will revisit the timeline at 7 p.m.

Prosecutor argues Trump didn't sign confidentiality agreement for a reason

Kyla Guilfoil

Prosecutor Joshua Steinglass tried to turn one of defense attorney Todd Blanche’s better arguments on its head.

Steinglass said that Trump didn’t sign the agreement because that was the point: The agreement was no less enforceable without his signature.

The timing of the payment on Oct. 27, 2016, Steinglass argued, further showed that Trump's primary concern was not his family but the election.

Prosecutor seems to say for first time there were 2 calls between Cohen and Weisselberg in late October 2016

Rebecca Shabad is in Washington, D.C.

Joshua Steinglass mentioned that in the phone records they have, prosecutors saw six calls between Michael Cohen and Allen Weisselberg over three years, two of which were in late October 2016, right before the Stormy Daniels deal was reached.

This appears to be the first time the calls have been mentioned in the case.

Steinglass also emphasized that Trump and Cohen spoke twice on the morning of Oct. 26, 2016, right before Cohen went to First Republic to submit paperwork to open his new account and to send the wire transfer to Keith Davidson on Daniels’ behalf.

Prosecutor walks through Michael Cohen's bank papers

Prosecutor Joshua Steinglass is now going through the false claims and omissions in former Trump lawyer Michael Cohen’s paperwork to First Republic to open an account in the name of his new LLC.

Those forms could serve as the “unlawful means” through which the alleged conspiracy to promote Trump’s election was acted on.

Prosecutor: Stormy Daniels' testimony shows Trump was 'not just words'

Prosecutor Joshua Steinglass is going at Trump now, referring to Story Daniels' testimony to argue that Trump is "not just words."

"Stormy Daniels was a walking, talking reminder that Trump was not just words" at a time when Trump was trying to distinguish between his words and the actions of both Clintons, Steinglass said.

He also noted that Daniels' story got little to no traction until the day after the "Access Hollywood" tape became national news, with phone traffic exploding among Keith Davidson, Dylan Howard, Michael Cohen and Trump.

Prosecutor describes ramifications of the 'Access Hollywood' tape

After a brief break, prosecutor Joshua Steinglass resumed his closing argument by describing the "Access Hollywood" tape, which multiple witnesses during the trial described as catastrophic for Trump's 2016 campaign.

Steinglass said the tape eclipsed coverage of a Category 4 hurricane, according to Hope Hicks; debate prep at Trump Tower was disrupted as campaign leadership discussed how to respond; and elected Republicans raced to disavow Trump's comments on the tape, with some withdrawing their endorsements.

Trump aide Madeleine Westerhout testified that senior Republican National Committee officials were even discussing dropping Trump from the 2016 ticket, Steinglass said.

“The video was vulgar, to say the least," he added.

Prosecution's closing arguments are one-third of the way done

Asked by Judge Juan Merchan "how much longer" the prosecution's closing arguments would take, Joshua Steinglass replied that there was still a lot to get through.

"We’re about a third of the way through," he said.

The prosecution's closing arguments began today shortly before 2:15 p.m.

Prosecutor says Cohen-Trump call shows effort to influence 2016 election

Prosecutor Joshua Steinglass told the jurors that it's their decision what the tape between Michael Cohen and Trump from Sept. 6, 2016, said.

Steinglass said it showed Trump suggested paying in cash — whether it means no financing, lump sum, it doesn’t matter, he said. Steinglass said they were trying to take steps that would not get noticed.

“This tape unequivocally shows a presidential candidate actively engaging in a scheme to influence the election," Steinglass said.

Prosecutor defends Michael Cohen's phone records

Prosecutor Joshua Steinglass is making arguments to defend Michael Cohen's phone records after the defense questioned their integrity.

Steinglass said that Cohen had no idea the Manhattan district attorney's office would ask for phone records again in January of last year, and there would be no conceivable reason for him to delete evidence of a crime he’d already been convicted and served time for.

Prosecution using graphics to illustrate points during closing arguments

The graphics that the Manhattan district attorney's team is using during their summation are high-tech and modern.

In presenting them, prosecutors are isolating certain calls and using zoom functions to highlight them. The graphics offer a clean and accessible way for the attorneys to illustrate their points to the jury.

Prosecutor: Call between David Pecker and Trump makes it 'impossible' to claim Cohen acted independently

Prosecutor Joshua Steinglass discussed a call between David Pecker and Trump in which Pecker apprised him that Michael Cohen had told Trump about Karen McDougal coming forward.

"This call makes it impossible for the defense to claim that Cohen was acting on his own here," Steinglass said.

He said the transaction was an unlawful corporate contribution to the Trump campaign — and not only did Trump know about it, Steinglass said, but he participated as well.

Prosecutor details Karen McDougal catch-and-kill scheme

Prosecutor Joshua Steinglass is going through the Karen McDougal catch-and-kill scheme in minute detail — call by call, text by text and day by day.

Virtually no testimony is needed to illustrate the negotiations — and to the extent that testimony is used, it’s not from key witness and former Trump lawyer Michael Cohen. It's from David Pecker, former publisher of the National Enquirer.

Analysis: Steinglass pokes hole in defense's argument around National Enquirer

Steinglass makes a very good point about the Dino Sajuddin story and corresponding payment.

Sajuddin is the former Trump Tower doorman who claims Trump fathered a child out of wedlock, a claim the former president has denied.

Given that everyone believed Sajuddin's claim to be false, purchasing the story was not something David Pecker did because of his fiduciary duty to shareholders; there was no reason to do it other than to benefit the 2016 Trump campaign.

Steinglass calls 2015 meeting at Trump Tower a 'subversion of democracy'

Steinglass characterized a meeting at Trump Tower almost a decade ago as a “subversion of democracy.”

He said the entire purpose of the August 2015 meeting was to “pull the wool over [voters’] eyes” before they made their decisions.

He also pointed out that while NDAs are not unlawful, nor are contracts illegal, a contract to kill your wife is illegal, and therefore an NDA designed to prevent certain information from becoming public during a political campaign is also illegal.

Steinglass tells jurors to think of Cohen as a 'tour guide'

Daniel Arkin

Trump's lawyers repeatedly attempted to make Cohen's trustworthiness and motives a focal point of the trial — a strategy that Steinglass flat-out rejected in his summations. "This case is not about Michael Cohen," Steinglass told the jury. "This case is about Donald Trump."

Steinglass encouraged the jury to instead think of Cohen as a "tour guide" through the evidence introduced during the proceedings, including what the state has presented as falsified business records aimed at covering up an election law violation. Cohen, according to Steinglass, "provides context and color to the documents" — but he is not the trial's main character.

Steinglass begins touching on campaign finance violations

Steinglass is teasing the crux of the prosecution’s argument, saying, “Once money starts changing hands on behalf of the campaign, that’s election law — that’s federal election campaign finance violation.”

“We’ll get back to that,” he adds.

Prosecution argues there is a 'mountain of evidence' against Trump

Steinglass is fighting back against the defense's rhetoric that the only evidence in this case came from Michael Cohen's testimony.

The prosecutor told the jury that Judge Merchan will say Cohen is an accomplice because he participated in these crimes, but you cannot convict Trump on Cohen’s word alone — unless there is corroborating evidence.

Steinglass said that there is a mountain of evidence in the case, saying "it’s difficult to conceive of a case with more corroboration than this one.”

Steinglass looks to counter questions on details of Cohen's stories

Steinglass is now using an imaginary conversation to explain Cohen’s retelling of some of the stories or dates he’d recounted to the jury that Trump’s lawyers had questioned.

“These guys know each other well, they speak in code. A better explanation is that Cohen could have gotten the time and place of the call wrong. This is one date in many, he spoke to the defendant 20 times in the month of October,” Steinglass said.

“Let’s say you had dinner at a restaurant with an old friend and the friend says they were getting married. Later you find a receipt and think that was the night they told you they were getting married, but found out the friend was actually in California on that night. That does not mean that you are lying about the fact that you had dinner with the friend or about the fact that your friend told you they were getting married,” Steinglass said.

Steinglass: We didn't pick Cohen at the 'witness store'

Steinglass is forcefully pushing back on the Trump team's attempts to tarnish Cohen's character and motives, reminding the jury that the ex-fixer was once a valued member of the former president's inner circle: "We didn't choose Michael Cohen. We didn't pick him up at the witness store. Mr. Trump chose Mr. Cohen for the same qualities his attorneys now urge you to reject."

Cohen's top quality was loyalty to his former boss, Steinglass said. Cohen was "drawn to the defendant like a moth to a flame, and he wasn't the only one. David Pecker saw Mr. Trump as a mentor; Mr. Trump saw David Pecker as a useful tool."

On Trump attacks on Cohen: 'That is what some people might call chutzpah'

Steinglass is explaining that Cohen had lied at Trump’s direction and that Trump was now using those lies to harm Cohen’s credibility in the trial.

“The defense also tells you you should reject his testimony because he lied and took pleas in federal court. He has had some trouble accepting responsibility,” Steinglass said. “For bank fraud conviction and his tax law violation, he said he admitted to you that he did the things. He pleaded guilty.”

“He feels like he was treated unfairly and as a first offender he should have been able to pay a fine and back taxes and he believes the Trump Justice Department did him dirty. Whether that is true or not, he accepted responsibility and went to prison for it,” Steinglass added.

“You should consider all of this for his credibility” he continued. “The lies he told to Congress had to do with the Mueller investigation and the Russia probe, and what he lied about was the number of dealings the defendant had with Russia, and the only benefit was he stayed in the defendant’s good graces.”

“Those lies that he told are being used by the same defendant to undermine his credibility,” Steinglass said. 

“That is what some people might call chutzpah,” he added, using a Yiddish word meaning audacity.

Prosecution is careful to repeatedly call Trump 'the defendent'

There’s subtle but notable rhetorical move happening in this closing by the prosecution.

Steinglass is repeatedly referring to Trump as “the defendant” instead of “Mr. Trump” or “the former president.” This contrasts greatly from the defense's language, as Trump's lawyers almost always refer to him as "the president."

It will be important to watch for Steinglass to argue at some point that no one is above the law, even the former president of the United States -- something we’ve seen other state and federal prosecutors say about Trump over the last year.

Steinglass focuses on inconsistencies in defense argument

Steinglass zeroed in on an example of what the prosecution considers an inconsistency in the defense team's case. He told the jury that if the $420,000 payment for Cohen was for legal services, as the defense argued, Cohen could not have stolen $60,000 from the Trump Organization, as the defense also argued. It's either one or the other, the prosecutor argues — not both.

Steinglass: 'I'm not asking you to feel bad for Michael Cohen'

Steinglass is trying to reason with the jury, telling the jurors that they don't need to feel bad for Cohen, but they should understand where Cohen is coming from.

“I am not asking you to feel bad for Michael Cohen. He made his bed," Steinglass said.

“But you can hardly blame him that he’s making money for the one thing he has left," he added, referencing Cohen's knowledge of the inner workings of the Trump organization.

Steinglass admits that Daniels’ testimony was “messy” — but 'Stormy Daniels is the motive'

Steinglass is laying out how “the defense has gone to great lengths to shame Stormy Daniels, saying that she changed her story” but adds that “her false denials have been thoroughly discussed and explained.”

“She lived 2017 in pure silence, Michael Cohen came out and said sex never happened” and Daniels “felt compelled to set the record straight,” he said.

Steinglass said that “parts of her testimony” were “cringeworthy” and “uncomfortable.”

But details like “what the suite” at Harrah’s “looked like” and how the toiletry bag appeared “ring true.”

“They’re the kind of details you’d expect someone to remember,” Steinglass explained, adding that, “fortunately, she was not asked or did she volunteer specific details of the sexual act itself.”

“It certainly is true you don’t have to prove that sex took place — that is not an element of the crime, the defendant knew what happened and reinforces the incentive to buy her silence,” explained Steinglass.

“Her story is messy,” he said. “But that’s kind of the point. That’s the display the defendant didn’t want the American voter to see.”

“If her testimony were so irrelevant, why did they work so hard to discredit her?” he added. “In the simplest terms, Stormy Daniels is the motive.”

Steinglass undercuts defense argument that Trump was totally in the dark on Daniels payment

Steinglass displayed quotes from one of the state's exhibits: a phone call in which Cohen — well before he started cooperating with prosecutors — tells Davidson that Trump hates the fact that his team settled with Daniels.

The quotes undercut the defense team's insistence that Trump knew nothing about the hush money payments to Daniels.

Steinglass to jury: You don't need to believe Cohen to find there was a conspiracy

Steinglass defended the state's witnesses against the Trump team's accusations of lying, but he added that the jury does not necessarily need to believe every word of Cohen's testimony to find that there was a conspiracy to unlawfully influence the 2016 election.

"You don’t need Michael Cohen to prove that one bit," Steinglass said, referring to the state's accusation of a conspiracy.

He added that Hope Hicks, Rhona Graff, Madeleine Westerhout, Jeffrey McConney and Deborah Tarasoff were all witnesses who like Trump but confirmed Cohen's testimony.

Steinglass: 'You don't get to commit election fraud or falsify your business records'

Steinglass is appealing to the jury by explaining to them that it doesn't really matter why Trump broke the law, as long as they feel he did break the law. The argument appears to be a response to the claim by Blanche, during his own closing arguments, that Stormy Daniels had attempted to extort Trump.

"In the end it doesn’t really matter, because you don't get to commit election fraud or falsify your business records because you think you’ve been victimized," he said.

"In other words, extortion is not a defense for falsifying business records," he added.

"You've got to use your common sense, here," Steinglass continued. "Consider the utterly damning testimony of David Pecker."

Steinglass rebuts defense arguments about phone records

"The defense seems to be questioning our integrity,” Steinglass told the jury near the top of his summation.

But, he argued, it was the defense that didn't properly depict phone records.

The call summaries were made to help guide you, the prosecutor explained to the jury. The phone records are all in evidence and you can look through them at your leisure, he added.

It’s also an interesting accusation, Steinglass points out, given that the defense’s summary of calls between Cohen and Costello double-counts their calls. He also reminds them that not every phone call is accounted for in their phone records. Cohen had 11 phone numbers for Trump; they had records corresponding to two of them.

Prosecution kicks off closing arguments

The prosecution is now kicking off its closing arguments. Joshua Steinglass will give them.

Merchan told jury to disregard Blanche's 'prison' comment

Merchan, who chastised Blanche for imploring jurors not to send Trump to prison, told the jury that the lawyer's comment was "improper, and you must disregard it."

"If there is a verdict of guilty," the judge added, "it will be up to me to impose a sentence."

He went on to explain that a "prison sentence is not required in the event of a guilty verdict."

We are back

Merchan is at the bench. Trump is seated at the defense table.

Trump's family shows support outside the courthouse

Trump's sons Donald Trump Jr. and Eric Trump along with Eric's wife, Lara Trump, the co-chair of the Republican National Committee, slammed the proceedings in remarks to reporters outside the courthouse during the lunch break.

"Michael Cohen is the embodiment of reasonable doubt," Donald Jr. said. "This entire case hinges on someone who has quite literally lied to every single person and body he's ever been in front of in his life before."

Both he and Eric Trump echoed their father's often repeated characterization of the trial, calling it a "political witch hunt" and a "sham."

Eric went on to say that the district attorney's office is ignoring crimes across the city and using the trial to attack Trump.

"They're sitting there, they're laughing, they're giggling," Eric said. "This was their moment. This is how they embarrass Donald Trump."

Laura Trump added that the trial has been "banana republic-type stuff."

"This is a case about politics, pure and simple," she said.  

After walking away from the news conference, Donald Trump Jr. added that Democrats "talk about democracy but are laughing about it like it's a soundbite," and claimed they are “trying to scare anyone who has any kind of belief that doesn’t go 100% with what they believe.”

Merchan says he will give curative instructions after Blanche's 'prison' comment

Merchan appeared to chastise Blanche after the defense lawyer implored jurors not to send Trump to prison — an unlikely outcome in this case

"I think that statement was outrageous, Mr. Blanche," Merchan said after jurors were excused for their daily lunch break, later adding: "It's simply not allowed. Period. It’s hard for me to imagine how that was accidental in any way."

Merchan told the court that he plans to give jurors a curative instruction — in other words, general direction that is aimed at clearing up an erroneous statement.

Prosecutor slams Blanche's 'prison' comment

Joshua Steinglass, the prosecutor who is expected to deliver the state's closing arguments, blasted Blanche's comment to the jury about prison time as a "blatant and wholly inappropriate move" by the defense.

Steinglass asked Merchan to provide a curative instruction, a direction given by a judge to correct an erroneous statement.

Trump lawyer tells jurors that 'this isn't a referendum on your views of' Trump

Summer Concepcion

Toward the end of his closing arguments, Trump lawyer Todd Blanche told jurors that the verdict “isn’t a referendum on your views of” Trump, or “a referendum on the ballot box,” stressing the importance of basing their decision on evidence that emerged throughout the trial.

“If you focus just on the evidence you heard in this courtroom, this is a very very quick and easy not guilty verdict. Thank you,” he said.

‘You are gangsters!’: Robert De Niro clashes with Trump supporters in New York

Katherine Doyle

Amanda Terkel Politics Managing Editor

President Joe Biden’s campaign held a news conference outside the Manhattan courtroom where  Donald Trump is on trial  in his hush money case, with actor Robert De Niro and  two officers who defended the Capitol  from the Jan. 6 mob warning about the dangers of re-electing the former president.

“The Twin Towers fell just over here, just over there. This part of the city was like a ghost town, but we vowed we would not allow terrorists to change our way of life. ... I love this city. I don’t want to destroy it. Donald Trump wants to destroy not only the city, but the country, and eventually he can destroy the world,” De Niro said.

Afterward, on the way back to his car, De Niro mixed it up with some pro-Trump protesters, who yelled that he’s a “wannabe,” “paid sell-out” to the Democratic National Committee, “nobody” and a “little punk” whose “movies suck.”

“You’re not going to intimidate,” De Niro replied. “That’s what Trump does. ... We are going to fight back. We’re trying to be gentlemen in this world, the Democrats. You are gangsters. You are gangsters!”

Read the full story here.

Blanche finishes summation

Blanche finished his summation at 12:49 p.m. ET, about three hours after he began the closing arguments.

Blanche refers to jail time

Blanche told the jurors: "You cannot send someone to prison based on the words of Michael Cohen."

It's worth noting that it's unlikely the former president will be sentenced to prison in this case.

Blanche says Michael Cohen is the 'GLOAT'

Blanche says that Michael Cohen is the "greatest liar of all time."

“Michael Cohen is the GLOAT. He’s literally the greatest liar of all time," Blanche said, a play on the sports term GOAT "greatest of all time. “He has lied to every single branch of Congress.”

He added, “He has lied to the Department of Justice.”

Blanche outlines 10 reasons why he believes jury should have reasonable doubt

Blanche presented jurors with a list:

• The invoices. Blanche argues Cohen created the invoices, Trump had no intent to defraud, and prosecutors did not present evidence that Trump knew about them.
• Valentine's Day 2017 vouchers. Blanche argues there is no proof Trump ever saw the vouchers.
• No evidence of intent to defraud.
• No evidence to commit or conceal a crime. "There is no falsification of business records, period," Blanche argued.
• No evidence Trump was involved in illegal agreement to influence election.
• AMI would have run Sajudin's story. Dino Sajudin is the former Trump Tower doorman who tried to sell a story about Trump fathering a child out of wedlock.
• McDougal did not want her story published .
• Daniels' story was already public .
• Alleged manipulation of evidence .
• Cohen is the "embodiment of reasonable doubt." "He lied to you repeatedly," Blanche said. "He is biased and motivated to tell you a story that is not true."

Blanche insists there was no felony because even if there was a conspiracy, it wasn't through 'unlawful means'

Blanche is insisting that there can be no felony falsification of business records because even if there was a conspiracy to influence the election, it was not carried out through any “unlawful means.”

To support his “no unlawful means” argument, Blanche said there is no proof Trump ever knew, for example, about certain paperwork Michael Cohen submitted to his bank or paperwork prepared to transfer Karen McDougal’s life rights from AMI to Trump.

Trump’s knowledge, however, is not required. All that matters legally is that a member of the conspiracy undertook those “unlawful means.”

Trump lawyer plays audio of Cohen screaming on his podcast

After playing audio of Cohen excitedly talking about the prospect of Trump being convicted, Trump lawyer Todd Blanche then played two excerpts of Cohen screaming on his podcast in a tone virtually unrecognizable to anyone who has encountered him only here.

This was more effective than most moments today.

Blanche says Michael Cohen is the 'MVP of liars'

Blanche said that Michael Cohen has lied to his family, including his wife and kids, his banker, the Federal Election Commission, reporters, Congress, prosecutors, business associates and bosses.

"He's literally like the MVP of liars," Blanche said.

Blanche raises his voice in accusing Cohen of lying

Blanche began shouting as he again accused Cohen of lying under oath. He reminded jurors that Cohen testified that he called Trump on Oct. 24, 2016, to provide an update on the Daniels situation, "It was a lie!" he said, pointing out that the call was actually to Trump's bodyguard, Keith Schiller.

"That was a lie and he got caught red-handed,” Blanche added.

Blanche accuses the prosecution of using Stormy Daniels to inflame jury

Over objections by prosecutor Joshua Steinglass, Blanche is accusing the prosecution of calling Stormy Daniels as a witness at trial, but not calling her as a grand jury witness.

Blanche is arguing it was intended to inflame the jury’s emotions and to embarrass the former president.

The jury didn't appear to react to that statement.

Trump lawyer portrays Trump as the victim of the infamous 'Access Hollywood' tape

Jonathan Allen

Blanche may be the first person to portray Trump as the victim of the “Access Hollywood” tape .

Though Blanche says it was not “so catastrophic” as to motivate Trump to break the law — more precisely, that there’s “no evidence” that it was — he says this of the release of the video Oct. 7, 2016: “This was an extremely personal event for President Trump. Nobody wants their family to be subjected to that sort of thing.”

(The video had Trump on a hot mic discussing getting away with assaulting women because he was famous.)

Blanche accuses Daniels of 'extortion,' and the prosecution stays mum (for now)

Blanche just said of Daniels’ nondisclosure agreement: “This started out as an extortion and it ended up very well for Ms. Daniels, there’s no doubt about that.”

The prosecution has not objected to Blanche’s repeated use of the word “extortion,” which suggests a crime was committed. That could be a strategic choice, because what they say in refuting that characterization during their own summation could be more memorable and powerful than a sustained objection.

Blanche claims that threats against Stormy Daniels never happened

Blanche said that Stormy Daniels decided to go public with her story supposedly because she was trying to protect herself from threats in a parking lot that she received five years earlier.

Blanche said, however, that there are recordings that show that's not true. He said Michael Avenatti, Gina Rodriguez and Daniels were lying about these threats.

“They never happened," Blanche said. “The recording makes clear that Ms. Daniels lied to you.”

Blanche has resumed his summation

The morning break is over and Trump's defense team is continuing with its closing arguments.

Blanche said he expects about 30 to 40 more minutes.

Trial takes a break

The trial took a quick break starting at 11:35 a.m.

Blanche questions why no one in Trump campaign addressed Stormy Daniels issue in April 2016

Blanche questions why no one in the campaign did anything about Stormy Daniels in April 2016 when her manager reached out about it.

But Blanche's point ignores the impact that the leak of the "Access Hollywood" tape in October 2016 had on the campaign. Trump's campaign was beleaguered by accusations of sexism as a result of the tape, so Daniels' claim may have had more of an impact.

Fight appears to break out between pro-Trump supporters outside the courthouse

Elizabeth Maline

A fight appears to have broken out between pro-Trump supporters in Columbus Park across the street from the courthouse.

New York City Police Department officers were seen hopping over the fence into the park to respond to the clash.

Blanche tries to impress upon jury that Cohen's recording of Trump call is unreliable

Blanche wants the jury to believe that Michael Cohen's recording of the call with Trump is unreliable because it cuts off early.

But more than that, Blanche is trying to tell the jury that the transcript of what they have is unreliable because while the recording discussed AMI and Pecker, there is doubt that they are talking about Karen McDougal, whose name is never mentioned, or any payment of $150,000, which cannot be heard on the tape.

Blanche says they were “talking past each other,” and that Cohen’s invocation of “financing” shocked Trump, who had no idea what was going on, and that Cohen’s interpretation of “cash” to mean actual bills is a fiction designed to make the conversation sound more sinister.

Trump team responds outside courthouse immediately following Biden campaign

Moments after the Biden campaign finished its remarks outside the courthouse, Trump campaign members went to the microphone to speak.

Jason Miller, a senior adviser to the campaign, called the Biden campaign's decision to have Robert De Niro — whom he called a "washed-up actor" — speak today as a way to "try to change the subject" from Biden's "falling" poll numbers.

Karoline Leavitt, a Trump campaign spokesperson, called the Biden team's conference "a full-blown confession that this trial is a witch hunt."

"This is a disgrace. President Trump has been locked up in that courtroom for six weeks," Leavitt said. "But guess what, the American people see through this witch hunt, this scam, and that's why President Trump continues to rise in the polls."

Leavitt added that Biden is "weak" and "pathetic" and is using "elitist, out-of-touch Hollywood actors like Robert De Niro who have no idea the real problems that people in this city and across this country are facing." 

Blanche accuses Cohen of lying about Pecker lunch. Pecker didn't dispute it, though.

Blanche is continuing his effort to convince jurors that Cohen is a shameless liar. "Remember when Cohen told you he had lunch with Pecker?" Blanche told the jury. "Pecker said he was really frustrated that he was not getting paid for the McDougal story. Ladies and gentlemen, that lunch did not happen. Cohen made it up."

However, Blanche and Trump's other lawyers never entered any evidence backing up that claim — and Pecker during his testimony did not dispute that the lunch happened.

Blanche appears to want to have it both ways regarding David Pecker

Blanche appears to want it both ways regarding Pecker.

On one hand, he has characterized David Pecker as a “truth teller” and someone who, because of Pecker's immunity deal with the Manhattan DA, had no incentive to lie.

But Blanche also tells the jury that Pecker’s explanation that if the story from Trump Tower doorman Dino Sajuddin had been true, he would have published it — but only after the election — is not entirely credible because such a major story would have been published immediately.

Blanche argues the effort to silence Karen McDougal wasn't a 'catch and kill'

Blanche argued that the effort to silence Karen McDougal "is not a catch and kill either" because she didn't want her story published.

Blanche said McDougal wanted to kick-start her career, be on the cover of magazines and write articles. He said it wasn't McDougal's intention to publish her story.

"She didn't want her story published," he said.

Former Capitol police officers campaign for Biden outside courthouse

Harry Dunn, a former Capitol Police officer, and Michael Fanone, a former D.C. Metropolitan Police officer, who defended the Capitol during the Jan. 6 attack, spoke in support of the Biden campaign outside of the courthouse today.

Fanone, who suffered a brain injury and a heart attack in the assault, recounted the attack adding that "if Jan. 6 didn't happen, we wouldn't be here right now, I'd still be at work."

Dunn went on to say that Trump is "the greatest threat to our democracy and to the safety of communities across the country today."

"Trump does whatever will get him votes and helps Donald Trump," he said.

Blanche mixes up details in 'catch and kill' cases

Reporting from Manhattan criminal court

Blanche has been walking through each of the stories that were caught and killed. But he is mixing up details. He mentioned, for example, that Karen McDougal’s business manager was Gina Rodriguez. But Rodriguez worked for Stormy Daniels, not McDougal.

Analysis: Blanche's assertions about the Enquirer don't really hold up to scrutiny

Blanche is arguing that the Enquirer’s reach was not wide enough to influence the election. But especially in today’s social media-fueled age, the idea that a story’s reach is limited to the publication’s own distribution is simply untrue. More significantly, however, the Enquirer’s influence here was in preventing certain stories from ever seeing the light of day.

Blanche pushes back on idea that the Enquirer could influence an election

Blanche, attempting to undercut one of the key planks of the prosecution's narrative, told the jury that it's absurd to believe that positive stories in the National Enquirer could affect the outcome of an American election.

"The idea that even sophisticated people like President Trump and David Pecker believed that positive stories in the National Enquirer could influence the 2016 election is preposterous," Blanche said, referring to the former publisher of the tabloid magazine. He went on to say that many of the articles published in the Enquirer were recycled from other outlets.

Pecker testified earlier in the trial that he purchased potentially damaging stories about Trump and then made sure they never saw the light of day — a practice known as "catch and kill." He also testified that his editorial team attempted to run more glowing stories about Trump in the lead-up to the 2016 election.

Robert De Niro condemns Trump in fiery remarks outside courthouse: 'He could destroy the world'

Actor Robert De Niro spoke to the press as a surrogate for the Biden campaign outside the courthouse, railing against Trump.

"I love this city. I don’t want to destroy it," De Niro, a native New Yorker, said.

"Donald Trump wants to destroy not only the city, but the country and eventually he could destroy the world," he continued.

De Niro, who has also appeared in ads for the Biden campaign, condemned Trump for the violence that occurred Jan. 6 at the Capitol, arguing that if Trump wins in November, "he will never leave."

At the end of De Niro's remarks, a Trump supporter in the crowd called the two former police officers standing with De Niro — both present at the Capitol on Jan. 6, 2021 — "traitors."

The actor engaged in a back-and-forth with the man in the crowd, defending the officers, Harry Dunn and Michael Fanone.

"They stood there. They didn’t have to," De Niro said. "They stood there and fought for us. They fought for you, buddy. You’re able to stand right here."

"They are the true heroes. I’m honored to be with these two heroes today," De Niro continued.

Blanche says every campaign is a 'conspiracy to promote a candidate'

Blanche said that the prosecution wants the jury to believe that the entire scheme was to promote Trump's successful candidacy in 2016.

“Even if you find that’s true, that’s still not enough. It doesn’t matter — as I said to you in the opening statement — it doesn’t matter if there was a conspiracy to win the election," Blanche said. “Every campaign is a conspiracy to promote a candidate.”

Blanche hammers on the question of Trump's intent to defraud

Blanche asked the jury: "Where is the intent to defraud on the part of President Trump?" He then showed a slide labeled "No Intent to Defraud."

The exact language of the charges against Trump in this case accuse the former president of breaking various laws with the "intent to defraud and intent to commit another crime and aid and conceal the commission thereof."

Biden campaign arrives with Robert De Niro outside courthouse

Biden campaign members have arrived outside the courthouse with actor Robert De Niro and Capitol Police Officer Harry Dunn, who was attacked in the Jan. 6 attack on the U.S. Capitol.

Blanche again suggests Cohen was bitter

Blanche asked the jurors whether they "believe for a second that, after getting stiffed on his bonus in 2016, when he thought he worked so hard," Cohen would then "want to work for free" for Trump.

"Was that the man who testified," Blanche asked rhetorically, "or was that a lie?"

Cohen did indeed testify that he was upset after he did not receive a holiday season bonus after the 2016 presidential election, but he repeatedly rejected the defense team's suggestions that bitterness and vindictiveness drove him to cooperate with prosecutors.

Blanche then argued it was "absurd" that Trump would agree to pay Cohen $420,000 even though the former president owed him only $130,000.

Blanche suggests Trump, as president, was too busy to be part of 'scheme'

Blanche repeatedly refers to Trump being in the White House when the repayments were made. He was very busy, Blanche said. That he was somehow in on a “scheme” to conceal a repayment is “absurd,” he added

His argument also reminds the jury this is no normal defendant: It’s the former president of the United States.

It’s an interesting line to to walk: Trump is so careful about his finances that he would never overpay, but he was also so busy in the White House that he was sometimes careless and wouldn’t know what he was paying for.

Blanche says prosecutors asked jury to believe Michael Cohen

Trump lawyer Todd Blanche said, “What the people have done, what the government did for the last five weeks, at the end of the day, is ask you to believe the man who testified two weeks ago, Michael Cohen.”

Blanche rejects assertion that Trump had full knowledge

Blanche told jurors it was "a stretch" that Trump always "had full knowledge of what was happening" inside the Trump Organization and his other business enterprises.

"That is reasonable doubt, ladies and gentlemen," he said.

Trump lawyer says there's nothing 'sinister or criminal' about the word 'retainer'

Blanche commented on the fact that retainer was listed as the reason for the reimbursement checks from Trump to Cohen.

"There's nothing sinister or criminal about that word," Blanche said.

Blanche said it wasn't put there by Trump or Allen Weisselberg but by Trump Organization accounting employee Deb Tarasoff, who testified earlier in the trial.

What was missing from the chart put up on the screen

When Blanche put up a visual aid for the jurors showing invoices, vouchers and checks, the most glaringly noticeable line on any of the documents was the very familiar, thick-lettered signature of Donald Trump.

Blanche calls attention to the fact that Don Jr. and Eric Trump weren't called as witnesses

As Blanche is calling attention to the fact that Don Jr. and Eric Trump were not called as witnesses, they are sitting in the front row of the courtroom behind their dad.

“The burden is always on the government, they make decisions about who to call," Blanche said, adding, “They did not call Don or Eric.”

The jury did not look over at the Trump children.

Blanche tries to steer jury away from old Trump books

Blanche tells the jury to be wary if the prosecution starts reading from an old Trump book to help prove how involved the former president was in his company’s accounting system.

Those books were co-written by ghostwriters, Blanche says, implying the ghostwriters did the due diligence of figuring out the system in lieu of Trump’s personal knowledge.

Blanche tries to address toughest evidence before prosecution gets to it

Blanche is working hard to try to pre-empt certain arguments the jury is likely to hear from the prosecution after he sits down. Because he goes first and the prosecution will have the last word -- per New York law -- he can’t afford not to address the toughest evidence for his client. 

Blanche pushes back on hush money argument

Blanche appeared to suggest that Cohen received retainer payments not because of the hush money arrangement but because he was Trump's personal attorney.

"There’s a reason why in life usually the simplest answer is the right one, and that’s certainly the case here. That the story Mr. Cohen told you on that witness stand is not true.” 

Cohen was paid $35,000 a month by Trump to be his attorney, Blanche said.

Blanche planting the seeds of reasonable doubt

Blanche is doing two things simultaneously to plant seeds of reasonable doubt early in this closing argument — establish that the internal records at the heart of this case weren't falsified and that Michael Cohen is a liar.

Blanche argues Michael Cohen was working as Trump's personal lawyer

Trump attorney Todd Blanche argued that Michael Cohen was serving as Trump's personal attorney, which he said was not in dispute.

“He talked to every reporter that he could, pushing the fact that he was going to be the personal attorney to President Trump," Blanche said. “This was not a secret. Michael Cohen was President Trump’s personal attorney. Period.”

Biden's campaign set to hold press conference outside the courthouse

President Joe Biden's campaign is scheduled to hold a news conference outside the court this morning at 10:15 ET.

The news conference is set to include the campaign team and "special guests," although the news release didn't say who they would be.

Trump lawyer argues invoices were false and there was no intent to defraud

Trump lawyer Todd Blanche argued that the invoices weren't false and there was no intent to defraud — and that if the jurors are so convinced, they don't have to go further.

As a matter of law, Blanche is correct, but it is also the case that the requisite intent to defraud is defined as including the intent to commit or conceal another crime.

Put another way, if the jurors believe the documents are false, they do have to confront whether Trump intended to conceal the underlying alleged conspiracy.

Jury sees chart that won't be put into evidence

Blanche displayed a chart on the courtroom screen showing what it presented as various financial records, including Cohen's invoices (which were then turned into vouchers, and then turned into checks).

The chart will not be put into evidence, so the jury can't refer back to it — and the general public may never see it publicly produced.

Trump lawyer accuses Michael Cohen of lying for likely the first of many times today

It's 9:48 a.m. and Trump lawyer Todd Blanche just accused Michael Cohen of lying — the first of many times we're likely to hear that claim today.

Blanche: 'This is a paper case'

Blanche continues his sentiment that the testimony that the jury has heard thus far is not enough to convict Trump. Instead, Blanche argues the true evidence for this case lies in documents.

"This case is about documents, it’s a paper case," Blanche said.

Blanche went on to argue that the case is not about Stormy Daniels, but instead about the payments Trump made to Michael Cohen.

“Were those bookings done with an intent to defraud? That’s why you’re here. And the answer to that — to those questions is absolutely positively not," Blanche said.

"The bookings were accurate, and there was absolutely no intent to defraud. And beyond that, there was no conspiracy," he continued.

Blanche tries to undercut Cohen and Daniels testimony

Blanche tells the jury members that “they should want and expect more than the testimony of Michael Cohen. ... You should want and expect more than the word of a woman who claims something happened in 2006.”

He continues by saying they should want and expect more than the testimony of Keith Davidson, who was trying to extort Trump. Notably, the district attorney's office does not object to the characterization of what happened as attempted or actual extortion.

Trump lawyer reiterates to jury that his client is innocent

Trump attorney Todd Blanche told the jury that they, as a group of citizens, decide the facts and decide whether Trump is guilty or not guilty. He said he wanted to repeat what he told them five weeks ago.

“President Trump is innocent," Blanche said. "He did not commit any crimes, and the district attorney has not met their burden of proof. Period.”

Blanche starts his closing arguments

Trump's lawyer Todd Blanche began giving his closing arguments at about 9:40 a.m. ET. He said that he expects he'll need 2½ hours to deliver the end of the defense's case.

He briefly put up a PowerPoint presentation and then took it down.

Merchan to jurors: You are the judges of the facts

Merchan is giving jurors an overview of what they're going to hear today from lawyers on both sides of the case. He explained that the summations "provide each lawyer the opportunity to review the evidence and give you the conclusions that can be drawn."

"You are the finders of fact, and it is for you and for you alone to determine the facts from the evidence," the judge told the jury.

He reminded the jury that the "lawyers are not witnesses," adding that nothing they say in their summations constitutes "evidence."

"You and you alone are the judges of the facts in this case," Merchan said.

Judge tells prosecution and defense: Don't go into the law

Before the jury entered, Judge Merchan told both the prosecution and defense teams that they shouldn't explain the law to the jurors during summation.

"Please do not go into the law. Stay away from the law," he said. "That'll be my job. I'll take care of it."

District attorney staff members are watching from the overflow room

As proceedings begin today, more than eight secondary members of the prosecution team have come into the overflow room to watch the trial.

The members present appear to be senior leadership from the district attorney’s office, including First Assistant District Attorney Meg Reiss and former Executive Assistant District Attorney Peter Pope, who led the investigation of this case leading to the grand jury’s indictment.

The staff members are seated in the jury box in the overflow room -- an area we have not seen used before for seating.

How long will summations last?

Todd Blanche, Trump's lawyer, estimates he'll need around 2½ hours to deliver his closing argument. He goes first.

Joshua Steinglass, one of the prosecutors, says he'll need "somewhere in the vicinity of 4 to 4½ hours."

Trump says 'this is a dark day in America' before heading into courtroom for closing arguments

Shortly before heading into the courtroom for closing arguments, Trump repeated his claims that he was forced to attend courtroom proceedings in the hush money trial because of President Joe Biden, without providing evidence.

The presumptive Republican presidential nominee griped that the trial is “election hunting, election interfering” because it is an effort to go after Biden’s political opponent.

Trump again accused Judge Merchan of being “highly conflicted” and “corrupt” and read aloud quotes from legal analysts who support his assertions that the former president did not commit wrongdoing in the case.

Trump also complained about Merchan’s gag order that bars him from making disparaging comments against his family members and others involved in the case, saying that it’s an “unconstitutional thing” to impose on a presidential candidate.

“This is not a trial that should happen. It’s a very sad day. This is a dark day in America,” he said. “We have a rigged court case that should have never been brought, and it should have been brought in another jurisdiction.”

Jury instructions set in stone

Judge Merchan says that he provided the jury instructions to the defense and prosecution on Thursday afternoon and that neither side has commented on them. They are now final.

Merchan is on the stand and they're ready on go

The judge has taken his seat and proceedings are about to get underway.

The prosecution and defense in Trump’s criminal hush money trial will begin making their closing arguments to the jury today as the first criminal trial of a former president enters its final phase. NBC’s Laura Jarrett reports and Hallie Jackson provides analysis for "TODAY."

‘Phony’ checks and hush money payments: Breaking down Trump’s 34 charges in his New York criminal trial

JoElla Carman

Trump faces 34 felony counts in the New York hush money trial that is expected to potentially wrap up as early as this week.

Here's what to know about the charges.

Biden campaign preps for a Trump trial verdict: From the Politics Desk

Monica Alba

Natasha Korecki

Mike Memoli

President Joe Biden has largely steered clear of Trump’s legal woes. But with a verdict in the  hush money trial  coming as soon as this week, Biden’s campaign is exploring a shift to a new, more aggressive posture, according to two people familiar with the strategy. 

Regardless of the outcome, top Biden campaign officials plan to stress to voters that Trump will be on the ballot in the fall and that no potential court proceeding will change that fact.

A person familiar with the discussions summed it up this way: “Donald Trump’s legal troubles are not going to keep him out of the White House. Only one thing will do that: voting this November for Joe Biden.” 

Trump has departed for the courthouse

Brittany Kubicko

The former president has left Trump Tower for the courthouse downtown.

Rudy Giuliani's son argues with anti-Israel protester outside court

Former New York gubernatorial candidate Andrew Giuliani started a heated argument with a protester who was shouting antisemitic tropes outside the courthouse this morning.

Giuliani, a former Trump White House official and the son of former New York Mayor Rudy Giuliani, followed the demonstrator who was wearing a ski mask around a protest zone and yelled at the man about the Oct. 7 terrorist attack on Israel.

The protester carried a sign with numbers representing Gazans who have been killed in the ensuing conflict and voiced canards about Jews controlling the U.S. government and the entertainment industry.

Trump's guests in court today

Jake Traylor

Matt Korade

Several of Trump's children will be in court for closing arguments, including Donald Trump Jr., Eric Trump and his wife, Lara Trump, who is the co-chair of the Republican National Committee, as well as Tiffany Trump, the former president's only daughter with his ex-wife Marla Maples, and her husband, Michael Boulos.

Also in attendance will be Trump's longtime friend Steve Witkoff, a real-estate investor who testified as a defense expert in Trump’s Manhattan civil fraud trial , Will Scharf, a lawyer for Trump who is running for attorney general in Missouri against Republican incumbent Andrew Bailey, and Deroy Murdock, a contributing editor for National Review Online.

Trump lawyer says she has 'zero confidence' Judge Merchan will issue jury instructions 'in an appropriate manner'

Trump legal spokesperson Alina Habba on Sunday expressed concerns about jury instructions in the hush money trial against the former president and the jurors not being sequestered over the holiday weekend.

“Generally, as an attorney, as an American who understands the law and how to apply to laws to facts, there are no facts that support this alleged crime,” Habba said during an interview on Fox News “Sunday Morning Futures.” “We’re not even sure what the crime is. So it’s a books and records issue.”

Habba echoed Trump’s claims that Merchan is “severely conflicted” without evidence, noting the judge’s gag order that bars Trump from issuing disparaging comments on his family members and others involved in the case. Trump has repeatedly accused Merchan of being “conflicted,” often citing his daughter’s work at a digital fundraising and advertising firm that often collaborates with Democratic politicians.

“This judge is the judge that determines the jury instructions. The jury instructions are the road map for non-attorneys and jurors to follow the law,” she said. “It’s going to be critical, and frankly, at this point, I have zero confidence in the fact that this person, who should not be sitting on the bench right now, will do the right thing and give jury instructions that are in an appropriate manner without any persuasion towards the prosecution.”

Habba then raised concerns about jurors not being sequestered over the holiday weekend, arguing that they could be swayed by family and friends who have certain opinions.

“They should have been sequestered because, in my opinion, these jurors are handling something that is completely unprecedented and unwarranted in America, and for them to be able to be out and about on a holiday weekend with friends and families who have opinions, who are watching the news TVs on the background at the pool party — I have serious concerns,” she said.

Trump blasts Merchan and District Attorney Alvin Bragg in Truth Social posts over the weekend

Alexandra Marquez is based in Washington, D.C.

Isabelle Schmeler

In a series of social media posts over the holiday weekend, Trump attacked Manhattan District Attorney Alvin Bragg, who brought the charges in this case against him, attacked Judge Juan Merchan and said the case was about a "legal expense" and a "bookkeeping error."

"I have a great case, but with a rigged and conflicted judge," Trump said in one post, before adding in another one, "The City of New York’s D.A., Alvin Bragg, is trying to prosecute a Federal case, which cannot be done, and where there is NO CRIME."

One post blasted the case for blowing a "legal expense" out of proportion, saying, "Let’s put the President in jail for 150 years because a LEGAL EXPENSE to a lawyer was called, by a bookkeeper."

Another post yesterday accused Merchan, without evidence, of being a "corrupt and conflicted" judge and claimed that Bragg is backed by liberal billionaire megadonor George Soros, who has been a target of antisemitic conspiracy theories .

Trump’s lawyers are preparing for the final stretch of the former president’s hush money trial in New York. NBC News’ Gabe Gutierrez reports on Trump’s busy weekend ahead of closing arguments in court.

Closing arguments set to begin in Trump’s criminal trial

Dareh Gregorian

Closing arguments will begin today in the People of the State of New York v. Donald J. Trump , as the first criminal trial of a former president enters its final phase.

After the prosecution and the defense deliver their concluding arguments, the judge will give instructions to the jury. Then, the 12 ordinary New Yorkers who sit on the jury will begin deliberations on whether or not the former president is guilty of the charges against him.

After 20 days in a courtroom, here's what you missed in the Trump hush money trial

Ahead of this week's closing arguments, catch up on what you missed over the last few weeks of the first criminal trial of a former president.

In sometimes explosive testimony, former Trump "fixer" Michael Cohen said that he did call Trump a "Cheeto-dusted" villain but admitted to past lies and theft upon questioning by Trump's attorneys.

Despite promising to testify, Trump did not ultimately take the stand and pushed back on media reports that he fell asleep multiple times during the trial. On his Truth Social account, the former president claimed he was simply resting his “beautiful blue eyes” while listening “intensely” to the proceedings.