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Fundamental Analysis

Hypothesis to Be Tested: Definition and 4 Steps for Testing with Example

What Is Hypothesis Testing?

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 from 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, with the goal of providing evidence on the plausibility of the null hypothesis.

Statistical analysts test a hypothesis by measuring and examining 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 (e.g., the population mean return is not equal to zero). 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.

4 Steps of Hypothesis Testing

All hypotheses are tested using a four-step process:

The first step is for the analyst to state the hypotheses.
The second step is to formulate an analysis plan, which outlines how the data will be evaluated.
The third step is to carry out the plan and analyze the sample data.
The final step is to analyze the results and either reject the null hypothesis, or state that the null hypothesis is plausible, given the data.

Real-World Example of Hypothesis Testing

If, for example, a person 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 would be represented as Ho: P = 0.5. The alternative hypothesis would be denoted as "Ha" and be 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 then 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, on the other hand, 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."

Some staticians 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 is Hypothesis Testing?

Hypothesis testing refers to a process used by analysts to assess the plausibility of a hypothesis by using sample data. In hypothesis testing, statisticians formulate two hypotheses: the null hypothesis and the alternative hypothesis. A null hypothesis determines there is no difference between two groups or conditions, while the alternative hypothesis determines that there is a difference. Researchers evaluate the statistical significance of the test based on the probability that the null hypothesis is true.

What are the Four Key Steps Involved in Hypothesis Testing?

Hypothesis testing begins with an analyst stating two hypotheses, with only one that can be right. The analyst then formulates an analysis plan, which outlines how the data will be evaluated. Next, they move to the testing phase and analyze the sample data. Finally, the analyst analyzes the results and either rejects the null hypothesis or states that the null hypothesis is plausible, given the data.

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.

The Bottom Line

Hypothesis testing refers to a statistical process that helps researchers and/or analysts 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. There are different types of hypothesis testing, each with their own set of rules and procedures. However, 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. Hypothesis testing plays a vital part of the scientific process, helping to test assumptions and make better data-based decisions.

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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Statistics Made Easy

Introduction to Hypothesis Testing

A statistical hypothesis is an assumption about a population parameter .

For example, we may assume that the mean height of a male in the U.S. is 70 inches.

The assumption about the height is the statistical hypothesis and the true mean height of a male in the U.S. is the population parameter .

A hypothesis test is a formal statistical test we use to reject or fail to reject a statistical hypothesis.

The Two Types of Statistical Hypotheses

To test whether a statistical hypothesis about a population parameter is true, we obtain a random sample from the population and perform a hypothesis test on the sample data.

There are two types of statistical hypotheses:

The null hypothesis , denoted as H 0 , is the hypothesis that the sample data occurs purely from chance.

The alternative hypothesis , denoted as H 1 or H a , is the hypothesis that the sample data is influenced by some non-random cause.

Hypothesis Tests

A hypothesis test consists of five steps:

1. State the hypotheses.

State the null and alternative hypotheses. These two hypotheses need to be mutually exclusive, so if one is true then the other must be false.

2. Determine a significance level to use for the hypothesis.

Decide on a significance level. Common choices are .01, .05, and .1.

3. Find the test statistic.

Find the test statistic and the corresponding p-value. Often we are analyzing a population mean or proportion and the general formula to find the test statistic is: (sample statistic – population parameter) / (standard deviation of statistic)

4. Reject or fail to reject the null hypothesis.

Using the test statistic or the p-value, determine if you can reject or fail to reject the null hypothesis based on the significance level.

The p-value tells us the strength of evidence in support of a null hypothesis. If the p-value is less than the significance level, we reject the null hypothesis.

5. Interpret the results.

Interpret the results of the hypothesis test in the context of the question being asked.

The Two Types of Decision Errors

There are two types of decision errors that one can make when doing a hypothesis test:

Type I error: You reject the null hypothesis when it is actually true. The probability of committing a Type I error is equal to the significance level, often called alpha , and denoted as α.

Type II error: You fail to reject the null hypothesis when it is actually false. The probability of committing a Type II error is called the Power of the test or Beta , denoted as β.

One-Tailed and Two-Tailed Tests

A statistical hypothesis can be one-tailed or two-tailed.

A one-tailed hypothesis involves making a “greater than” or “less than ” statement.

For example, suppose we assume the mean height of a male in the U.S. is greater than or equal to 70 inches. The null hypothesis would be H0: µ ≥ 70 inches and the alternative hypothesis would be Ha: µ < 70 inches.

A two-tailed hypothesis involves making an “equal to” or “not equal to” statement.

For example, suppose we assume the mean height of a male in the U.S. is equal to 70 inches. The null hypothesis would be H0: µ = 70 inches and the alternative hypothesis would be Ha: µ ≠ 70 inches.

Note: The “equal” sign is always included in the null hypothesis, whether it is =, ≥, or ≤.

Related: What is a Directional Hypothesis?

Types of Hypothesis Tests

There are many different types of hypothesis tests you can perform depending on the type of data you’re working with and the goal of your analysis.

The following tutorials provide an explanation of the most common types of hypothesis tests:

Introduction to the One Sample t-test Introduction to the Two Sample t-test Introduction to the Paired Samples t-test Introduction to the One Proportion Z-Test Introduction to the Two Proportion Z-Test

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A Complete Guide on Hypothesis Testing in Statistics

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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Avijeet is a Senior Research Analyst at Simplilearn. Passionate about Data Analytics, Machine Learning, and Deep Learning, Avijeet is also interested in politics, cricket, and football.

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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.

Probability and Statistics
Data Handling

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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9.1: Introduction to Hypothesis Testing

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Kyle Siegrist
University of Alabama in Huntsville via Random Services

Basic Theory

Preliminaries.

As usual, our starting point is a random experiment with an underlying sample space and a probability measure $\P$. In the basic statistical model, we have an observable random variable $\bs{X}$ taking values in a set $S$. In general, $\bs{X}$ can have quite a complicated structure. For example, if the experiment is to sample $n$ objects from a population and record various measurements of interest, then \[ \bs{X} = (X_1, X_2, \ldots, X_n) \] where $X_i$ is the vector of measurements for the $i$th object. The most important special case occurs when $(X_1, X_2, \ldots, X_n)$ are independent and identically distributed. In this case, we have a random sample of size $n$ from the common distribution.

The purpose of this section is to define and discuss the basic concepts of statistical hypothesis testing . Collectively, these concepts are sometimes referred to as the Neyman-Pearson framework, in honor of Jerzy Neyman and Egon Pearson, who first formalized them.

A statistical hypothesis is a statement about the distribution of $\bs{X}$. Equivalently, a statistical hypothesis specifies a set of possible distributions of $\bs{X}$: the set of distributions for which the statement is true. A hypothesis that specifies a single distribution for $\bs{X}$ is called simple ; a hypothesis that specifies more than one distribution for $\bs{X}$ is called composite .

In hypothesis testing , the goal is to see if there is sufficient statistical evidence to reject a presumed null hypothesis in favor of a conjectured alternative hypothesis . The null hypothesis is usually denoted $H_0$ while the alternative hypothesis is usually denoted $H_1$.

An hypothesis test is a statistical decision ; the conclusion will either be to reject the null hypothesis in favor of the alternative, or to fail to reject the null hypothesis. The decision that we make must, of course, be based on the observed value $\bs{x}$ of the data vector $\bs{X}$. Thus, we will find an appropriate subset $R$ of the sample space $S$ and reject $H_0$ if and only if $\bs{x} \in R$. The set $R$ is known as the rejection region or the critical region . Note the asymmetry between the null and alternative hypotheses. This asymmetry is due to the fact that we assume the null hypothesis, in a sense, and then see if there is sufficient evidence in $\bs{x}$ to overturn this assumption in favor of the alternative.

An hypothesis test is a statistical analogy to proof by contradiction, in a sense. Suppose for a moment that $H_1$ is a statement in a mathematical theory and that $H_0$ is its negation. One way that we can prove $H_1$ is to assume $H_0$ and work our way logically to a contradiction. In an hypothesis test, we don't prove anything of course, but there are similarities. We assume $H_0$ and then see if the data $\bs{x}$ are sufficiently at odds with that assumption that we feel justified in rejecting $H_0$ in favor of $H_1$.

Often, the critical region is defined in terms of a statistic $w(\bs{X})$, known as a test statistic , where $w$ is a function from $S$ into another set $T$. We find an appropriate rejection region $R_T \subseteq T$ and reject $H_0$ when the observed value $w(\bs{x}) \in R_T$. Thus, the rejection region in $S$ is then $R = w^{-1}(R_T) = \left\{\bs{x} \in S: w(\bs{x}) \in R_T\right\}$. As usual, the use of a statistic often allows significant data reduction when the dimension of the test statistic is much smaller than the dimension of the data vector.

The ultimate decision may be correct or may be in error. There are two types of errors, depending on which of the hypotheses is actually true.

Types of errors:

A type 1 error is rejecting the null hypothesis $H_0$ when $H_0$ is true.
A type 2 error is failing to reject the null hypothesis $H_0$ when the alternative hypothesis $H_1$ is true.

Similarly, there are two ways to make a correct decision: we could reject $H_0$ when $H_1$ is true or we could fail to reject $H_0$ when $H_0$ is true. The possibilities are summarized in the following table:

Of course, when we observe $\bs{X} = \bs{x}$ and make our decision, either we will have made the correct decision or we will have committed an error, and usually we will never know which of these events has occurred. Prior to gathering the data, however, we can consider the probabilities of the various errors.

If $H_0$ is true (that is, the distribution of $\bs{X}$ is specified by $H_0$), then $\P(\bs{X} \in R)$ is the probability of a type 1 error for this distribution. If $H_0$ is composite, then $H_0$ specifies a variety of different distributions for $\bs{X}$ and thus there is a set of type 1 error probabilities.

The maximum probability of a type 1 error, over the set of distributions specified by $ H_0 $, is the significance level of the test or the size of the critical region.

The significance level is often denoted by $\alpha$. Usually, the rejection region is constructed so that the significance level is a prescribed, small value (typically 0.1, 0.05, 0.01).

If $H_1$ is true (that is, the distribution of $\bs{X}$ is specified by $H_1$), then $\P(\bs{X} \notin R)$ is the probability of a type 2 error for this distribution. Again, if $H_1$ is composite then $H_1$ specifies a variety of different distributions for $\bs{X}$, and thus there will be a set of type 2 error probabilities. Generally, there is a tradeoff between the type 1 and type 2 error probabilities. If we reduce the probability of a type 1 error, by making the rejection region $R$ smaller, we necessarily increase the probability of a type 2 error because the complementary region $S \setminus R$ is larger.

The extreme cases can give us some insight. First consider the decision rule in which we never reject $H_0$, regardless of the evidence $\bs{x}$. This corresponds to the rejection region $R = \emptyset$. A type 1 error is impossible, so the significance level is 0. On the other hand, the probability of a type 2 error is 1 for any distribution defined by $H_1$. At the other extreme, consider the decision rule in which we always rejects $H_0$ regardless of the evidence $\bs{x}$. This corresponds to the rejection region $R = S$. A type 2 error is impossible, but now the probability of a type 1 error is 1 for any distribution defined by $H_0$. In between these two worthless tests are meaningful tests that take the evidence $\bs{x}$ into account.

If $H_1$ is true, so that the distribution of $\bs{X}$ is specified by $H_1$, then $\P(\bs{X} \in R)$, the probability of rejecting $H_0$ is the power of the test for that distribution.

Thus the power of the test for a distribution specified by $ H_1 $ is the probability of making the correct decision.

Suppose that we have two tests, corresponding to rejection regions $R_1$ and $R_2$, respectively, each having significance level $\alpha$. The test with region $R_1$ is uniformly more powerful than the test with region $R_2$ if \[ \P(\bs{X} \in R_1) \ge \P(\bs{X} \in R_2) \text{ for every distribution of } \bs{X} \text{ specified by } H_1 \]

Naturally, in this case, we would prefer the first test. Often, however, two tests will not be uniformly ordered; one test will be more powerful for some distributions specified by $H_1$ while the other test will be more powerful for other distributions specified by $H_1$.

If a test has significance level $\alpha$ and is uniformly more powerful than any other test with significance level $\alpha$, then the test is said to be a uniformly most powerful test at level $\alpha$.

Clearly a uniformly most powerful test is the best we can do.

$P$-value

In most cases, we have a general procedure that allows us to construct a test (that is, a rejection region $R_\alpha$) for any given significance level $\alpha \in (0, 1)$. Typically, $R_\alpha$ decreases (in the subset sense) as $\alpha$ decreases.

The $P$-value of the observed value $\bs{x}$ of $\bs{X}$, denoted $P(\bs{x})$, is defined to be the smallest $\alpha$ for which $\bs{x} \in R_\alpha$; that is, the smallest significance level for which $H_0$ is rejected, given $\bs{X} = \bs{x}$.

Knowing $P(\bs{x})$ allows us to test $H_0$ at any significance level for the given data $\bs{x}$: If $P(\bs{x}) \le \alpha$ then we would reject $H_0$ at significance level $\alpha$; if $P(\bs{x}) \gt \alpha$ then we fail to reject $H_0$ at significance level $\alpha$. Note that $P(\bs{X})$ is a statistic . Informally, $P(\bs{x})$ can often be thought of as the probability of an outcome as or more extreme than the observed value $\bs{x}$, where extreme is interpreted relative to the null hypothesis $H_0$.

Analogy with Justice Systems

There is a helpful analogy between statistical hypothesis testing and the criminal justice system in the US and various other countries. Consider a person charged with a crime. The presumed null hypothesis is that the person is innocent of the crime; the conjectured alternative hypothesis is that the person is guilty of the crime. The test of the hypotheses is a trial with evidence presented by both sides playing the role of the data. After considering the evidence, the jury delivers the decision as either not guilty or guilty . Note that innocent is not a possible verdict of the jury, because it is not the point of the trial to prove the person innocent. Rather, the point of the trial is to see whether there is sufficient evidence to overturn the null hypothesis that the person is innocent in favor of the alternative hypothesis of that the person is guilty. A type 1 error is convicting a person who is innocent; a type 2 error is acquitting a person who is guilty. Generally, a type 1 error is considered the more serious of the two possible errors, so in an attempt to hold the chance of a type 1 error to a very low level, the standard for conviction in serious criminal cases is beyond a reasonable doubt .

Tests of an Unknown Parameter

Hypothesis testing is a very general concept, but an important special class occurs when the distribution of the data variable $\bs{X}$ depends on a parameter $\theta$ taking values in a parameter space $\Theta$. The parameter may be vector-valued, so that $\bs{\theta} = (\theta_1, \theta_2, \ldots, \theta_n)$ and $\Theta \subseteq \R^k$ for some $k \in \N_+$. The hypotheses generally take the form \[ H_0: \theta \in \Theta_0 \text{ versus } H_1: \theta \notin \Theta_0 \] where $\Theta_0$ is a prescribed subset of the parameter space $\Theta$. In this setting, the probabilities of making an error or a correct decision depend on the true value of $\theta$. If $R$ is the rejection region, then the power function $ Q $ is given by \[ Q(\theta) = \P_\theta(\bs{X} \in R), \quad \theta \in \Theta \] The power function gives a lot of information about the test.

The power function satisfies the following properties:

$Q(\theta)$ is the probability of a type 1 error when $\theta \in \Theta_0$.
$\max\left\{Q(\theta): \theta \in \Theta_0\right\}$ is the significance level of the test.
$1 - Q(\theta)$ is the probability of a type 2 error when $\theta \notin \Theta_0$.
$Q(\theta)$ is the power of the test when $\theta \notin \Theta_0$.

If we have two tests, we can compare them by means of their power functions.

Suppose that we have two tests, corresponding to rejection regions $R_1$ and $R_2$, respectively, each having significance level $\alpha$. The test with rejection region $R_1$ is uniformly more powerful than the test with rejection region $R_2$ if $ Q_1(\theta) \ge Q_2(\theta)$ for all $ \theta \notin \Theta_0 $.

Most hypothesis tests of an unknown real parameter $\theta$ fall into three special cases:

Suppose that $ \theta $ is a real parameter and $ \theta_0 \in \Theta $ a specified value. The tests below are respectively the two-sided test , the left-tailed test , and the right-tailed test .

$H_0: \theta = \theta_0$ versus $H_1: \theta \ne \theta_0$
$H_0: \theta \ge \theta_0$ versus $H_1: \theta \lt \theta_0$
$H_0: \theta \le \theta_0$ versus $H_1: \theta \gt \theta_0$

Thus the tests are named after the conjectured alternative. Of course, there may be other unknown parameters besides $\theta$ (known as nuisance parameters ).

Equivalence Between Hypothesis Test and Confidence Sets

There is an equivalence between hypothesis tests and confidence sets for a parameter $\theta$.

Suppose that $C(\bs{x})$ is a $1 - \alpha$ level confidence set for $\theta$. The following test has significance level $\alpha$ for the hypothesis $ H_0: \theta = \theta_0 $ versus $ H_1: \theta \ne \theta_0 $: Reject $H_0$ if and only if $\theta_0 \notin C(\bs{x})$

By definition, $\P[\theta \in C(\bs{X})] = 1 - \alpha$. Hence if $H_0$ is true so that $\theta = \theta_0$, then the probability of a type 1 error is $P[\theta \notin C(\bs{X})] = \alpha$.

Equivalently, we fail to reject $H_0$ at significance level $\alpha$ if and only if $\theta_0$ is in the corresponding $1 - \alpha$ level confidence set. In particular, this equivalence applies to interval estimates of a real parameter $\theta$ and the common tests for $\theta$ given above .

In each case below, the confidence interval has confidence level $1 - \alpha$ and the test has significance level $\alpha$.

Suppose that $\left[L(\bs{X}, U(\bs{X})\right]$ is a two-sided confidence interval for $\theta$. Reject $H_0: \theta = \theta_0$ versus $H_1: \theta \ne \theta_0$ if and only if $\theta_0 \lt L(\bs{X})$ or $\theta_0 \gt U(\bs{X})$.
Suppose that $L(\bs{X})$ is a confidence lower bound for $\theta$. Reject $H_0: \theta \le \theta_0$ versus $H_1: \theta \gt \theta_0$ if and only if $\theta_0 \lt L(\bs{X})$.
Suppose that $U(\bs{X})$ is a confidence upper bound for $\theta$. Reject $H_0: \theta \ge \theta_0$ versus $H_1: \theta \lt \theta_0$ if and only if $\theta_0 \gt U(\bs{X})$.

Pivot Variables and Test Statistics

Recall that confidence sets of an unknown parameter $\theta$ are often constructed through a pivot variable , that is, a random variable $W(\bs{X}, \theta)$ that depends on the data vector $\bs{X}$ and the parameter $\theta$, but whose distribution does not depend on $\theta$ and is known. In this case, a natural test statistic for the basic tests given above is $W(\bs{X}, \theta_0)$.

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Unit 12: Significance tests (hypothesis testing)

About this unit.

Significance tests give us a formal process for using sample data to evaluate the likelihood of some claim about a population value. Learn how to conduct significance tests and calculate p-values to see how likely a sample result is to occur by random chance. You'll also see how we use p-values to make conclusions about hypotheses.

The idea of significance tests

Simple hypothesis testing (Opens a modal)
Idea behind hypothesis testing (Opens a modal)
Examples of null and alternative hypotheses (Opens a modal)
P-values and significance tests (Opens a modal)
Comparing P-values to different significance levels (Opens a modal)
Estimating a P-value from a simulation (Opens a modal)
Using P-values to make conclusions (Opens a modal)
Simple hypothesis testing Get 3 of 4 questions to level up!
Writing null and alternative hypotheses Get 3 of 4 questions to level up!
Estimating P-values from simulations Get 3 of 4 questions to level up!

Error probabilities and power

Introduction to Type I and Type II errors (Opens a modal)
Type 1 errors (Opens a modal)
Examples identifying Type I and Type II errors (Opens a modal)
Introduction to power in significance tests (Opens a modal)
Examples thinking about power in significance tests (Opens a modal)
Consequences of errors and significance (Opens a modal)
Type I vs Type II error Get 3 of 4 questions to level up!
Error probabilities and power Get 3 of 4 questions to level up!

Tests about a population proportion

Constructing hypotheses for a significance test about a proportion (Opens a modal)
Conditions for a z test about a proportion (Opens a modal)
Reference: Conditions for inference on a proportion (Opens a modal)
Calculating a z statistic in a test about a proportion (Opens a modal)
Calculating a P-value given a z statistic (Opens a modal)
Making conclusions in a test about a proportion (Opens a modal)
Writing hypotheses for a test about a proportion Get 3 of 4 questions to level up!
Conditions for a z test about a proportion Get 3 of 4 questions to level up!
Calculating the test statistic in a z test for a proportion Get 3 of 4 questions to level up!
Calculating the P-value in a z test for a proportion Get 3 of 4 questions to level up!
Making conclusions in a z test for a proportion Get 3 of 4 questions to level up!

Tests about a population mean

Writing hypotheses for a significance test about a mean (Opens a modal)
Conditions for a t test about a mean (Opens a modal)
Reference: Conditions for inference on a mean (Opens a modal)
When to use z or t statistics in significance tests (Opens a modal)
Example calculating t statistic for a test about a mean (Opens a modal)
Using TI calculator for P-value from t statistic (Opens a modal)
Using a table to estimate P-value from t statistic (Opens a modal)
Comparing P-value from t statistic to significance level (Opens a modal)
Free response example: Significance test for a mean (Opens a modal)
Writing hypotheses for a test about a mean Get 3 of 4 questions to level up!
Conditions for a t test about a mean Get 3 of 4 questions to level up!
Calculating the test statistic in a t test for a mean Get 3 of 4 questions to level up!
Calculating the P-value in a t test for a mean Get 3 of 4 questions to level up!
Making conclusions in a t test for a mean Get 3 of 4 questions to level up!

8.1: The Elements of Hypothesis Testing

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Learning Objectives

To understand the logical framework of tests of hypotheses.
To learn basic terminology connected with hypothesis testing.
To learn fundamental facts about hypothesis testing.

Types of Hypotheses

A hypothesis about the value of a population parameter is an assertion about its value. As in the introductory example we will be concerned with testing the truth of two competing hypotheses, only one of which can be true.

Definition: null hypothesis and alternative hypothesis

The null hypothesis , denoted $H_0$, is the statement about the population parameter that is assumed to be true unless there is convincing evidence to the contrary.
The alternative hypothesis , denoted $H_a$, is a statement about the population parameter that is contradictory to the null hypothesis, and is accepted as true only if there is convincing evidence in favor of it.

Definition: statistical procedure

Hypothesis testing is a statistical procedure in which a choice is made between a null hypothesis and an alternative hypothesis based on information in a sample.

The end result of a hypotheses testing procedure is a choice of one of the following two possible conclusions:

Reject $H_0$ (and therefore accept $H_a$), or
Fail to reject $H_0$ (and therefore fail to accept $H_a$).

The null hypothesis typically represents the status quo, or what has historically been true. In the example of the respirators, we would believe the claim of the manufacturer unless there is reason not to do so, so the null hypotheses is $H_0:\mu =75$. The alternative hypothesis in the example is the contradictory statement $H_a:\mu <75$. The null hypothesis will always be an assertion containing an equals sign, but depending on the situation the alternative hypothesis can have any one of three forms: with the symbol $<$, as in the example just discussed, with the symbol $>$, or with the symbol $\neq$. The following two examples illustrate the latter two cases.

Example $\PageIndex{1}$

A publisher of college textbooks claims that the average price of all hardbound college textbooks is $\$127.50$. A student group believes that the actual mean is higher and wishes to test their belief. State the relevant null and alternative hypotheses.

The default option is to accept the publisher’s claim unless there is compelling evidence to the contrary. Thus the null hypothesis is $H_0:\mu =127.50$. Since the student group thinks that the average textbook price is greater than the publisher’s figure, the alternative hypothesis in this situation is $H_a:\mu >127.50$.

Example $\PageIndex{2}$

The recipe for a bakery item is designed to result in a product that contains $8$ grams of fat per serving. The quality control department samples the product periodically to insure that the production process is working as designed. State the relevant null and alternative hypotheses.

The default option is to assume that the product contains the amount of fat it was formulated to contain unless there is compelling evidence to the contrary. Thus the null hypothesis is $H_0:\mu =8.0$. Since to contain either more fat than desired or to contain less fat than desired are both an indication of a faulty production process, the alternative hypothesis in this situation is that the mean is different from $8.0$, so $H_a:\mu \neq 8.0$.

In Example $\PageIndex{1}$, the textbook example, it might seem more natural that the publisher’s claim be that the average price is at most $\$127.50$, not exactly $\$127.50$. If the claim were made this way, then the null hypothesis would be $H_0:\mu \leq 127.50$, and the value $\$127.50$ given in the example would be the one that is least favorable to the publisher’s claim, the null hypothesis. It is always true that if the null hypothesis is retained for its least favorable value, then it is retained for every other value.

Thus in order to make the null and alternative hypotheses easy for the student to distinguish, in every example and problem in this text we will always present one of the two competing claims about the value of a parameter with an equality. The claim expressed with an equality is the null hypothesis. This is the same as always stating the null hypothesis in the least favorable light. So in the introductory example about the respirators, we stated the manufacturer’s claim as “the average is $75$ minutes” instead of the perhaps more natural “the average is at least $75$ minutes,” essentially reducing the presentation of the null hypothesis to its worst case.

The first step in hypothesis testing is to identify the null and alternative hypotheses.

The Logic of Hypothesis Testing

Although we will study hypothesis testing in situations other than for a single population mean (for example, for a population proportion instead of a mean or in comparing the means of two different populations), in this section the discussion will always be given in terms of a single population mean $\mu$.

The null hypothesis always has the form $H_0:\mu =\mu _0$ for a specific number $\mu _0$ (in the respirator example $\mu _0=75$, in the textbook example $\mu _0=127.50$, and in the baked goods example $\mu _0=8.0$). Since the null hypothesis is accepted unless there is strong evidence to the contrary, the test procedure is based on the initial assumption that $H_0$ is true. This point is so important that we will repeat it in a display:

The test procedure is based on the initial assumption that $H_0$ is true.

The criterion for judging between $H_0$ and $H_a$ based on the sample data is: if the value of $\overline{X}$ would be highly unlikely to occur if $H_0$ were true, but favors the truth of $H_a$, then we reject $H_0$ in favor of $H_a$. Otherwise we do not reject $H_0$.

Supposing for now that $\overline{X}$ follows a normal distribution, when the null hypothesis is true the density function for the sample mean $\overline{X}$ must be as in Figure $\PageIndex{1}$: a bell curve centered at $\mu _0$. Thus if $H_0$ is true then $\overline{X}$ is likely to take a value near $\mu _0$ and is unlikely to take values far away. Our decision procedure therefore reduces simply to:

if $H_a$ has the form $H_a:\mu <\mu _0$ then reject $H_0$ if $\bar{x}$ is far to the left of $\mu _0$;
if $H_a$ has the form $H_a:\mu >\mu _0$ then reject $H_0$ if $\bar{x}$ is far to the right of $\mu _0$;
if $H_a$ has the form $H_a:\mu \neq \mu _0$ then reject $H_0$ if $\bar{x}$ is far away from $\mu _0$ in either direction.

Think of the respirator example, for which the null hypothesis is $H_0:\mu =75$, the claim that the average time air is delivered for all respirators is $75$ minutes. If the sample mean is $75$ or greater then we certainly would not reject $H_0$ (since there is no issue with an emergency respirator delivering air even longer than claimed).

If the sample mean is slightly less than $75$ then we would logically attribute the difference to sampling error and also not reject $H_0$ either.

Values of the sample mean that are smaller and smaller are less and less likely to come from a population for which the population mean is $75$. Thus if the sample mean is far less than $75$, say around $60$ minutes or less, then we would certainly reject $H_0$, because we know that it is highly unlikely that the average of a sample would be so low if the population mean were $75$. This is the rare event criterion for rejection: what we actually observed $(\overline{X}<60)$ would be so rare an event if $\mu =75$ were true that we regard it as much more likely that the alternative hypothesis $\mu <75$ holds.

In summary, to decide between $H_0$ and $H_a$ in this example we would select a “rejection region” of values sufficiently far to the left of $75$, based on the rare event criterion, and reject $H_0$ if the sample mean $\overline{X}$ lies in the rejection region, but not reject $H_0$ if it does not.

The Rejection Region

Each different form of the alternative hypothesis Ha has its own kind of rejection region:

if (as in the respirator example) $H_a$ has the form $H_a:\mu <\mu _0$, we reject $H_0$ if $\bar{x}$ is far to the left of $\mu _0$, that is, to the left of some number $C$, so the rejection region has the form of an interval $(-\infty ,C]$;
if (as in the textbook example) $H_a$ has the form $H_a:\mu >\mu _0$, we reject $H_0$ if $\bar{x}$ is far to the right of $\mu _0$, that is, to the right of some number $C$, so the rejection region has the form of an interval $[C,\infty )$;
if (as in the baked good example) $H_a$ has the form $H_a:\mu \neq \mu _0$, we reject $H_0$ if $\bar{x}$ is far away from $\mu _0$ in either direction, that is, either to the left of some number $C$ or to the right of some other number $C′$, so the rejection region has the form of the union of two intervals $(-\infty ,C]\cup [C',\infty )$.

The key issue in our line of reasoning is the question of how to determine the number $C$ or numbers $C$ and $C′$, called the critical value or critical values of the statistic, that determine the rejection region.

Definition: critical values

The critical value or critical values of a test of hypotheses are the number or numbers that determine the rejection region.

Suppose the rejection region is a single interval, so we need to select a single number $C$. Here is the procedure for doing so. We select a small probability, denoted $\alpha$, say $1\%$, which we take as our definition of “rare event:” an event is “rare” if its probability of occurrence is less than $\alpha$. (In all the examples and problems in this text the value of $\alpha$ will be given already.) The probability that $\overline{X}$ takes a value in an interval is the area under its density curve and above that interval, so as shown in Figure $\PageIndex{2}$ (drawn under the assumption that $H_0$ is true, so that the curve centers at $\mu _0$) the critical value $C$ is the value of $\overline{X}$ that cuts off a tail area $\alpha$ in the probability density curve of $\overline{X}$. When the rejection region is in two pieces, that is, composed of two intervals, the total area above both of them must be $\alpha$, so the area above each one is $\alpha /2$, as also shown in Figure $\PageIndex{2}$.

The number $\alpha$ is the total area of a tail or a pair of tails.

Example $\PageIndex{3}$

In the context of Example $\PageIndex{2}$, suppose that it is known that the population is normally distributed with standard deviation $\alpha =0.15$ gram, and suppose that the test of hypotheses $H_0:\mu =8.0$ versus $H_a:\mu \neq 8.0$ will be performed with a sample of size $5$. Construct the rejection region for the test for the choice $\alpha =0.10$. Explain the decision procedure and interpret it.

If $H_0$ is true then the sample mean $\overline{X}$ is normally distributed with mean and standard deviation

\[\begin{align} \mu _{\overline{X}} &=\mu \nonumber \\[5pt] &=8.0 \nonumber \end{align} \nonumber \]

\[\begin{align} \sigma _{\overline{X}}&=\dfrac{\sigma}{\sqrt{n}} \nonumber \\[5pt] &= \dfrac{0.15}{\sqrt{5}} \nonumber\\[5pt] &=0.067 \nonumber \end{align} \nonumber \]

Since $H_a$ contains the $\neq$ symbol the rejection region will be in two pieces, each one corresponding to a tail of area $\alpha /2=0.10/2=0.05$. From Figure 7.1.6, $z_{0.05}=1.645$, so $C$ and $C′$ are $1.645$ standard deviations of $\overline{X}$ to the right and left of its mean $8.0$:

\[C=8.0-(1.645)(0.067) = 7.89 \; \; \text{and}\; \; C'=8.0 + (1.645)(0.067) = 8.11 \nonumber \]

The result is shown in Figure $\PageIndex{3}$. α = 0.1

The decision procedure is: take a sample of size $5$ and compute the sample mean $\bar{x}$. If $\bar{x}$ is either $7.89$ grams or less or $8.11$ grams or more then reject the hypothesis that the average amount of fat in all servings of the product is $8.0$ grams in favor of the alternative that it is different from $8.0$ grams. Otherwise do not reject the hypothesis that the average amount is $8.0$ grams.

The reasoning is that if the true average amount of fat per serving were $8.0$ grams then there would be less than a $10\%$ chance that a sample of size $5$ would produce a mean of either $7.89$ grams or less or $8.11$ grams or more. Hence if that happened it would be more likely that the value $8.0$ is incorrect (always assuming that the population standard deviation is $0.15$ gram).

Because the rejection regions are computed based on areas in tails of distributions, as shown in Figure $\PageIndex{2}$, hypothesis tests are classified according to the form of the alternative hypothesis in the following way.

Definitions: Test classifications

If $H_a$ has the form $\mu \neq \mu _0$ the test is called a two-tailed test .
If $H_a$ has the form $\mu < \mu _0$ the test is called a left-tailed test .
If $H_a$ has the form $\mu > \mu _0$the test is called a right-tailed test .

Each of the last two forms is also called a one-tailed test .

Two Types of Errors

The format of the testing procedure in general terms is to take a sample and use the information it contains to come to a decision about the two hypotheses. As stated before our decision will always be either

reject the null hypothesis $H_0$ in favor of the alternative $H_a$ presented, or
do not reject the null hypothesis $H_0$ in favor of the alternative $H_0$ presented.

There are four possible outcomes of hypothesis testing procedure, as shown in the following table:

As the table shows, there are two ways to be right and two ways to be wrong. Typically to reject $H_0$ when it is actually true is a more serious error than to fail to reject it when it is false, so the former error is labeled “ Type I ” and the latter error “ Type II ”.

Definition: Type I and Type II errors

In a test of hypotheses:

A Type I error is the decision to reject $H_0$ when it is in fact true.
A Type II error is the decision not to reject $H_0$ when it is in fact not true.

Unless we perform a census we do not have certain knowledge, so we do not know whether our decision matches the true state of nature or if we have made an error. We reject $H_0$ if what we observe would be a “rare” event if $H_0$ were true. But rare events are not impossible: they occur with probability $\alpha$. Thus when $H_0$ is true, a rare event will be observed in the proportion $\alpha$ of repeated similar tests, and $H_0$ will be erroneously rejected in those tests. Thus $\alpha$ is the probability that in following the testing procedure to decide between $H_0$ and $H_a$ we will make a Type I error.

Definition: level of significance

The number $\alpha$ that is used to determine the rejection region is called the level of significance of the test. It is the probability that the test procedure will result in a Type I error .

The probability of making a Type II error is too complicated to discuss in a beginning text, so we will say no more about it than this: for a fixed sample size, choosing $alpha$ smaller in order to reduce the chance of making a Type I error has the effect of increasing the chance of making a Type II error . The only way to simultaneously reduce the chances of making either kind of error is to increase the sample size.

Standardizing the Test Statistic

Hypotheses testing will be considered in a number of contexts, and great unification as well as simplification results when the relevant sample statistic is standardized by subtracting its mean from it and then dividing by its standard deviation. The resulting statistic is called a standardized test statistic . In every situation treated in this and the following two chapters the standardized test statistic will have either the standard normal distribution or Student’s $t$-distribution.

Definition: hypothesis test

A standardized test statistic for a hypothesis test is the statistic that is formed by subtracting from the statistic of interest its mean and dividing by its standard deviation.

For example, reviewing Example $\PageIndex{3}$, if instead of working with the sample mean $\overline{X}$ we instead work with the test statistic

\[\frac{\overline{X}-8.0}{0.067} \nonumber \]

then the distribution involved is standard normal and the critical values are just $\pm z_{0.05}$. The extra work that was done to find that $C=7.89$ and $C′=8.11$ is eliminated. In every hypothesis test in this book the standardized test statistic will be governed by either the standard normal distribution or Student’s $t$-distribution. Information about rejection regions is summarized in the following tables:

Every instance of hypothesis testing discussed in this and the following two chapters will have a rejection region like one of the six forms tabulated in the tables above.

No matter what the context a test of hypotheses can always be performed by applying the following systematic procedure, which will be illustrated in the examples in the succeeding sections.

Systematic Hypothesis Testing Procedure: Critical Value Approach

Identify the null and alternative hypotheses.
Identify the relevant test statistic and its distribution.
Compute from the data the value of the test statistic.
Construct the rejection region.
Compare the value computed in Step 3 to the rejection region constructed in Step 4 and make a decision. Formulate the decision in the context of the problem, if applicable.

The procedure that we have outlined in this section is called the “Critical Value Approach” to hypothesis testing to distinguish it from an alternative but equivalent approach that will be introduced at the end of Section 8.3.

Key Takeaway

A test of hypotheses is a statistical process for deciding between two competing assertions about a population parameter.
The testing procedure is formalized in a five-step procedure.

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Hypothesis Testing

What is Hypothesis Testing?

Hypothesis testing in statistics refers to analyzing an assumption about a population parameter. It is used to make an educated guess about an assumption using statistics. With the use of sample data, hypothesis testing makes an assumption about how true the assumption is for the entire population from where the sample is being taken.

Any hypothetical statement we make may or may not be valid, and it is then our responsibility to provide evidence for its possibility. To approach any hypothesis, we follow these four simple steps that test its validity.

First, we formulate two hypothetical statements such that only one of them is true. By doing so, we can check the validity of our own hypothesis.

The next step is to formulate the statistical analysis to be followed based upon the data points.

Then we analyze the given data using our methodology.

The final step is to analyze the result and judge whether the null hypothesis will be rejected or is true.

Let’s look at several hypothesis testing examples:

It is observed that the average recovery time for a knee-surgery patient is 8 weeks. A physician believes that after successful knee surgery if the patient goes for physical therapy twice a week rather than thrice a week, the recovery period will be longer. Conduct hypothesis for this statement.

David is a ten-year-old who finishes a 25-yard freestyle in the meantime of 16.43 seconds. David’s father bought goggles for his son, believing that it would help him to reduce his time. He then recorded a total of fifteen 25-yard freestyle for David, and the average time came out to be 16 seconds. Conduct a hypothesis.

A tire company claims their A-segment of tires have a running life of 50,000 miles before they need to be replaced, and previous studies show a standard deviation of 8,000 miles. After surveying a total of 28 tires, the mean run time came to be 46,500 miles with a standard deviation of 9800 miles. Is the claim made by the tire company consistent with the given data? Conduct hypothesis testing.

All of the hypothesis testing examples are from real-life situations, which leads us to believe that hypothesis testing is a very practical topic indeed. It is an integral part of a researcher's study and is used in every research methodology in one way or another.

Inferential statistics majorly deals with hypothesis testing. The research hypothesis states there is a relationship between the independent variable and dependent variable. Whereas the null hypothesis rejects this claim of any relationship between the two, our job as researchers or students is to check whether there is any relation between the two.

Hypothesis Testing in Research Methodology

Now that we are clear about what hypothesis testing is? Let's look at the use of hypothesis testing in research methodology. Hypothesis testing is at the centre of research projects.

What is Hypothesis Testing and Why is it Important in Research Methodology?

Often after formulating research statements, the validity of those statements need to be verified. Hypothesis testing offers a statistical approach to the researcher about the theoretical assumptions he/she made. It can be understood as quantitative results for a qualitative problem.

(Image will be uploaded soon)

Hypothesis testing provides various techniques to test the hypothesis statement depending upon the variable and the data points. It finds its use in almost every field of research while answering statements such as whether this new medicine will work, a new testing method is appropriate, or if the outcomes of a random experiment are probable or not.

Procedure of Hypothesis Testing

To find the validity of any statement, we have to strictly follow the stepwise procedure of hypothesis testing. After stating the initial hypothesis, we have to re-write them in the form of a null and alternate hypothesis. The alternate hypothesis predicts a relationship between the variables, whereas the null hypothesis predicts no relationship between the variables.

After writing them as H 0 (null hypothesis) and H a (Alternate hypothesis), only one of the statements can be true. For example, taking the hypothesis that, on average, men are taller than women, we write the statements as:

H 0 : On average, men are not taller than women.

H a : On average, men are taller than women.

Our next aim is to collect sample data, what we call sampling, in a way so that we can test our hypothesis. Your data should come from the concerned population for which you want to make a hypothesis.

What is the p value in hypothesis testing? P-value gives us information about the probability of occurrence of results as extreme as observed results.

You will obtain your p-value after choosing the hypothesis testing method, which will be the guiding factor in rejecting the hypothesis. Usually, the p-value cutoff for rejecting the null hypothesis is 0.05. So anything below that, you will reject the null hypothesis.

A low p-value means that the between-group variance is large enough that there is almost no overlapping, and it is unlikely that these came about by chance. A high p-value suggests there is a high within-group variance and low between-group variance, and any difference in the measure is due to chance only.

What is statistical hypothesis testing?

When forming conclusions through research, two sorts of errors are common: A hypothesis must be set and defined in statistics during a statistical survey or research. A statistical hypothesis is what it is called. It is, in fact, a population parameter assumption. However, it is unmistakable that this idea is always proven correct. Hypothesis testing refers to the predetermined formal procedures used by statisticians to determine whether hypotheses should be accepted or rejected. The process of selecting hypotheses for a given probability distribution based on observable data is known as hypothesis testing. Hypothesis testing is a fundamental and crucial issue in statistics.

Why do I Need to Test it? Why not just prove an alternate one?

The quick answer is that you must as a scientist; it is part of the scientific process. Science employs a variety of methods to test or reject theories, ensuring that any new hypothesis is free of errors. One protection to ensure your research is not incorrect is to include both a null and an alternate hypothesis. The scientific community considers not incorporating the null hypothesis in your research to be poor practice. You are almost certainly setting yourself up for failure if you set out to prove another theory without first examining it. At the very least, your experiment will not be considered seriously.

Types of Hypothesis Testing

There are several types of hypothesis testing, and they are used based on the data provided. Depending on the sample size and the data given, we choose among different hypothesis testing methodologies. Here starts the use of hypothesis testing tools in research methodology.

Normality- This type of testing is used for normal distribution in a population sample. If the data points are grouped around the mean, the probability of them being above or below the mean is equally likely. Its shape resembles a bell curve that is equally distributed on either side of the mean.

T-test- This test is used when the sample size in a normally distributed population is comparatively small, and the standard deviation is unknown. Usually, if the sample size drops below 30, we use a T-test to find the confidence intervals of the population.

Chi-Square Test- The Chi-Square test is used to test the population variance against the known or assumed value of the population variance. It is also a better choice to test the goodness of fit of a distribution of data. The two most common Chi-Square tests are the Chi-Square test of independence and the chi-square test of variance.

ANOVA- Analysis of Variance or ANOVA compares the data sets of two different populations or samples. It is similar in its use to the t-test or the Z-test, but it allows us to compare more than two sample means. ANOVA allows us to test the significance between an independent variable and a dependent variable, namely X and Y, respectively.

Z-test- It is a statistical measure to test that the means of two population samples are different when their variance is known. For a Z-test, the population is assumed to be normally distributed. A z-test is better suited in the case of large sample sizes greater than 30. This is due to the central limit theorem that as the sample size increases, the samples are considered to be distributed normally.

FAQs on Hypothesis Testing

1. Mention the types of hypothesis Tests.

There are two types of a hypothesis tests:

Null Hypothesis: It is denoted as H₀.

Alternative Hypothesis: IT is denoted as H₁ or Hₐ.

2. What are the two errors that can be found while performing the null Hypothesis test?

While performing the null hypothesis test there is a possibility of occurring two types of errors,

Type-1: The type-1 error is denoted by (α), it is also known as the significance level. It is the rejection of the true null hypothesis. It is the error of commission.

Type-2: The type-2 error is denoted by (β). (1 - β) is known as the power test. The false null hypothesis is not rejected. It is the error of the omission.

3. What is the p-value in hypothesis testing?

During hypothetical testing in statistics, the p-value indicates the probability of obtaining the result as extreme as observed results. A smaller p-value provides evidence to accept the alternate hypothesis. The p-value is used as a rejection point that provides the smallest level of significance at which the null hypothesis is rejected. Often p-value is calculated using the p-value tables by calculating the deviation between the observed value and the chosen reference value.

It may also be calculated mathematically by performing integrals on all the values that fall under the curve and areas far from the reference value as the observed value relative to the total area of the curve. The p-value determines the evidence to reject the null hypothesis in hypothesis testing.

4. What is a null hypothesis?

The null hypothesis in statistics says that there is no certain difference between the population. It serves as a conjecture proposing no difference, whereas the alternate hypothesis says there is a difference. When we perform hypothesis testing, we have to state the null hypothesis and alternative hypotheses such that only one of them is ever true.

By determining the p-value, we calculate whether the null hypothesis is to be rejected or not. If the difference between groups is low, it is merely by chance, and the null hypothesis, which states that there is no difference among groups, is true. Therefore, we have no evidence to reject the null hypothesis.

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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

$\mu$

Key Terms of Hypothesis Testing

$\alpha$

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:

$\mu \geq 50$

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.

$\mu =$

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.

$\alpha$

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

$\alpha$

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,

$p\leq\alpha$

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.

$z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}$

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:

$t=\frac{x̄-μ}{s/\sqrt{n}}$

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:

$\chi^2 = \sum \frac{(O_{ij} - E_{ij})^2}{E_{ij}}$

i,j are the rows and columns index respectively.

$E_{ij}$

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.

$(203.8 - 200) / (5 \div \sqrt{25})$

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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Hypothesis Test for a Mean

This lesson explains how to conduct a hypothesis test of a mean, when the following conditions are met:

The sampling method is simple random sampling .
The sampling distribution is normal or nearly normal.

Generally, the sampling distribution will be approximately normally distributed if any of the following conditions apply.

The population distribution is normal.
The population distribution is symmetric , unimodal , without outliers , and the sample size is 15 or less.
The population distribution is moderately skewed , unimodal, without outliers, and the sample size is between 16 and 40.
The sample size is greater than 40, without outliers.

This approach consists of four steps: (1) state the hypotheses, (2) formulate an analysis plan, (3) analyze sample data, and (4) interpret results.

State the Hypotheses

Every hypothesis test requires the analyst to state a null hypothesis and an alternative hypothesis . The hypotheses are stated in such a way that they are mutually exclusive. That is, if one is true, the other must be false; and vice versa.

The table below shows three sets of hypotheses. Each makes a statement about how the population mean μ is related to a specified value M . (In the table, the symbol ≠ means " not equal to ".)

The first set of hypotheses (Set 1) is an example of a two-tailed test , since an extreme value on either side of the sampling distribution would cause a researcher to reject the null hypothesis. The other two sets of hypotheses (Sets 2 and 3) are one-tailed tests , since an extreme value on only one side of the sampling distribution would cause a researcher to reject the null hypothesis.

Formulate an Analysis Plan

The analysis plan describes how to use sample data to accept or reject the null hypothesis. It should specify the following elements.

Significance level. Often, researchers choose significance levels equal to 0.01, 0.05, or 0.10; but any value between 0 and 1 can be used.
Test method. Use the one-sample t-test to determine whether the hypothesized mean differs significantly from the observed sample mean.

Analyze Sample Data

Using sample data, conduct a one-sample t-test. This involves finding the standard error, degrees of freedom, test statistic, and the P-value associated with the test statistic.

SE = s * sqrt{ ( 1/n ) * [ ( N - n ) / ( N - 1 ) ] }

SE = s / sqrt( n )

Degrees of freedom. The degrees of freedom (DF) is equal to the sample size (n) minus one. Thus, DF = n - 1.

t = ( x - μ) / SE

P-value. The P-value is the probability of observing a sample statistic as extreme as the test statistic. Since the test statistic is a t statistic, use the t Distribution Calculator to assess the probability associated with the t statistic, given the degrees of freedom computed above. (See sample problems at the end of this lesson for examples of how this is done.)

Sample Size Calculator

As you probably noticed, the process of hypothesis testing can be complex. When you need to test a hypothesis about a mean score, consider using the Sample Size Calculator. The calculator is fairly easy to use, and it is free. You can find the Sample Size Calculator in Stat Trek's main menu under the Stat Tools tab. Or you can tap the button below.

Interpret Results

If the sample findings are unlikely, given the null hypothesis, the researcher rejects the null hypothesis. Typically, this involves comparing the P-value to the significance level , and rejecting the null hypothesis when the P-value is less than the significance level.

Test Your Understanding

In this section, two sample problems illustrate how to conduct a hypothesis test of a mean score. The first problem involves a two-tailed test; the second problem, a one-tailed test.

Problem 1: Two-Tailed Test

An inventor has developed a new, energy-efficient lawn mower engine. He claims that the engine will run continuously for 5 hours (300 minutes) on a single gallon of regular gasoline. From his stock of 2000 engines, the inventor selects a simple random sample of 50 engines for testing. The engines run for an average of 295 minutes, with a standard deviation of 20 minutes. Test the null hypothesis that the mean run time is 300 minutes against the alternative hypothesis that the mean run time is not 300 minutes. Use a 0.05 level of significance. (Assume that run times for the population of engines are normally distributed.)

Solution: The solution to this problem takes four steps: (1) state the hypotheses, (2) formulate an analysis plan, (3) analyze sample data, and (4) interpret results. We work through those steps below:

Null hypothesis: μ = 300

Alternative hypothesis: μ ≠ 300

Formulate an analysis plan . For this analysis, the significance level is 0.05. The test method is a one-sample t-test .

SE = s / sqrt(n) = 20 / sqrt(50) = 20/7.07 = 2.83

DF = n - 1 = 50 - 1 = 49

t = ( x - μ) / SE = (295 - 300)/2.83 = -1.77

where s is the standard deviation of the sample, x is the sample mean, μ is the hypothesized population mean, and n is the sample size.

Since we have a two-tailed test , the P-value is the probability that the t statistic having 49 degrees of freedom is less than -1.77 or greater than 1.77. We use the t Distribution Calculator to find P(t < -1.77) is about 0.04.

If you enter 1.77 as the sample mean in the t Distribution Calculator, you will find the that the P(t < 1.77) is about 0.04. Therefore, P(t > 1.77) is 1 minus 0.96 or 0.04. Thus, the P-value = 0.04 + 0.04 = 0.08.
Interpret results . Since the P-value (0.08) is greater than the significance level (0.05), we cannot reject the null hypothesis.

Note: If you use this approach on an exam, you may also want to mention why this approach is appropriate. Specifically, the approach is appropriate because the sampling method was simple random sampling, the population was normally distributed, and the sample size was small relative to the population size (less than 5%).

Problem 2: One-Tailed Test

Bon Air Elementary School has 1000 students. The principal of the school thinks that the average IQ of students at Bon Air is at least 110. To prove her point, she administers an IQ test to 20 randomly selected students. Among the sampled students, the average IQ is 108 with a standard deviation of 10. Based on these results, should the principal accept or reject her original hypothesis? Assume a significance level of 0.01. (Assume that test scores in the population of engines are normally distributed.)

Null hypothesis: μ >= 110

Alternative hypothesis: μ < 110

Formulate an analysis plan . For this analysis, the significance level is 0.01. The test method is a one-sample t-test .

SE = s / sqrt(n) = 10 / sqrt(20) = 10/4.472 = 2.236

DF = n - 1 = 20 - 1 = 19

t = ( x - μ) / SE = (108 - 110)/2.236 = -0.894

Here is the logic of the analysis: Given the alternative hypothesis (μ < 110), we want to know whether the observed sample mean is small enough to cause us to reject the null hypothesis.

The observed sample mean produced a t statistic test statistic of -0.894. We use the t Distribution Calculator to find P(t < -0.894) is about 0.19.

This means we would expect to find a sample mean of 108 or smaller in 19 percent of our samples, if the true population IQ were 110. Thus the P-value in this analysis is 0.19.
Interpret results . Since the P-value (0.19) is greater than the significance level (0.01), we cannot reject the null hypothesis.

FOR INSTRUCTOR
FOR INSTRUCTORS

8.4.3 Hypothesis Testing for the Mean

$\quad$ $H_0$: $\mu=\mu_0$, $\quad$ $H_1$: $\mu \neq \mu_0$.

$\quad$ $H_0$: $\mu \leq \mu_0$, $\quad$ $H_1$: $\mu > \mu_0$.

$\quad$ $H_0$: $\mu \geq \mu_0$, $\quad$ $H_1$: $\mu \lt \mu_0$.

Two-sided Tests for the Mean:

Therefore, we can suggest the following test. Choose a threshold, and call it $c$. If $|W| \leq c$, accept $H_0$, and if $|W|>c$, accept $H_1$. How do we choose $c$? If $\alpha$ is the required significance level, we must have

As discussed above, we let \begin{align}%\label{} W(X_1,X_2, \cdots,X_n)=\frac{\overline{X}-\mu_0}{\sigma / \sqrt{n}}. \end{align} Note that, assuming $H_0$, $W \sim N(0,1)$. We will choose a threshold, $c$. If $|W| \leq c$, we accept $H_0$, and if $|W|>c$, accept $H_1$. To choose $c$, we let \begin{align} P(|W| > c \; | \; H_0) =\alpha. \end{align} Since the standard normal PDF is symmetric around $0$, we have \begin{align} P(|W| > c \; | \; H_0) = 2 P(W>c | \; H_0). \end{align} Thus, we conclude $P(W>c | \; H_0)=\frac{\alpha}{2}$. Therefore, \begin{align} c=z_{\frac{\alpha}{2}}. \end{align} Therefore, we accept $H_0$ if \begin{align} \left|\frac{\overline{X}-\mu_0}{\sigma / \sqrt{n}} \right| \leq z_{\frac{\alpha}{2}}, \end{align} and reject it otherwise.
We have \begin{align} \beta (\mu) &=P(\textrm{type II error}) = P(\textrm{accept }H_0 \; | \; \mu) \\ &= P\left(\left|\frac{\overline{X}-\mu_0}{\sigma / \sqrt{n}} \right| \lt z_{\frac{\alpha}{2}}\; | \; \mu \right). \end{align} If $X_i \sim N(\mu,\sigma^2)$, then $\overline{X} \sim N(\mu, \frac{\sigma^2}{n})$. Thus, \begin{align} \beta (\mu)&=P\left(\left|\frac{\overline{X}-\mu_0}{\sigma / \sqrt{n}} \right| \lt z_{\frac{\alpha}{2}}\; | \; \mu \right)\\ &=P\left(\mu_0- z_{\frac{\alpha}{2}} \frac{\sigma}{\sqrt{n}} \leq \overline{X} \leq \mu_0+ z_{\frac{\alpha}{2}} \frac{\sigma}{\sqrt{n}}\right)\\ &=\Phi\left(z_{\frac{\alpha}{2}}+\frac{\mu_0-\mu}{\sigma / \sqrt{n}}\right)-\Phi\left(-z_{\frac{\alpha}{2}}+\frac{\mu_0-\mu}{\sigma / \sqrt{n}}\right). \end{align}
Let $S^2$ be the sample variance for this random sample. Then, the random variable $W$ defined as \begin{equation} W(X_1,X_2, \cdots, X_n)=\frac{\overline{X}-\mu_0}{S / \sqrt{n}} \end{equation} has a $t$-distribution with $n-1$ degrees of freedom, i.e., $W \sim T(n-1)$. Thus, we can repeat the analysis of Example 8.24 here. The only difference is that we need to replace $\sigma$ by $S$ and $z_{\frac{\alpha}{2}}$ by $t_{\frac{\alpha}{2},n-1}$. Therefore, we accept $H_0$ if \begin{align} |W| \leq t_{\frac{\alpha}{2},n-1}, \end{align} and reject it otherwise. Let us look at a numerical example of this case.

$\quad$ $H_0$: $\mu=170$, $\quad$ $H_1$: $\mu \neq 170$.

Let's first compute the sample mean and the sample standard deviation. The sample mean is \begin{align}%\label{} \overline{X}&=\frac{X_1+X_2+X_3+X_4+X_5+X_6+X_7+X_8+X_9}{9}\\ &=165.8 \end{align} The sample variance is given by \begin{align}%\label{} {S}^2=\frac{1}{9-1} \sum_{k=1}^9 (X_k-\overline{X})^2&=68.01 \end{align} The sample standard deviation is given by \begin{align}%\label{} S&= \sqrt{S^2}=8.25 \end{align} The following MATLAB code can be used to obtain these values: x=[176.2,157.9,160.1,180.9,165.1,167.2,162.9,155.7,166.2]; m=mean(x); v=var(x); s=std(x); Now, our test statistic is \begin{align} W(X_1,X_2, \cdots, X_9)&=\frac{\overline{X}-\mu_0}{S / \sqrt{n}}\\ &=\frac{165.8-170}{8.25 / 3}=-1.52 \end{align} Thus, $|W|=1.52$. Also, we have \begin{align} t_{\frac{\alpha}{2},n-1} = t_{0.025,8} \approx 2.31 \end{align} The above value can be obtained in MATLAB using the command $\mathtt{tinv(0.975,8)}$. Thus, we conclude \begin{align} |W| \leq t_{\frac{\alpha}{2},n-1}. \end{align} Therefore, we accept $H_0$. In other words, we do not have enough evidence to conclude that the average height in the city is different from the average height in the country.

Let us summarize what we have obtained for the two-sided test for the mean.

One-sided Tests for the Mean:

As before, we define the test statistic as \begin{align}%\label{} W(X_1,X_2, \cdots,X_n)=\frac{\overline{X}-\mu_0}{\sigma / \sqrt{n}}. \end{align} If $H_0$ is true (i.e., $\mu \leq \mu_0$), we expect $\overline{X}$ (and thus $W$) to be relatively small, while if $H_1$ is true, we expect $\overline{X}$ (and thus $W$) to be larger. This suggests the following test: Choose a threshold, and call it $c$. If $W \leq c$, accept $H_0$, and if $W>c$, accept $H_1$. How do we choose $c$? If $\alpha$ is the required significance level, we must have \begin{align} P(\textrm{type I error}) &= P(\textrm{Reject }H_0 \; | \; H_0) \\ &= P(W > c \; | \; \mu \leq \mu_0) \leq \alpha. \end{align} Here, the probability of type I error depends on $\mu$. More specifically, for any $\mu \leq \mu_0$, we can write \begin{align} P(\textrm{type I error} \; | \; \mu) &= P(\textrm{Reject }H_0 \; | \; \mu) \\ &= P(W > c \; | \; \mu)\\ &=P \left(\frac{\overline{X}-\mu_0}{\sigma / \sqrt{n}}> c \; | \; \mu\right)\\ &=P \left(\frac{\overline{X}-\mu}{\sigma / \sqrt{n}}+\frac{\mu-\mu_0}{\sigma / \sqrt{n}}> c \; | \; \mu\right)\\ &=P \left(\frac{\overline{X}-\mu}{\sigma / \sqrt{n}}> c+\frac{\mu_0-\mu}{\sigma / \sqrt{n}} \; | \; \mu\right)\\ &\leq P \left(\frac{\overline{X}-\mu}{\sigma / \sqrt{n}}> c \; | \; \mu\right) \quad (\textrm{ since }\mu \leq \mu_0)\\ &=1-\Phi(c) \quad \big(\textrm{ since given }\mu, \frac{\overline{X}-\mu}{\sigma / \sqrt{n}} \sim N(0,1) \big). \end{align} Thus, we can choose $\alpha=1-\Phi(c)$, which results in \begin{align} c=z_{\alpha}. \end{align} Therefore, we accept $H_0$ if \begin{align} \frac{\overline{X}-\mu_0}{\sigma / \sqrt{n}} \leq z_{\alpha}, \end{align} and reject it otherwise.

$\quad$ $H_0$: $\mu \geq \mu_0$, $\quad$ $H_1$: $\mu \lt \mu_0$,

What is Hypothesis?
Post last modified: 20 April 2021
Reading time: 20 mins read
Post category: Research Methodology

Hypothesis is a proposition which can be put to a test to determine validity and is useful for further research.

Hypothesis is a statement which can be proved or disproved. It is a statement capable of being tested. In a sense, hypothesis is a question which definitely has an answer. Hypothesis aids us a great deal while collecting, tabulating and analyzing data and other relevant information.

Table of Content

1 What is Hypothesis?
2 Hypothesis Definition
3 Meaning of Hypothesis
4.1 Conceptual Clarity
4.2 Need of the empirical referents
4.3 Hypothesis should be specific
4.4 Hypothesis should be within the ambit of the available research techniques
4.5 Hypothesis should be consistent with the theory
4.6 Hypothesis should be concerned with observable facts and empirical events
4.7 Hypothesis should be simple
5 Formulation of Hypothesis
6 Null Hypothesis
7.1 Stating the hypothesis of interest
7.2 Collection of relevant data and information
7.3 Formation of null hypothesis
7.4 Alternative Hypothesis
7.5 Selection of suitable test statistic
7.6 Determine the level of significance
7.7 Decision

Hypothesis thus is inevitable in any kind of research, if it is to be carried out successfully. The meaning and exact nature of hypothesis will become clear from the following definitions.

Hypothesis Definition

Meaning of hypothesis.

From the above mentioned definitions of hypothesis, its meaning can be explained in the following ways:

At the primary level, a hypothesis is the possible and probable explanation of the sequence of happenings or data.
Sometimes, hypothesis may emerge from an imagination, common sense or a sudden event.
Hypothesis can be a probable answer to the research problem undertaken for study.
Hypothesis may not always be true. It can get disproven. In other words, hypothesis need not always be a true proposition.
Hypothesis, in a sense, is an attempt to present the interrelations that exist in the available data or information.
Hypothesis is not an individual opinion or community thought. Instead, it is a philosophical means which is to be used for research purpose. Hypothesis is not to be considered as the ultimate objective; rather it is to be taken as the means of explaining scientifically the prevailing situation.

Characteristics of Hypothesis

Not all the hypotheses are good and useful from the point of view of research. It is only the few hypotheses satisfying certain criteria that are good, useful and directive in the research work undertaken.

The characteristics of a good hypothesis can be listed as below.

Conceptual Clarity

Need of the empirical referents, hypothesis should be specific, hypothesis should be within the ambit of the available research techniques, hypothesis should be consistent with the theory, hypothesis should be concerned with observable facts and empirical events, hypothesis should be simple.

The concepts used while framing hypothesis should be crystal clear and unambiguous. Such concepts must be clearly defined so that they become lucid and acceptable to everyone.

How are the newly developed concepts interrelated and how are they linked with the old one is to be very clear so that the hypothesis framed on their basis also carries the same clarity. A hypothesis embodying unclear and ambiguous concepts can to a great extent undermine the successful completion of the research work.

A hypothesis can be useful in the research work undertaken only when it has links with some empirical referents. Hypothesis based on moral values and ideals are useless as they cannot be tested. Similarly, hypothesis containing opinions as good and bad or expectation with respect to something are not testable and therefore useless.

For example, ‘current account deficit can be lowered if people change their attitude towards gold’ is a hypothesis encompassing expectation. In case of such a hypothesis, the attitude towards gold is something which cannot clearly be described and therefore a hypothesis which embodies such an unclear thing cannot be tested and proved or disproved. In short, the hypothesis should be linked with some testable referents.

For the successful conduction of research, it is necessary that the hypothesis is specific and presented in a precise manner. Hypothesis which is general, too ambitious and grandiose in scope is not to be made as such hypothesis cannot be easily put to test. A hypothesis is to be based on such concepts which are precise and empirical in nature. A hypothesis should give a clear idea about the indicators which are to be used.

For example, a hypothesis that economic power is increasingly getting concentrated in few hands in India should enable us to define the concept of economic power. It should be explicated in terms of the measurable indicator like income, wealth, etc. Such specificity in the formulation of hypothesis ensures that the research is practicable and significant.

While framing the hypothesis, the researcher should be aware of the available research techniques and should see that the hypothesis framed is testable on the basis of them.

In other words, a hypothesis should be researchable and for this, it is important that a due thought has been given to the methods and techniques which can be used to measure the concepts and variables embodied in the hypothesis.

It does not, however, mean that hypotheses which are not testable with the available techniques of research are not to be made. If the problem is too significant and therefore the hypothesis framed becomes too ambitious and complex, it’s testing becomes possible with the development of new research techniques or the hypothesis itself leads to the development of new research techniques.

A hypothesis must be related to the existing theory or should have a theoretical orientation. The growth of the knowledge takes place in the sequence of facts, hypothesis, theory and law or principles.

It means the hypothesis should have a correspondence with the existing facts and theory. If the hypothesis is related to some theory, the research work will enable us to support, modify or refute the existing theory. Theoretical orientation of the hypothesis ensures that it becomes scientifically useful.

According to Prof. Goode and Prof. Hatt, research work can contribute to the existing knowledge only when the hypothesis is related to some theory.

This enables us to explain the observed facts and situations and also verify the framed hypothesis.

In the words of Prof. Cohen and Prof. Nagel, “hypothesis must be formulated in such a manner that deduction can be made from it and that consequently a decision can be reached as to whether it does or does not explain the facts considered.

If the research work based on a hypothesis is to be successful, it is necessary that the later is as simple and easy as possible. An ambition of finding out something new may lead the researcher to frame an unrealistic and unclear hypothesis. Such a temptation is to be avoided.

Framing a simple, easy and testable hypothesis requires that the researcher is well acquainted with the related concepts.

Formulation of Hypothesis

The real beginning of any research is made with the formulation of hypothesis. In a sense, research is nothing but accepting the hypothesis by proving it or rejecting it if it is disproved or modifying it.

Moreover, in any type of research work, the information and data is to be collected with reference to the hypothesis and the concepts embodied in it. Hypothesis therefore occupies an important place in any type of research.

Formulation of hypothesis, however, requires that the difficulties encountered are overcome. A researcher may suffer from a number of difficulties at the stage of formulating a good hypothesis

The researcher should have a thorough knowledge of the accepted theories and basic concepts of that research area where he has decided to work in.
The researcher should also acquire the logical and scientific thinking power to frame a hypothesis based on the theories and basic concepts known to him.
The researcher should also be well acquainted with the available research methods and techniques.

Normally, the hypothesis made in the beginning of research is of crude or working nature. Such a working hypothesis is to be made while planning a research work. As the research work proceeds with the working hypothesis, new information, data and evidence becomes available. In the light of new information and evidence, the working hypothesis is to be modified and revised.

Sometimes, the working hypothesis changes in a significant way after the modifications are made. In some researches, the hypothesis is formulated not in the beginning but at the time of classification and analysis of data and information.

In the case of such a hypothesis also it becomes necessary that new or additional information is collected. It thus implies that every hypothesis is subject to change. In order to put the research work in an operative mode, several alternative hypotheses are made in the beginning.

While framing such hypotheses utmost care is to be taken while using the concepts. The nature of the hypothesis should be such that it enables the researcher to find out something new, something which is previously unknown.

In the context of research work and while performing the hypothesis testing exercise, both the alternative hypothesis which is to be proved and accepted and null hypothesis, which is to be disproved, are important and required.

The main hypothesis of the research work is the research hypothesis or the alternative hypothesis. Researcher’s job is to collect information and data so as to prove the alternative hypothesis so that it can be accepted. Null hypothesis on the other hand is the exact opposite of research or alternative hypothesis.

Null hypothesis is also called a hypothesis with no difference. Like the research or alternative hypothesis, the null hypothesis is also a statement.

The logic behind formulating a null hypothesis is that it is always easy to prove that a statement is wrong than to prove that a statement (research hypothesis) is cent percent true.

In short, while framing hypothesis for research work, it is important that at least two hypotheses are framed, one of which is a null hypothesis and the other one is an alternative hypothesis.

For instance, a null hypothesis and alternative hypothesis can be as below.

Null Hypothesis

The average age of entry in to the labour market of commerce graduates is 22 years.

However, the collected data and information, when analysed, reveals that Hypothesis the average entry age is greater than or less than 22 years, then the null hypothesis gets rejected.

In such a case the alternative hypothesis can be as under

The average age of entry into the labour market is greater than 22 years (> 22)
The average age of entry into the labour market is less than 22 years (< 22)
The average age of entry into the labour market is not 22 years (‘“ 22)

Test of Hypothesis

As stated in the beginning, the hypothesis formulation marks the beginning of any research. After the hypothesis is formulated in the context of a research problem, next process involves a collection of relevant data and information and analysis of the same using an appropriate statistical technique, which proves or disproves the hypothesis formulated in the beginning.

The testing of hypothesis thus represents the end of the research work. Testing of hypothesis can be considered as the most important step in any type of research work as it determines the fruitfulness of the research work.

Unless the hypothesis is tested, it will only remain an inference or a proposition. The act of determining the validity of the hypothesis based on the collected data is called the testing of hypothesis.

The exercise of hypothesis testing is a systematic work and normally involves following stages or steps:

Stating the hypothesis of interest

Collection of relevant data and information, formation of null hypothesis, alternative hypothesis, selection of suitable test statistic, determine the level of significance.

Based on the research problem and a primitive understanding of the relationship between the variables involved, a researcher formulates a hypothesis of interest or a research hypothesis which he wants to prove.

Given the research problem and the formulated hypothesis of interest, the next step is to collect the relevant data and information to proceed further towards the end objective (i.e. proving the research hypothesis).

For the testing purpose, a null hypothesis is formed based on the statistical data. The null hypothesis is also called as the hypothesis with no difference.

In other words, null hypothesis states that there is no difference between the variables involved in the hypothesis or the variables are not related.

For example, if the research hypothesis is that the commerce graduates are more employable than the arts graduates, then the null hypothesis will be that both are equally employable or that there is no difference in the employment opportunities available to both.

If in research hypothesis, price and demand are said to be inversely related, the null hypothesis assumes them independent or states that price and demand are not related.

After the formulation of null hypothesis, alternative hypothesis can be derived. Alternative hypothesis is the negation of null hypothesis and can be more than one and conform to the research hypothesis.

In the example of employability, the alternative hypothesis can be

commerce graduates are more employable or arts graduates are more employable
commerce graduates are having more employability
arts graduates are having more employability.

The next step in the hypothesis testing exercise is that of selecting an appropriate statistical test. It can be chi-square test, t-test or f-test or any other test. Such a test is carried out at a given level of significance.

As stated in the above step a statistical test is conducted at a given level of significance

A level of significance indicates the probability of rejecting or accepting the null hypothesis.

The last step in testing hypothesis is that of taking a decision on the basis of the given level of significance

It is seen whether the null hypothesis falls in the accepting region or in rejecting region and accordingly a decision is taken. In this way, the acceptance or rejection of null hypothesis determines the acceptance or rejection of the initial research hypothesis.

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Biden administration plans to drastically change federal rules on marijuana

The Biden administration is poised to make a landmark change to the federal government's position on marijuana with a proposed plan that would no longer consider marijuana among the most dangerous and addictive substances .

In what would be the biggest change in marijuana policy the federal government has taken since pot was first outlawed, the Drug Enforcement Administration will take public comments on a plan to recategorize marijuana under the Controlled Substances Act, according to a source familiar with the process. The news was first reported by The Associated Press .

The Department of Justice will send its recommendation to reclassify marijuana from a Schedule I drug to a Schedule III drug to the White House Office of Management and Budget, according to the source, who was not authorized to speak publicly. The Justice Department is expected to transmit the recommendation today, the source said.

More: Trucker failed drug test after taking CBD supplement. Supreme Court to decide if he can sue

The plan wouldn't legalize marijuana at the federal level outright, but it would reclassify it from a Schedule I drug – believed highly dangerous, addictive and without medical use – to a Schedule III drug that can be lawfully prescribed as medication. Marijuana has been a Schedule I drug since the Controlled Substances Act was signed in 1970.

“It is significant for these federal agencies, and the DEA and FDA in particular, to acknowledge publicly for the first time what many patients and advocates have known for decades: that cannabis is a safe and effective therapeutic agent for tens of millions of Americans," said Paul Armentano, deputy director of the National Organization for the Reform of Marijuana Laws, or NORML, which advocates for cannabis to be removed altogether from the list of controlled substances.

This bureaucratic move is only a small step toward what advocates hope will be full legalization of the drug. However, the new proposed classification does not fully address the inconsistencies between federal restrictions and the laws in a growing number of states that have authorized medical and recreational use of pot.

Twenty-four states and Washington, D.C., have legalized the recreational use of marijuana, and 14 other states authorize it for medical use, according to the Pew Research Center .

“Rescheduling the cannabis plant to Schedule III fails to adequately address this conflict, as existing state legalization laws – both adult use and medical – will continue to be in conflict with federal regulations, thereby perpetuating the existing divide between state and federal marijuana policies," Armentano said in a statement.

The federal proposal to reschedule marijuana would have broad support among voters. A nationwide survey last fall commissioned by the Coalition for Cannabis Rescheduling Reform found nearly 60% of likely voters supported rescheduling, with 65% of younger voters 18 to 25 favoring it, the highest of any demographic group polled. Overall, the number of Americans who think marijuana should be legal reached a record high at 70%, according to a Gallup poll in the fall.

For decades, marijuana has been listed under the Controlled Substances Act as a Schedule I drug, alongside heroin, LSD and ecstasy. The act categorizes drugs based on their potential for abuse, addiction and medical use. Schedule I drugs are outlawed under federal law level and deemed to be without accepted medical use.

In 2022, President Joe Biden directed the Department of Health and Human Services to conduct a review of how marijuana is classified; and last year HHS recommended it be rescheduled to Schedule III, alongside drugs like Tylenol with codeine and anabolic steroids. The Justice Department did its own analysis and reached the same conclusion, the source said.

The proposal will undergo a public review period; the source did not say when the proposed rule would be open to public comment.

Rep. Andy Harris, R-Md., has previously criticized federal efforts to change Marijuana's classification . Harris was a physician at the Johns Hopkins Hospital, according to his online biography .

"Removing restrictions on an addictive gateway drug like Marijuana is a dangerous mistake. Numerous studies, including a recent and reputable study published by JAMA, points to the negative impact recreational marijuana has on the body and brain," Harris said in a Tuesday social media post on X, formerly known as Twitter.

Experts previously told USA TODAY that marijuana’s placement on Schedule I was not based on credible scientific evidence of its perils, but once it was listed, researchers and advocates faced a heavy burden trying to prove it shouldn’t face such stiff restrictions.

What exactly does rescheduling cannabis mean?

Placing marijuana in Schedule III puts it on par with drugs, such as ketamine, testosterone, anabolic steroids or Tylenol with codeine, that have “moderate to low potential for physical and psychological dependence,” according to the DEA.

Schedule III drugs can be legally prescribed by licensed health care providers and dispensed by licensed pharmacies. Rescheduling could also help resolve a massive federal tax burden that has been placed on cannabis companies – which were effectively seen as drug traffickers for tax purposes.

But rescheduling marijuana doesn’t make it legal to use recreationally, and it doesn’t change much about current state cannabis programs, said Jay Wexler, who teaches a seminar about marijuana laws at Boston University. It would still a controlled substance even with the new announcement

Wexler and other policy experts and advocates say rescheduling is not a solution, but it could be a sign the federal government is catching up with public opinion and consensus in the medical field that there are therapeutic benefits to marijuana, along with some risks.

"Rescheduling is a step forward, but it is not nearly enough. And there's no reason to keep cannabis in the Controlled Substances Act,” Wexler previously told USA TODAY.

What are the possible risks of marijuana?

Because of its classification, marijuana has been hard to study. But the move to reschedule marijuana is due in large part to its lower public health risk, federal scientists have said.

In a leaked HHS document , officials wrote to the DEA to support lowering its classification to Schedule III. Its risk for addiction was lower than other drugs and it had medical benefits, unlike Schedule I and II drugs, HHS researchers said.

Still, scientists said, users develop moderate to low physical dependence on it, and there is some risk of psychological dependence. However, they noted, the withdrawal symptoms are “relatively mild” compared with alcohol. Marijuana is more comparable to tobacco, they said.

There are no known deaths from a marijuana overdose, according to the National Institute on Drug Abuse , or NIDA. But it does affect physical and mental health.

Marijuana can cause permanent IQ loss for people who begin using it at a young age, the institute said. Additionally, long-term use has been associated with temporary paranoia and hallucinations, and it can exacerbate symptoms with disorders such as schizophrenia, NIDA said.

Marijuana smoke has a similar health impact to tobacco smoke. NIDA found people who smoke marijuana frequently develop issues with breathing, akin to tobacco smokers.

Smoking cannabis, the most common way to consume the drug, may have additional risks because of particulate matter a person inhales, according to a recent study in the Journal of the American Heart Association . Researchers noted cannabis smoke isn’t all that different than tobacco smoke, the only difference being the added effect of the psychoactive drug THC in marijuana rather than nicotine in tobacco.

Respiratory issues include daily cough, phlegm and a higher risk of lung infections, however, the institute said it’s unclear if marijuana causes a greater risk of lung cancer.

Smoking marijuana also increases heart rate, which can increase the chance of heart attack, especially among older people and people with heart conditions. The Heart Association journal study linked increased cannabis use with an increased risk of heart attack and stroke.

“Despite common use, little is known about the risks of cannabis use and, in particular, the cardiovascular disease risks,” the study’s lead author, Abra Jeffers, a data analyst at Boston’s Massachusetts General Hospital, said in a statement. “The perceptions of the harmfulness of smoking cannabis are decreasing, and people have not considered cannabis use dangerous to their health. However, previous research suggested that cannabis could be associated with cardiovascular disease.” She noted that smoking cannabis, which is the predominant way it is used, could pose other risks because it involves inhaling particulate matter.

In the study published in late February, researchers examined Centers for Disease Control and Prevention survey data of over 400,000 adults from 2016 to 2020, looking at self-reported cannabis use with cardiovascular outcomes, such as heart disease, heart attacks and strokes.

People who used marijuana daily had a 25% higher chance of having a heart attack and a 42% higher chance of stroke than those who didn’t use it at all.

Proposal reflects potential for health benefits

The cannabis plant has been used for medicinal purposes for centuries if not millennia. It appears to help with treating pain , insomnia, anxiety, and glaucoma, among other health conditions. Still, evidence is mixed and more research into its health benefits is needed, researchers at Johns Hopkins Bloomberg School of Public Health said in August.

While the FDA hasn’t approved the cannabis plant for any medical use, federal regulators have approved several drugs containing cannabinoids, or substances such as THC or CBD found in the cannabis plant, according to the National Institutes of Health .

These include Epidiolex, a purified form of CBD ingested orally, that is FDA-approved to treat seizures associated with two severe forms of epilepsy. Marinol and Syndros both contain synthetic THC and are used to treat nausea and vomiting caused by chemotherapy. Nabilone, another synthetic similar to THC, is approved as the brand name drug Cesamet for people with HIV/AIDS who experiencing weight loss and appetite loss.

A 2017 federal report found cannabis or cannabinoids were more likely to reduce pain symptoms for patients with chronic pain. Additionally, there is some evidence that cannabis is effective in treating symptoms of multiple sclerosis, particularly addressing the stiff or rigid muscles caused by the disease. One cannabinoid drug, nabiximol, a mouth spray that has both THC and CBD, has been approved in several countries but not in the U.S. Under the brand name Sativex, it has shown pain relief for people with cancer or multiple sclerosis.

Other research has examined cannabis’ uses to treat post-traumatic stress disorder, but the NIH said the evidence is mixed.

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What the New Overtime Rule Means for Workers

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One of the basic principles of the American workplace is that a hard day’s work deserves a fair day’s pay. Simply put, every worker’s time has value. A cornerstone of that promise is the Fair Labor Standards Act ’s (FLSA) requirement that when most workers work more than 40 hours in a week, they get paid more. The Department of Labor ’s new overtime regulation is restoring and extending this promise for millions more lower-paid salaried workers in the U.S.

Overtime protections have been a critical part of the FLSA since 1938 and were established to protect workers from exploitation and to benefit workers, their families and our communities. Strong overtime protections help build America’s middle class and ensure that workers are not overworked and underpaid.

Some workers are specifically exempt from the FLSA’s minimum wage and overtime protections, including bona fide executive, administrative or professional employees. This exemption, typically referred to as the “EAP” exemption, applies when:

1. An employee is paid a salary,

2. The salary is not less than a minimum salary threshold amount, and

3. The employee primarily performs executive, administrative or professional duties.

While the department increased the minimum salary required for the EAP exemption from overtime pay every 5 to 9 years between 1938 and 1975, long periods between increases to the salary requirement after 1975 have caused an erosion of the real value of the salary threshold, lessening its effectiveness in helping to identify exempt EAP employees.

The department’s new overtime rule was developed based on almost 30 listening sessions across the country and the final rule was issued after reviewing over 33,000 written comments. We heard from a wide variety of members of the public who shared valuable insights to help us develop this Administration’s overtime rule, including from workers who told us: “I would love the opportunity to...be compensated for time worked beyond 40 hours, or alternately be given a raise,” and “I make around $40,000 a year and most week[s] work well over 40 hours (likely in the 45-50 range). This rule change would benefit me greatly and ensure that my time is paid for!” and “Please, I would love to be paid for the extra hours I work!”

The department’s final rule, which will go into effect on July 1, 2024, will increase the standard salary level that helps define and delimit which salaried workers are entitled to overtime pay protections under the FLSA.

Starting July 1, most salaried workers who earn less than $844 per week will become eligible for overtime pay under the final rule. And on Jan. 1, 2025, most salaried workers who make less than $1,128 per week will become eligible for overtime pay. As these changes occur, job duties will continue to determine overtime exemption status for most salaried employees.

Who will become eligible for overtime pay under the final rule? Currently most salaried workers earning less than $684/week. Starting July 1, 2024, most salaried workers earning less than $844/week. Starting Jan. 1, 2025, most salaried workers earning less than $1,128/week. Starting July 1, 2027, the eligibility thresholds will be updated every three years, based on current wage data. DOL.gov/OT

The rule will also increase the total annual compensation requirement for highly compensated employees (who are not entitled to overtime pay under the FLSA if certain requirements are met) from $107,432 per year to $132,964 per year on July 1, 2024, and then set it equal to $151,164 per year on Jan. 1, 2025.

Starting July 1, 2027, these earnings thresholds will be updated every three years so they keep pace with changes in worker salaries, ensuring that employers can adapt more easily because they’ll know when salary updates will happen and how they’ll be calculated.

The final rule will restore and extend the right to overtime pay to many salaried workers, including workers who historically were entitled to overtime pay under the FLSA because of their lower pay or the type of work they performed.

We urge workers and employers to visit our website to learn more about the final rule.

Jessica Looman is the administrator for the U.S. Department of Labor’s Wage and Hour Division. Follow the Wage and Hour Division on Twitter at  @WHD_DOL and LinkedIn . Editor's note: This blog was edited to correct a typo (changing "administrator" to "administrative.")

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Marijuana | The US will reclassify marijuana. What does…

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Marijuana | the latest | 4 officers injured removing protesters at university of wisconsin in madison, marijuana | the us will reclassify marijuana. what does that mean, the proposal would move marijuana from the “schedule i” group to the less tightly regulated “schedule iii”.

File photo: The Justice Department proposal would recognize the medical uses of cannabis, but wouldn’t legalize it nationally for recreational use.

WASHINGTON — The U.S. Drug Enforcement Administration is moving toward reclassifying marijuana as a less dangerous drug. The Justice Department proposal would recognize the medical uses of cannabis, but wouldn’t legalize it for recreational use.

The proposal would move marijuana from the “Schedule I” group to the less tightly regulated “Schedule III.”

So what does that mean, and what are the implications?

WHAT HAS ACTUALLY CHANGED? WHAT HAPPENS NEXT?

Still, the switch is considered “paradigm-shifting, and it’s very exciting,” Vince Sliwoski, a Portland, Oregon-based cannabis and psychedelics attorney who runs well-known legal blogs on those topics, told The Associated Press when the federal Health and Human Services Department recommended the change.

“I can’t emphasize enough how big of news it is,” he said.

It came after President Joe Biden asked both HHS and the attorney general, who oversees the DEA, last year to review how marijuana was classified. Schedule I put it on par, legally, with heroin, LSD, quaaludes and ecstasy, among others.

Biden, a Democrat, supports legalizing medical marijuana for use “where appropriate, consistent with medical and scientific evidence,” White House press secretary Karine Jean-Pierre said Thursday. “That is why it is important for this independent review to go through.”

IF MARIJUANA GETS RECLASSIFIED, WOULD IT LEGALIZE RECREATIONAL CANNABIS NATIONWIDE?

No. Schedule III drugs — which include ketamine, anabolic steroids and some acetaminophen-codeine combinations — are still controlled substances.

They’re subject to various rules that allow for some medical uses, and for federal criminal prosecution of anyone who traffics in the drugs without permission.

No changes are expected to the medical marijuana programs now licensed in 38 states or the legal recreational cannabis markets in 23 states, but it’s unlikely they would meet the federal production, record-keeping, prescribing and other requirements for Schedule III drugs.

There haven’t been many federal prosecutions for simply possessing marijuana in recent years, even under marijuana’s current Schedule I status, but the reclassification wouldn’t have an immediate impact on people already in the criminal justice system.

“Put simple, this move from Schedule I to Schedule III is not getting people out of jail,” said David Culver, senior vice president of public affairs at the U.S. Cannabis Council. But rescheduling in itself would have some impact, particularly on research and marijuana business taxes.

WHAT WOULD THIS MEAN FOR RESEARCH?

Because marijuana is on Schedule I, it’s been very difficult to conduct authorized clinical studies that involve administering the drug. That has created something of a Catch-22: calls for more research, but barriers to doing it. (Scientists sometimes rely instead on people’s own reports of their marijuana use.)

Schedule III drugs are easier to study, though the reclassification wouldn’t immediately reverse all barriers to study, Culver said.

WHAT ABOUT TAXES (AND BANKING)?

Under the federal tax code, businesses involved in “trafficking” in marijuana or any other Schedule I or II drug can’t deduct rent, payroll or various other expenses that other businesses can write off. (Yes, at least some cannabis businesses, particularly state-licensed ones, do pay taxes to the federal government, despite its prohibition on marijuana.) Industry groups say the tax rate often ends up at 70% or more.

The deduction rule doesn’t apply to Schedule III drugs, so the proposed change would cut cannabis companies’ taxes substantially.

They say it would treat them like other industries and help them compete against illegal competitors that are frustrating licensees and officials in places such as New York.

“You’re going to make these state-legal programs stronger,” says Adam Goers, an executive at medical and recreational cannabis giant Columbia Care. He co-chairs a coalition of corporate and other players that’s pushing for rescheduling.

It could also mean more cannabis promotion and advertising if those costs could be deducted, according to Beau Kilmer, co-director of the RAND Drug Policy Center.

Rescheduling wouldn’t directly affect another marijuana business problem: difficulty accessing banks, particularly for loans, because the federally regulated institutions are wary of the drug’s legal status. The industry has been looking instead to a measure called the SAFE Banking Act. It has repeatedly passed the House but stalled in the Senate.

ARE THERE CRITICS? WHAT DO THEY SAY?

Indeed, there are, including the national anti-legalization group Smart Approaches to Marijuana. President Kevin Sabet, a former Obama administration drug policy official, said the HHS recommendation “flies in the face of science, reeks of politics” and gives a regrettable nod to an industry “desperately looking for legitimacy.”

Some legalization advocates say rescheduling weed is too incremental. They want to keep the focus on removing it completely from the controlled substances list, which doesn’t include such items as alcohol or tobacco (they’re regulated, but that’s not the same).

Paul Armentano, the deputy director of the National Organization for the Reform of Marijuana Laws, said that simply reclassifying marijuana would be “perpetuating the existing divide between state and federal marijuana policies.” Minority Cannabis Business Association President Kaliko Castille said rescheduling just “re-brands prohibition,” rather than giving an all-clear to state licensees and putting a definitive close to decades of arrests that disproportionately pulled in people of color.

“Schedule III is going to leave it in this kind of amorphous, mucky middle where people are not going to understand the danger of it still being federally illegal,” he said.

Peltz reported from New York. Associated Press writer Colleen Long in Washington contributed to this report.

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It comes after President Joe Biden called for a review of federal marijuana law in October 2022 and moved to pardon thousands of Americans convicted federally of simple possession of the drug. He has also called on governors and local leaders to take similar steps to erase marijuana convictions.

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Apr 29, 2024; Phoenix, Arizona, USA; Arizona Diamondbacks manager Torey Lovullo (17) looks on during loss to Dodgers

Why Did Torey Lovullo Pull Tommy Henry Early in Loss to Dodgers?

What seemed like a questionable decision at the time had a surprising explanation

Author: Jack Sommers

In this story:

During last nights game against the Dodgers, an eventual 8-4 Diamondbacks loss, Torey Lovullo made what seemed to be a questionable decision that cost him the game. He pulled Tommy Henry after just four innings and 72 pitches and trailing by a score of just 2-1.

As is often the case however, once there was a chance to ask the Manager and the player about the situation, an underlying reason not evident to those watching emerged. In fact, Henry was struggling, fatigued, and "out of whack" by his own admission. He communicated that to his manager, who acted accordingly,

Henry had labored through the first couple of innings giving up four hits, two walks, and two runs. He was fortunate the damage was not worse, as the average exit velocity on eight balls in play was 100 MPH.

He managed to get double plays in each of the first two innings to get out of trouble, the second of which came on a scorched ground ball from Shohei Ohtani 107 MPH off the bat. It went right to Ketel Marte however who started a 4-6-3 double play.

Henry seemed to settle down after that, retiring the side in order in the third and giving up just a one out single in the fourth. But knowing his pitcher was fatigued and feeling out of whack, and with the top of the Dodgers order looming in the fifth inning, Lovullo decided to cut Henry's outing short.

"I clipped Tommy early." Lovullo said. "He threw the ball pretty well. I know he was laboring a little bit and he shared a little bit of information , and we ask our guys to do that. "

So what information did Henry share exactly? This is how he described his outing and what he was telling his manager.

"Tonight was a little grindy. I felt like I was pitching upstream at times. ……I just felt a little out of whack. It’s just kind of one of those days from that standpoint. I feel like they happen. You’ve just got to fully commit to not battling yourself. There’s a temptation there to try and find it, try and search and make a ton of changes, I just tried to throw all that out the window today and compete with the hitter. "

On follow up he was asked if he was indeed fatigued, like his manager had suggested. "Yeah, I mean pitching like that when it’s not necessarily 1-2-3 innings and you’re kind of grinding out there for most of the outing can be a little bit exhausting . "

What happened next is something we've seen all too often. The bullpen gave up multiple runs, something they've done immediately after replacing the starter in four of the last five games. (Hat tip to Jesse Friedman of PHNX for this information) Andrew Saalfrank and Scott McGough combined for five walks, two doubles, and four runs allowed giving the Dodgers a 6-1 lead.

With Henry seemingly settled down in the third in the fourth innings, it's fair to question why the manager made the decision he did, especially when it blows up in his face like it so frequently does with this group of reliever. One thing you learn when covering a team every day however is to wait until you've at least had a chance to ask the manager and the players what was going in before passing judgment on the decision.

At least half of the time there is a good explanation for a decision gone wrong, but by the time we get to learn what it was, most fans have turned off the television or left the stadium. Some turn to social media to express their anger or even disgust. That's fair, and understandable. It's up to us in the media to try to get to the bottom of it and relay that information to the reader. What the reader does with that information is up to them.

One thing to know about Torey Lovullo is that when there isn't a good explanation as there was last night, he will take responsibility for the mistakes he makes. This wasn't a mistake last night however. He removed a fatigued, struggling starting pitcher and the pitchers who followed him did not do their job.

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Arizona Diamondbacks left-hander Tommy Henry (32) pitches against the Kansas City Royals at Salt River Fields on March 14, 2024.

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Hypothesis Testing
Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is most often used by scientists to test specific predictions, called hypotheses, that arise from theories. ... Stating results in a statistics assignment In our comparison of mean height between men and women we found an average difference ...
Hypothesis to Be Tested: Definition and 4 Steps for Testing with Example
Hypothesis 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 ...
Introduction to Hypothesis Testing
A hypothesis test consists of five steps: 1. State the hypotheses. State the null and alternative hypotheses. These two hypotheses need to be mutually exclusive, so if one is true then the other must be false. 2. Determine a significance level to use for the hypothesis. Decide on a significance level.
What is Hypothesis Testing in Statistics? Types and Examples
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.
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.
7.1: Basics of Hypothesis Testing
And since there is only one sample, it is usually called a one-sample z-test. Example \ (\PageIndex {2}\) battery example revisited. State the random variable and the parameter in words. State the null and alternative hypothesis and the level of significance. State and check the assumptions for a hypothesis test.
Statistical hypothesis test
The above image shows a table with some of the most common test statistics and their corresponding tests or models.. A statistical hypothesis test is a method of statistical inference used to decide whether the data sufficiently support a particular hypothesis. A statistical hypothesis test typically involves a calculation of a test statistic.Then a decision is made, either by comparing the ...
9.1: Introduction to Hypothesis Testing
In hypothesis testing, the goal is to see if there is sufficient statistical evidence to reject a presumed null hypothesis in favor of a conjectured alternative hypothesis.The null hypothesis is usually denoted $H_0$ while the alternative hypothesis is usually denoted $H_1$. An hypothesis test is a statistical decision; the conclusion will either be to reject the null hypothesis in favor ...
A Complete Guide to Hypothesis Testing
Hypothesis testing is a method of statistical inference that considers the null hypothesis H ₀ vs. the alternative hypothesis H a, where we are typically looking to assess evidence against H ₀. Such a test is used to compare data sets against one another, or compare a data set against some external standard. The former being a two sample ...
Statistics
Hypothesis testing. Hypothesis testing is a form of statistical inference that uses data from a sample to draw conclusions about a population parameter or a population probability distribution.First, a tentative assumption is made about the parameter or distribution. This assumption is called the null hypothesis and is denoted by H 0.An alternative hypothesis (denoted H a), which is the ...
T-test and Hypothesis Testing (Explained Simply)
T-test definition, formula explanation, and assumptions. The T-test is the test, which allows us to analyze one or two sample means, depending on the type of t-test. Yes, the t-test has several types: One-sample t-test — compare the mean of one group against the specified mean generated from a population. For example, a manufacturer of mobile ...
Hypothesis Testing
Step 2: State the Alternate Hypothesis. The claim is that the students have above average IQ scores, so: H 1: μ > 100. The fact that we are looking for scores "greater than" a certain point means that this is a one-tailed test. Step 3: Draw a picture to help you visualize the problem. Step 4: State the alpha level.
Significance tests (hypothesis testing)
Unit test. Significance tests give us a formal process for using sample data to evaluate the likelihood of some claim about a population value. Learn how to conduct significance tests and calculate p-values to see how likely a sample result is to occur by random chance. You'll also see how we use p-values to make conclusions about hypotheses.
8.1: The Elements of Hypothesis Testing
Definition: statistical procedure. Hypothesis testing is a statistical procedure in which a choice is made between a null hypothesis and an alternative hypothesis based on information in a sample. The end result of a hypotheses testing procedure is a choice of one of the following two possible conclusions: Reject H0.
Hypothesis Testing
Hypothesis testing in statistics refers to analyzing an assumption about a population parameter. It is used to make an educated guess about an assumption using statistics. With the use of sample data, hypothesis testing makes an assumption about how true the assumption is for the entire population from where the sample is being taken.
Understanding 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.
Choosing the Right Statistical Test
Hypothesis testing is a formal procedure for investigating our ideas about the world. It allows you to statistically test your predictions. ... Test statistics | Definition, Interpretation, and Examples The test statistic is a number, calculated from a statistical test, used to find if your data could have occurred under the null hypothesis. 251.
How Hypothesis Tests Work: Significance Levels (Alpha) and P values
Hypothesis testing is a vital process in inferential statistics where the goal is to use sample data to draw conclusions about an entire population. In the testing process, you use significance levels and p-values to determine whether the test results are statistically significant.
Hypothesis Test for a Mean
The first set of hypotheses (Set 1) is an example of a two-tailed test, since an extreme value on either side of the sampling distribution would cause a researcher to reject the null hypothesis. The other two sets of hypotheses (Sets 2 and 3) are one-tailed tests, since an extreme value on only one side of the sampling distribution would cause a researcher to reject the null hypothesis.
Hypothesis Testing for the Mean
Table 8.3: One-sided hypothesis testing for the mean: H0: μ ≤ μ0, H1: μ > μ0. Note that the tests mentioned in Table 8.3 remain valid if we replace the null hypothesis by μ = μ0. The reason for this is that in choosing the threshold c, we assumed the worst case scenario, i.e, μ = μ0 .
What Is Hypothesis? Definition, Meaning, Test, Formulation
Hypothesis is a proposition which can be put to a test to determine validity and is useful for further research. Hypothesis is a statement which can be proved or disproved. It is a statement capable of being tested. In a sense, hypothesis is a question which definitely has an answer.
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Revised on June 22, 2023. A t test is a statistical test that is used to compare the means of two groups. It is often used in hypothesis testing to determine whether a process or treatment actually has an effect on the population of interest, or whether two groups are different from one another. t test example.
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What Is Hypothesis Testing in Statistics?

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Null Hypothesis and Alternate Hypothesis

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Hypothesis Testing Calculation With Examples

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Step 2: Gather Data

Step 3: Conduct a Statistical Test

Step 4: Determine Rejection Of Your Null Hypothesis

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Hypothesis Testing and Confidence Intervals

Simple and Composite Hypothesis Testing

One-Tailed and Two-Tailed Hypothesis Testing

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Right Tailed Hypothesis Testing

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Limitations of Hypothesis Testing

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