what is composite hypothesis

Simple Hypothesis and Composite Hypothesis

A simple hypothesis is one in which all parameters of the distribution are specified. For example, the heights of college students are normally distributed with $${\sigma ^2} = 4$$, and the hypothesis that its mean $$\mu $$ is, say, $$62”$$; that is, $${H_o}:\mu = 62$$. So we have stated a simple hypothesis, as the mean and variance together specify a normal distribution completely. A simple hypothesis, in general, states that $$\theta = {\theta _o}$$ where $${\theta _o}$$ is the specified value of a parameter $$\theta $$, ($$\theta $$ may represent $$\mu ,p,{\mu _1} – {\mu _2}$$ etc).

A hypothesis which is not simple (i.e. in which not all of the parameters are specified) is called a composite hypothesis. For instance, if we hypothesize that $${H_o}:\mu > 62$$ (and $${\sigma ^2} = 4$$) or$${H_o}:\mu = 62$$ and $${\sigma ^2} < 4$$, the hypothesis becomes a composite hypothesis because we cannot know the exact distribution of the population in either case. Obviously, the parameters $$\mu > 62”$$ and$${\sigma ^2} < 4$$ have more than one value and no specified values are being assigned. The general form of a composite hypothesis is $$\theta \leqslant {\theta _o}$$ or $$\theta \geqslant {\theta _o}$$; that is, the parameter $$\theta $$ does not exceed or does not fall short of a specified value $${\theta _o}$$. The concept of simple and composite hypotheses applies to both the null hypothesis and alternative hypothesis.

Hypotheses may also be classified as exact and inexact. A hypothesis is said to be an exact hypothesis if it selects a unique value for the parameter, such as $${H_o}:\mu = 62$$ or $$p > 0.5$$. A hypothesis is called an inexact hypothesis when it indicates more than one possible value for the parameter, such as $${H_o}:\mu \ne 62$$ or $${H_o}:p = 62$$. A simple hypothesis must be exact while an exact hypothesis is not necessarily a simple hypothesis. An inexact hypothesis is a composite hypothesis.

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What does "Composite Hypothesis" mean?

Definition of Composite Hypothesis in the context of A/B testing (online controlled experiments).

What is a Composite Hypothesis?

In hypothesis testing a composite hypothesis is a hypothesis which covers a set of values from the parameter space. For example, if the entire parameter space covers -∞ to +∞ a composite hypothesis could be μ ≤ 0. It could be any other number as well, such 1, 2 or 3,1245. The alternative hypothesis is always a composite hypothesis : either one-sided hypothesis if the null is composite or a two-sided one if the null is a point null. The "composite" part means that such a hypothesis is the union of many simple point hypotheses.

In a Null Hypothesis Statistical Test only the null hypothesis can be a point hypothesis. Also, a composite hypothesis usually spans from -∞ to zero or some value of practical significance or from such a value to +∞.

Testing a composite null is what is most often of interest in an A/B testing scenario as we are usually interested in detecting and estimating effects in only one direction: either an increase in conversion rate or average revenue per user, or a decrease in unsubscribe events would be of interest and not its opposite. In fact, running a test so long as to detect a statistically significant negative outcome can result in significant business harm.

Like this glossary entry? For an in-depth and comprehensive reading on A/B testing stats, check out the book "Statistical Methods in Online A/B Testing" by the author of this glossary, Georgi Georgiev.

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Composite hypothesis

What composite hypothesis is.

Composite hypothesis is a type of statistical hypothesis test that combines two or more simple hypotheses into a single test. It is used to test the overall effect of a set of variables on a response variable.

The steps for performing a composite hypothesis test are as follows:

• State the null and alternative hypotheses:

The null hypothesis states that there is no overall effect of the set of variables on the response variable. The alternative hypothesis states that there is an overall effect of the set of variables on the response variable.

• Specify the test statistic:

The test statistic used for a composite hypothesis test is a test statistic that measures the overall effect of the set of variables on the response variable.

• Specify the distribution of the test statistic:

The distribution of the test statistic depends on the type of composite hypothesis test being performed. For example, if the composite hypothesis test is a t-test, then the distribution of the test statistic is a t-distribution.

• Calculate the test statistic:

The test statistic is calculated by applying the appropriate formula to the data.

• Compare the test statistic to the critical value:

The critical value is determined by looking up the appropriate critical value in a table or by using software. The test statistic is then compared to the critical value to determine if the null hypothesis should be rejected or not.

• Make a decision:

If the test statistic is greater than the critical value, then the null hypothesis is rejected and the alternative hypothesis is accepted. If the test statistic is less than the critical value, then the null hypothesis is not rejected and the alternative hypothesis is not accepted.

An example of a composite hypothesis in statistics is a hypothesis which states that two or more distributions have the same mean.

Another example of a composite hypothesis in statistics is a hypothesis which states that two or more groups of data have the same variance.

A third example of a composite hypothesis in statistics is a hypothesis which states that two or more populations have the same correlation.

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ALTERNATIVE

DISTRIBUTION

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Table of Contents

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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26.2 - uniformly most powerful tests.

The Neyman Pearson Lemma is all well and good for deriving the best hypothesis tests for testing a simple null hypothesis against a simple alternative hypothesis, but the reality is that we typically are interested in testing a simple null hypothesis, such as \(H_0 \colon \mu = 10\) against a composite alternative hypothesis, such as \(H_A \colon \mu > 10\). The good news is that we can extend the Neyman Pearson Lemma to account for composite alternative hypotheses, providing we take into account each simple alternative specified in H_A. Doing so creates what is called a uniformly most powerful (or UMP ) test .

A test defined by a critical region C of size \(\alpha\) is a uniformly most powerful (UMP) test if it is a most powerful test against each simple alternative in the alternative hypothesis \(H_A\). The critical region C is called a uniformly most powerful critical region of size \(\alpha\) .

Let's demonstrate by returning to the normal example from the previous page, but this time specifying a composite alternative hypothesis.

Example 26-6 Section  

Suppose \(X_1, X_2, \colon, X_n\) is a random sample from a normal population with mean \(\mu\) and variance 16. Find the test with the best critical region, that is, find the uniformly most powerful test, with a sample size of \(n = 16\) and a significance level \(\alpha\) = 0.05 to test the simple null hypothesis \(H_0: \mu = 10\) against the composite alternative hypothesis \(H_A: \mu > 10\).

For each simple alternative in \(H_A , \mu = \mu_a\), say, the ratio of the likelihood functions is:

\( \dfrac{L(10)}{L(\mu_\alpha)}= \dfrac{(32\pi)^{-16/2} exp \left[ -(1/32)\sum_{i=1}^{16}(x_i -10)^2 \right]}{(32\pi)^{-16/2} exp \left[ -(1/32)\sum_{i=1}^{16}(x_i -\mu_\alpha)^2 \right]} \le k \)

Simplifying, we get:

\(exp \left[ - \left(\dfrac{1}{32} \right) \left(\sum_{i=1}^{16}(x_i -10)^2 - \sum_{i=1}^{16}(x_i -\mu_\alpha)^2 \right) \right] \le k \)

And, simplifying yet more, we get:

Taking the natural logarithm of both sides of the inequality, collecting like terms, and multiplying through by 32, we get:

\( -2(\mu_\alpha - 10) \sum x_i +16 (\mu_{\alpha}^{2} - 10^2) \le 32 ln(k) \)

Moving the constant term on the left-side of the inequality to the right-side, and dividing through by \(-16(2(\mu_\alpha - 10)) \), we get:

\( \dfrac{1}{16} \sum x_i \ge - \dfrac{1}{16(2(\mu_\alpha - 10))}(32 ln(k) - 16(\mu_{\alpha}^{2} - 10^2)) = k^* \)

In summary, we have shown that the ratio of the likelihoods is small, that is:

\(\dfrac{L(10)}{L(\mu_\alpha)} \le k \)

if and only if:

\( \bar{x} \ge k^*\)

Therefore, the best critical region of size \(\alpha\) for testing \(H_0: \mu = 10\) against each simple alternative \(H_A \colon \mu = \mu_a\), where \(\mu_a > 10\), is given by:

\( C= \left\{ (x_1, x_1, ... , x_n): \bar{x} \ge k^* \right\} \)

where \(k^*\) is selected such that the probability of committing a Type I error is \(\alpha\), that is:

\( \alpha = P(\bar{X} \ge k^*) \text{ when } \mu = 10 \)

Because the critical region C defines a test that is most powerful against each simple alternative \(\mu_a > 10\), this is a uniformly most powerful test, and C is a uniformly most powerful critical region of size \(\alpha\).

Definition: Simple and composite hypothesis

Definition: Let $H$ be a statistical hypothesis . Then,

$H$ is called a simple hypothesis, if it completely specifies the population distribution; in this case, the sampling distribution of the test statistic is a function of sample size alone.

$H$ is called a composite hypothesis, if it does not completely specify the population distribution; for example, the hypothesis may only specify one parameter of the distribution and leave others unspecified.

• Wikipedia (2021): "Exclusion of the null hypothesis" ; in: Wikipedia, the free encyclopedia , retrieved on 2021-03-19 ; URL: https://en.wikipedia.org/wiki/Exclusion_of_the_null_hypothesis#Terminology .

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Home » What is a Hypothesis – Types, Examples and Writing Guide

What is a Hypothesis – Types, Examples and Writing Guide

Table of Contents

Definition:

Hypothesis is an educated guess or proposed explanation for a phenomenon, based on some initial observations or data. It is a tentative statement that can be tested and potentially proven or disproven through further investigation and experimentation.

Hypothesis is often used in scientific research to guide the design of experiments and the collection and analysis of data. It is an essential element of the scientific method, as it allows researchers to make predictions about the outcome of their experiments and to test those predictions to determine their accuracy.

Types of Hypothesis

Types of Hypothesis are as follows:

Research Hypothesis

A research hypothesis is a statement that predicts a relationship between variables. It is usually formulated as a specific statement that can be tested through research, and it is often used in scientific research to guide the design of experiments.

Null Hypothesis

The null hypothesis is a statement that assumes there is no significant difference or relationship between variables. It is often used as a starting point for testing the research hypothesis, and if the results of the study reject the null hypothesis, it suggests that there is a significant difference or relationship between variables.

Alternative Hypothesis

An alternative hypothesis is a statement that assumes there is a significant difference or relationship between variables. It is often used as an alternative to the null hypothesis and is tested against the null hypothesis to determine which statement is more accurate.

Directional Hypothesis

A directional hypothesis is a statement that predicts the direction of the relationship between variables. For example, a researcher might predict that increasing the amount of exercise will result in a decrease in body weight.

Non-directional Hypothesis

A non-directional hypothesis is a statement that predicts the relationship between variables but does not specify the direction. For example, a researcher might predict that there is a relationship between the amount of exercise and body weight, but they do not specify whether increasing or decreasing exercise will affect body weight.

Statistical Hypothesis

A statistical hypothesis is a statement that assumes a particular statistical model or distribution for the data. It is often used in statistical analysis to test the significance of a particular result.

Composite Hypothesis

A composite hypothesis is a statement that assumes more than one condition or outcome. It can be divided into several sub-hypotheses, each of which represents a different possible outcome.

Empirical Hypothesis

An empirical hypothesis is a statement that is based on observed phenomena or data. It is often used in scientific research to develop theories or models that explain the observed phenomena.

Simple Hypothesis

A simple hypothesis is a statement that assumes only one outcome or condition. It is often used in scientific research to test a single variable or factor.

Complex Hypothesis

A complex hypothesis is a statement that assumes multiple outcomes or conditions. It is often used in scientific research to test the effects of multiple variables or factors on a particular outcome.

Applications of Hypothesis

Hypotheses are used in various fields to guide research and make predictions about the outcomes of experiments or observations. Here are some examples of how hypotheses are applied in different fields:

• Science : In scientific research, hypotheses are used to test the validity of theories and models that explain natural phenomena. For example, a hypothesis might be formulated to test the effects of a particular variable on a natural system, such as the effects of climate change on an ecosystem.
• Medicine : In medical research, hypotheses are used to test the effectiveness of treatments and therapies for specific conditions. For example, a hypothesis might be formulated to test the effects of a new drug on a particular disease.
• Psychology : In psychology, hypotheses are used to test theories and models of human behavior and cognition. For example, a hypothesis might be formulated to test the effects of a particular stimulus on the brain or behavior.
• Sociology : In sociology, hypotheses are used to test theories and models of social phenomena, such as the effects of social structures or institutions on human behavior. For example, a hypothesis might be formulated to test the effects of income inequality on crime rates.
• Business : In business research, hypotheses are used to test the validity of theories and models that explain business phenomena, such as consumer behavior or market trends. For example, a hypothesis might be formulated to test the effects of a new marketing campaign on consumer buying behavior.
• Engineering : In engineering, hypotheses are used to test the effectiveness of new technologies or designs. For example, a hypothesis might be formulated to test the efficiency of a new solar panel design.

How to write a Hypothesis

Here are the steps to follow when writing a hypothesis:

Identify the Research Question

The first step is to identify the research question that you want to answer through your study. This question should be clear, specific, and focused. It should be something that can be investigated empirically and that has some relevance or significance in the field.

Conduct a Literature Review

Before writing your hypothesis, it’s essential to conduct a thorough literature review to understand what is already known about the topic. This will help you to identify the research gap and formulate a hypothesis that builds on existing knowledge.

Determine the Variables

The next step is to identify the variables involved in the research question. A variable is any characteristic or factor that can vary or change. There are two types of variables: independent and dependent. The independent variable is the one that is manipulated or changed by the researcher, while the dependent variable is the one that is measured or observed as a result of the independent variable.

Formulate the Hypothesis

Based on the research question and the variables involved, you can now formulate your hypothesis. A hypothesis should be a clear and concise statement that predicts the relationship between the variables. It should be testable through empirical research and based on existing theory or evidence.

Write the Null Hypothesis

The null hypothesis is the opposite of the alternative hypothesis, which is the hypothesis that you are testing. The null hypothesis states that there is no significant difference or relationship between the variables. It is important to write the null hypothesis because it allows you to compare your results with what would be expected by chance.

Refine the Hypothesis

After formulating the hypothesis, it’s important to refine it and make it more precise. This may involve clarifying the variables, specifying the direction of the relationship, or making the hypothesis more testable.

Examples of Hypothesis

Here are a few examples of hypotheses in different fields:

• Psychology : “Increased exposure to violent video games leads to increased aggressive behavior in adolescents.”
• Biology : “Higher levels of carbon dioxide in the atmosphere will lead to increased plant growth.”
• Sociology : “Individuals who grow up in households with higher socioeconomic status will have higher levels of education and income as adults.”
• Education : “Implementing a new teaching method will result in higher student achievement scores.”
• Marketing : “Customers who receive a personalized email will be more likely to make a purchase than those who receive a generic email.”
• Physics : “An increase in temperature will cause an increase in the volume of a gas, assuming all other variables remain constant.”
• Medicine : “Consuming a diet high in saturated fats will increase the risk of developing heart disease.”

Purpose of Hypothesis

The purpose of a hypothesis is to provide a testable explanation for an observed phenomenon or a prediction of a future outcome based on existing knowledge or theories. A hypothesis is an essential part of the scientific method and helps to guide the research process by providing a clear focus for investigation. It enables scientists to design experiments or studies to gather evidence and data that can support or refute the proposed explanation or prediction.

The formulation of a hypothesis is based on existing knowledge, observations, and theories, and it should be specific, testable, and falsifiable. A specific hypothesis helps to define the research question, which is important in the research process as it guides the selection of an appropriate research design and methodology. Testability of the hypothesis means that it can be proven or disproven through empirical data collection and analysis. Falsifiability means that the hypothesis should be formulated in such a way that it can be proven wrong if it is incorrect.

In addition to guiding the research process, the testing of hypotheses can lead to new discoveries and advancements in scientific knowledge. When a hypothesis is supported by the data, it can be used to develop new theories or models to explain the observed phenomenon. When a hypothesis is not supported by the data, it can help to refine existing theories or prompt the development of new hypotheses to explain the phenomenon.

When to use Hypothesis

Here are some common situations in which hypotheses are used:

• In scientific research , hypotheses are used to guide the design of experiments and to help researchers make predictions about the outcomes of those experiments.
• In social science research , hypotheses are used to test theories about human behavior, social relationships, and other phenomena.
• I n business , hypotheses can be used to guide decisions about marketing, product development, and other areas. For example, a hypothesis might be that a new product will sell well in a particular market, and this hypothesis can be tested through market research.

Characteristics of Hypothesis

Here are some common characteristics of a hypothesis:

• Testable : A hypothesis must be able to be tested through observation or experimentation. This means that it must be possible to collect data that will either support or refute the hypothesis.
• Falsifiable : A hypothesis must be able to be proven false if it is not supported by the data. If a hypothesis cannot be falsified, then it is not a scientific hypothesis.
• Clear and concise : A hypothesis should be stated in a clear and concise manner so that it can be easily understood and tested.
• Based on existing knowledge : A hypothesis should be based on existing knowledge and research in the field. It should not be based on personal beliefs or opinions.
• Specific : A hypothesis should be specific in terms of the variables being tested and the predicted outcome. This will help to ensure that the research is focused and well-designed.
• Tentative: A hypothesis is a tentative statement or assumption that requires further testing and evidence to be confirmed or refuted. It is not a final conclusion or assertion.
• Relevant : A hypothesis should be relevant to the research question or problem being studied. It should address a gap in knowledge or provide a new perspective on the issue.

Advantages of Hypothesis

Hypotheses have several advantages in scientific research and experimentation:

• Guides research: A hypothesis provides a clear and specific direction for research. It helps to focus the research question, select appropriate methods and variables, and interpret the results.
• Predictive powe r: A hypothesis makes predictions about the outcome of research, which can be tested through experimentation. This allows researchers to evaluate the validity of the hypothesis and make new discoveries.
• Facilitates communication: A hypothesis provides a common language and framework for scientists to communicate with one another about their research. This helps to facilitate the exchange of ideas and promotes collaboration.
• Efficient use of resources: A hypothesis helps researchers to use their time, resources, and funding efficiently by directing them towards specific research questions and methods that are most likely to yield results.
• Provides a basis for further research: A hypothesis that is supported by data provides a basis for further research and exploration. It can lead to new hypotheses, theories, and discoveries.
• Increases objectivity: A hypothesis can help to increase objectivity in research by providing a clear and specific framework for testing and interpreting results. This can reduce bias and increase the reliability of research findings.

Limitations of Hypothesis

Some Limitations of the Hypothesis are as follows:

• Limited to observable phenomena: Hypotheses are limited to observable phenomena and cannot account for unobservable or intangible factors. This means that some research questions may not be amenable to hypothesis testing.
• May be inaccurate or incomplete: Hypotheses are based on existing knowledge and research, which may be incomplete or inaccurate. This can lead to flawed hypotheses and erroneous conclusions.
• May be biased: Hypotheses may be biased by the researcher’s own beliefs, values, or assumptions. This can lead to selective interpretation of data and a lack of objectivity in research.
• Cannot prove causation: A hypothesis can only show a correlation between variables, but it cannot prove causation. This requires further experimentation and analysis.
• Limited to specific contexts: Hypotheses are limited to specific contexts and may not be generalizable to other situations or populations. This means that results may not be applicable in other contexts or may require further testing.
• May be affected by chance : Hypotheses may be affected by chance or random variation, which can obscure or distort the true relationship between variables.

About the author

Muhammad Hassan

Researcher, Academic Writer, Web developer

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What is the difference between Simple and Composite Hypothesis.

What is a hypothesis?

A h ypothesis is an educated guess about how something works. In the scientific method, a hypothesis is an idea that can be tested. If the hypothesis is correct, then the experiment will support the hypothesis. If the hypothesis is incorrect, the experiment will not support the hypothesis.

A hypothesis is simple if it specifies the population completely, i.e., it specifies the population distribution uniquely, while a composite hypothesis leads to two or more possibilities.

Before diving further into their differences, let’s first define a few terms that are handy in understanding the concept of a hypothesis.

Let’s dive in;

Difference between hypothesis and theory.

A hypothesis is a proposed explanation for a phenomenon. A scientific theory is a well-substantiated explanation for an aspect of the natural world supported by a vast body of evidence. Theories are generally much broader in scope than hypotheses and are often not as specific.

The objective of statistics is to make inferences about a population based on information contained in the sample.

There are two major areas of statistical inference, namely;

• Estimation of parameter
• Hypothesis testing

We will develop general methods for testing hypotheses and then apply them to common problems.

Statistical hypothesis

A statistical hypothesis is a testable statement about a population parameter. The statement is based on an assumption about the population parameter. This assumption is usually made about the population parameters based on past research or experience. The statistical hypothesis is used to make predictions about future events. These predictions are based on the assumption that the population parameters will remain the same.

A statistical hypothesis is about a population parameter, usually denoted by some symbol, such as μ or θ.

Statistical hypothesis testing is a method of statistical inference. There are two types of statistical hypothesis tests:

• A point null hypothesis specifies that a population parameter (such as the mean) equals a specific value. For example, the null hypothesis could be that μ=0.
• A composite null hypothesis specifies that a population parameter is less than, greater than, or not equal to a specific value. For example, the null hypothesis could be that μ≠0.

The alternative hypothesis is the hypothesis that is being tested against the null hypothesis. The alternative hypothesis could be that μ>0 or μ<0.

A statistical hypothesis test determines whether or not to reject the null hypothesis. The null hypothesis is rejected if the test statistic is greater than or less than the critical value.

Hypothesis Testing

A hypothesis is a statement or claims about how two variables are related. Hypothesis testing is a statistical procedure used to assess whether the null hypothesis—a statement that there is no difference between two groups or no association between two variables—can be rejected based on sample data. There are four steps in hypothesis testing:

• State the null and alternative hypotheses.
• Select a significance level.
• Calculate the test statistic.
• Interpret the results.

The first step is to state the null and alternative hypotheses. The null hypothesis is that the two variables have no difference or association. The alternative hypothesis is the statement that there is a difference or an association between two variables.

The second step is to select a significance level. The significance level is the probability of rejecting the null hypothesis when it is true. The most common significance levels are 0.05 and 0.01.

The third step is to calculate the test statistic. The test statistic measures the difference between the null and alternative hypotheses. There are many different test statistics, and the choice of test statistic depends on the data type and hypothesis test.

The fourth and final step is to interpret the results. The results of a hypothesis test are either significant or not significant. A significant result means that the null hypothesis can be rejected. A non-significant result means that the null hypothesis cannot be rejected.

Null Hypothesis vs Alternative Hypothesis

In statistics, a null hypothesis is a statement one seeks to disprove, reject or nullify. Most commonly, it is a statement that the phenomenon being studied produces no effect or makes no difference. For example, if one were testing the efficacy of a new drug, the null hypothesis would be that the drug does not affect the treated condition.

The null hypothesis is usually denoted H0, and the alternate hypothesis is denoted H1. If the null hypothesis is rejected in favor of the alternative hypothesis, it is said to be “statistically significant.” The null hypothesis is often assumed to be true until it can be proved otherwise.

Many different types of tests can be used to test a null hypothesis. The most common is the Student’s t-test, which compares the means of two groups. If the t-test is significant, there is a statistically significant difference between the two groups.

Other tests that can be used to test the null hypothesis include the chi-square, Fisher’s exact, and Wilcoxon rank-sum tests.

The alternative hypothesis is the hypothesis that is being tested in a statistical test. This is the hypothesis that is the opposite of the null hypothesis. We are trying to find evidence for the alternative hypothesis in a test.

Simple and Composite Hypothesis

Simple hypothesis.

Hypotheses can be composite or simple, and both are useful depending on the research question and the available evidence.

A simple hypothesis is a straightforward statement that proposes a relationship between two variables. It is a clear, concise statement that is easy to test and evaluate. A simple hypothesis is often used in experimental research where the researcher wants to test the effect of one variable on another.

Examples of hypothesis :

An example of a simple hypothesis is “students who study more will get better grades.” This hypothesis proposes a direct relationship between the amount of time a student spends studying and their academic performance. This hypothesis is testable by comparing the grades of students who study more with those who study less.

Another example of a simple hypothesis is “increased exposure to sunlight will result in higher vitamin D levels.” This hypothesis proposes a direct relationship between sunlight exposure and vitamin D levels. This hypothesis is testable by measuring the vitamin D levels of individuals with varying levels of sunlight exposure.

Simple hypotheses are advantageous because they are easy to test and evaluate. They also allow researchers to focus on a specific research question and avoid unnecessary complexity. Simple hypotheses are particularly useful in experimental research where researchers manipulate one variable to observe its effect on another.

However, simple hypotheses also have limitations. They may oversimplify complex phenomena, and their results may not generalize to a larger population. The available evidence may also limit simple hypotheses, and additional research may be necessary to understand the relationship between variables fully.

In essence, a simple hypothesis is a straightforward statement that proposes a relationship between two variables. Simple hypotheses are useful in experimental research and allow researchers to focus on a specific research question. However, simple hypotheses also have limitations and should be evaluated in the context of the available evidence and research question.

Composite Hypothesis

A composite hypothesis, on the other hand, proposes multiple relationships between two or more variables. For example, a composite hypothesis might state that “there is a significant difference between the average heights of men and women, and there is also a significant difference between the average heights of people from different continents.”

Composite hypothesis testing is a statistical technique used to determine the probability of an event or phenomenon based on observed data. This technique is often used in scientific research, quality control, and decision-making processes where the outcome of a particular experiment or test is uncertain.

A composite hypothesis is an alternative hypothesis encompassing a range of possible outcomes. It is defined as a hypothesis with more than one parameter value. For example, if we are testing the hypothesis that the mean of a population is greater than a certain value, we could define the composite hypothesis as follows:

H1: μ > μ0, where μ is the population means, and μ0 is the hypothesized value of the mean.

The composite hypothesis, in this case, includes all values of μ greater than μ0. This means we are not specifying a specific value of μ, but rather a range of possible values.

Composite hypothesis testing involves evaluating the probability of observing a particular result under the null hypothesis and then comparing it to the probability of observing the same result under the composite hypothesis. The result is considered significant if the probability of observing it under the composite hypothesis is sufficiently low.

We use statistical tests such as the t-test, F-test, or chi-square test to test a composite hypothesis. Given the null hypothesis and the observed data, these tests allow us to calculate the probability of observing a particular result.

In conclusion, composite hypothesis testing is a valuable statistical technique used to determine the probability of an event or phenomenon based on observed data. It allows us to test hypotheses that encompass a range of possible outcomes and is an essential tool for scientific research, quality control , and decision-making processes.

Understanding composite hypothesis testing is essential for anyone working in these fields and can help ensure that decisions are made based on solid statistical evidence.

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Detecting 'Hawking radiation' from black holes using today's telescopes

by David Appell , Phys.org

In 1974 Stephen Hawking famously claimed that black holes should emit particles as well as absorb them. This so-called "Hawking radiation" has not yet been observed, but now a research group from Europe has found that Hawking radiation should be observable by existing telescopes that are capable of detecting very high energy particles of light.

When two massive black holes collide and merge, or a neutron star and black hole do so, they emit gravitational waves, undulations in the fabric of spacetime that travel outward. Some of these waves wash over Earth millions or billions of years later. These waves were predicted by Einstein in 1916 and first directly observed by the LIGO detectors in 2016. Dozens of gravitational waves from black hole mergers have been detected since .

These mergers also emit a number of "black hole morsels," smaller black holes with masses of the order of an asteroid, created in the resulting extremely strong gravitational field around the merger due to so-called "nonlinear," high velocity effects in general relativity. These nonlinearities arise due to the inherently complex solutions to Einstein's equations, as warped spacetime and masses feedback on one another and both respond to and create new spacetime and masses.

This complexity also generates gamma ray bursts of extremely energetic photons. These bursts have similar characteristics, with a time delay from the merger of the order of their evaporation time. A morsel mass of 20 kilotons has an evaporation lifetime of 16 years, but this number can change drastically since the evaporation time is proportional to the morsel mass cubed.

Heavier morsels will initially provide a steady gamma ray burst signal, characterized by reduced particle energies, proportional to the Hawking temperature. The Hawking temperature is inversely proportional to a black hole's mass.

The research team showed, through numerical calculations using an open source public code called BlackHawk that calculates the Hawking evaporation spectra for any distribution of black holes, that the Hawking radiation from the black hole morsels creates gamma ray bursts that have a distinctive fingerprint. The work is published on the arXiv preprint server.

Detecting such events, which have multiple signals— gravitational waves , electromagnetic radiation , neutrino emissions —is called multimessenger astronomy in the astrophysical community, and is part of the observing programs at the LIGO gravitational wave detectors in the US, VIRGO in Italy and, in Japan, the KAGRA gravitational wave telescope.

Visible signals from black hole evaporation always include photons above the TeV range (a trillion electron volts, about 0.2 microjoules; for example, the CERN Large Hadron Collider in Europe, the largest particle accelerator on the planet, collides protons head-on with a total energy of 13.6 TeV). This provides a "golden opportunity," the group writes, for so-called high energy atmospheric Cherenkov telescopes to detect this Hawking radiation.

These Cherenkov telescopes are ground-based antenna dishes that can detect very energetic photons (gamma rays) in the energy range of 50 GeV (billion electron volts) to 50 TeV. These antennae accomplish that by detecting Cherenkov radiation flashes that are produced as the gamma rays cascade through the Earth's atmosphere, traveling faster than the ordinary wave velocity of light in air.

Recall that light travels slightly slower in air than it does in a vacuum, because air has an index of refraction slightly greater than one. The Hawking gamma ray radiation cascading down through the atmosphere exceeds this slower value, creating Cerenkov radiation (also called braking radiation—Bremsstrahlung in German). The blue light seen in pools of water that surround reaction rods in a nuclear reactor is an example of Cerenkov radiation.

There are now four telescopes that can detect these cascades of Cerenkov radiation—the High Energy Stereoscopic System (HESS) in Namibia, the Major Atmospheric Gamma Imaging Cherenkov Telescopes (MAGIC) on one of the Canary Islands, the First G-APD Cherenkov Telescope (FACT), also on La Palma Island in the Canary archipelago, and Very Energetic Radiation Imaging Telescope Array System (VERITAS) in Arizona. Though each uses different technology, they all can detect Cerenkov photons in the GeV-TeV energy range.

Detecting such Hawking radiation would also shed light (ahem…) on the production of black hole morsels, as well as particle production at energies higher than can be attained on Earth, and may carry signs of new physics such as supersymmetry, extra dimensions, or the existence of composite particles based on the strong force.

"It was a surprise to find that black hole morsels can radiate above the detection capabilities of current high energy Cherenkov telescopes on Earth," said Giacomo Cacciapaglia, lead author from the Université Lyon Claude Bernard 1 in Lyon, France. Noting that direct detection of Hawking radiation from black hole morsels would be the first evidence of the quantum behavior of black holes, he said "if the proposed signal is observed, we will have to question the current knowledge of the nature of black holes" and morsel production.

Cacciapaglia said they plan to contact colleagues from experimental groups, then to use the data collected to search for the Hawking radiation they propose.

Journal information: arXiv

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