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• Null and Alternative Hypotheses | Definitions & Examples

Null & Alternative Hypotheses | Definitions, Templates & Examples

Published on May 6, 2022 by Shaun Turney . Revised on June 22, 2023.

The null and alternative hypotheses are two competing claims that researchers weigh evidence for and against using a statistical test :

• Null hypothesis ( H 0 ): There’s no effect in the population .
• Alternative hypothesis ( H a or H 1 ) : There’s an effect in the population.

Table of contents

Answering your research question with hypotheses, what is a null hypothesis, what is an alternative hypothesis, similarities and differences between null and alternative hypotheses, how to write null and alternative hypotheses, other interesting articles, frequently asked questions.

The null and alternative hypotheses offer competing answers to your research question . When the research question asks “Does the independent variable affect the dependent variable?”:

• The null hypothesis ( H 0 ) answers “No, there’s no effect in the population.”
• The alternative hypothesis ( H a ) answers “Yes, there is an effect in the population.”

The null and alternative are always claims about the population. That’s because the goal of hypothesis testing is to make inferences about a population based on a sample . Often, we infer whether there’s an effect in the population by looking at differences between groups or relationships between variables in the sample. It’s critical for your research to write strong hypotheses .

You can use a statistical test to decide whether the evidence favors the null or alternative hypothesis. Each type of statistical test comes with a specific way of phrasing the null and alternative hypothesis. However, the hypotheses can also be phrased in a general way that applies to any test.

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The null hypothesis is the claim that there’s no effect in the population.

If the sample provides enough evidence against the claim that there’s no effect in the population ( p ≤ α), then we can reject the null hypothesis . Otherwise, we fail to reject the null hypothesis.

Although “fail to reject” may sound awkward, it’s the only wording that statisticians accept . Be careful not to say you “prove” or “accept” the null hypothesis.

Null hypotheses often include phrases such as “no effect,” “no difference,” or “no relationship.” When written in mathematical terms, they always include an equality (usually =, but sometimes ≥ or ≤).

You can never know with complete certainty whether there is an effect in the population. Some percentage of the time, your inference about the population will be incorrect. When you incorrectly reject the null hypothesis, it’s called a type I error . When you incorrectly fail to reject it, it’s a type II error.

Examples of null hypotheses

The table below gives examples of research questions and null hypotheses. There’s always more than one way to answer a research question, but these null hypotheses can help you get started.

*Note that some researchers prefer to always write the null hypothesis in terms of “no effect” and “=”. It would be fine to say that daily meditation has no effect on the incidence of depression and p 1 = p 2 .

The alternative hypothesis ( H a ) is the other answer to your research question . It claims that there’s an effect in the population.

Often, your alternative hypothesis is the same as your research hypothesis. In other words, it’s the claim that you expect or hope will be true.

The alternative hypothesis is the complement to the null hypothesis. Null and alternative hypotheses are exhaustive, meaning that together they cover every possible outcome. They are also mutually exclusive, meaning that only one can be true at a time.

Alternative hypotheses often include phrases such as “an effect,” “a difference,” or “a relationship.” When alternative hypotheses are written in mathematical terms, they always include an inequality (usually ≠, but sometimes < or >). As with null hypotheses, there are many acceptable ways to phrase an alternative hypothesis.

Examples of alternative hypotheses

The table below gives examples of research questions and alternative hypotheses to help you get started with formulating your own.

Null and alternative hypotheses are similar in some ways:

• They’re both answers to the research question.
• They both make claims about the population.
• They’re both evaluated by statistical tests.

However, there are important differences between the two types of hypotheses, summarized in the following table.

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To help you write your hypotheses, you can use the template sentences below. If you know which statistical test you’re going to use, you can use the test-specific template sentences. Otherwise, you can use the general template sentences.

General template sentences

The only thing you need to know to use these general template sentences are your dependent and independent variables. To write your research question, null hypothesis, and alternative hypothesis, fill in the following sentences with your variables:

Does independent variable affect dependent variable ?

• Null hypothesis ( H 0 ): Independent variable does not affect dependent variable.
• Alternative hypothesis ( H a ): Independent variable affects dependent variable.

Test-specific template sentences

Once you know the statistical test you’ll be using, you can write your hypotheses in a more precise and mathematical way specific to the test you chose. The table below provides template sentences for common statistical tests.

Note: The template sentences above assume that you’re performing one-tailed tests . One-tailed tests are appropriate for most studies.

If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.

• Normal distribution
• Descriptive statistics
• Measures of central tendency
• Correlation coefficient

Methodology

• Cluster sampling
• Stratified sampling
• Types of interviews
• Cohort study
• Thematic analysis

Research bias

• Implicit bias
• Cognitive bias
• Survivorship bias
• Availability heuristic
• Nonresponse bias
• Regression to the mean

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is used by scientists to test specific predictions, called hypotheses , by calculating how likely it is that a pattern or relationship between variables could have arisen by chance.

Null and alternative hypotheses are used in statistical hypothesis testing . The null hypothesis of a test always predicts no effect or no relationship between variables, while the alternative hypothesis states your research prediction of an effect or relationship.

The null hypothesis is often abbreviated as H 0 . When the null hypothesis is written using mathematical symbols, it always includes an equality symbol (usually =, but sometimes ≥ or ≤).

The alternative hypothesis is often abbreviated as H a or H 1 . When the alternative hypothesis is written using mathematical symbols, it always includes an inequality symbol (usually ≠, but sometimes < or >).

A research hypothesis is your proposed answer to your research question. The research hypothesis usually includes an explanation (“ x affects y because …”).

A statistical hypothesis, on the other hand, is a mathematical statement about a population parameter. Statistical hypotheses always come in pairs: the null and alternative hypotheses . In a well-designed study , the statistical hypotheses correspond logically to the research hypothesis.

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What is The Null Hypothesis & When Do You Reject The Null Hypothesis

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On This Page:

A null hypothesis is a statistical concept suggesting no significant difference or relationship between measured variables. It’s the default assumption unless empirical evidence proves otherwise.

The null hypothesis states no relationship exists between the two variables being studied (i.e., one variable does not affect the other).

The null hypothesis is the statement that a researcher or an investigator wants to disprove.

Testing the null hypothesis can tell you whether your results are due to the effects of manipulating ​ the dependent variable or due to random chance. 

How to Write a Null Hypothesis

Null hypotheses (H0) start as research questions that the investigator rephrases as statements indicating no effect or relationship between the independent and dependent variables.

It is a default position that your research aims to challenge or confirm.

For example, if studying the impact of exercise on weight loss, your null hypothesis might be:

There is no significant difference in weight loss between individuals who exercise daily and those who do not.

Examples of Null Hypotheses

When Do We Reject The Null Hypothesis? 

We reject the null hypothesis when the data provide strong enough evidence to conclude that it is likely incorrect. This often occurs when the p-value (probability of observing the data given the null hypothesis is true) is below a predetermined significance level.

If the collected data does not meet the expectation of the null hypothesis, a researcher can conclude that the data lacks sufficient evidence to back up the null hypothesis, and thus the null hypothesis is rejected. 

Rejecting the null hypothesis means that a relationship does exist between a set of variables and the effect is statistically significant ( p > 0.05).

If the data collected from the random sample is not statistically significance , then the null hypothesis will be accepted, and the researchers can conclude that there is no relationship between the variables. 

You need to perform a statistical test on your data in order to evaluate how consistent it is with the null hypothesis. A p-value is one statistical measurement used to validate a hypothesis against observed data.

Calculating the p-value is a critical part of null-hypothesis significance testing because it quantifies how strongly the sample data contradicts the null hypothesis.

The level of statistical significance is often expressed as a  p  -value between 0 and 1. The smaller the p-value, the stronger the evidence that you should reject the null hypothesis.

Usually, a researcher uses a confidence level of 95% or 99% (p-value of 0.05 or 0.01) as general guidelines to decide if you should reject or keep the null.

When your p-value is less than or equal to your significance level, you reject the null hypothesis.

In other words, smaller p-values are taken as stronger evidence against the null hypothesis. Conversely, when the p-value is greater than your significance level, you fail to reject the null hypothesis.

In this case, the sample data provides insufficient data to conclude that the effect exists in the population.

Because you can never know with complete certainty whether there is an effect in the population, your inferences about a population will sometimes be incorrect.

When you incorrectly reject the null hypothesis, it’s called a type I error. When you incorrectly fail to reject it, it’s called a type II error.

Why Do We Never Accept The Null Hypothesis?

The reason we do not say “accept the null” is because we are always assuming the null hypothesis is true and then conducting a study to see if there is evidence against it. And, even if we don’t find evidence against it, a null hypothesis is not accepted.

A lack of evidence only means that you haven’t proven that something exists. It does not prove that something doesn’t exist. 

It is risky to conclude that the null hypothesis is true merely because we did not find evidence to reject it. It is always possible that researchers elsewhere have disproved the null hypothesis, so we cannot accept it as true, but instead, we state that we failed to reject the null. 

One can either reject the null hypothesis, or fail to reject it, but can never accept it.

Why Do We Use The Null Hypothesis?

We can never prove with 100% certainty that a hypothesis is true; We can only collect evidence that supports a theory. However, testing a hypothesis can set the stage for rejecting or accepting this hypothesis within a certain confidence level.

The null hypothesis is useful because it can tell us whether the results of our study are due to random chance or the manipulation of a variable (with a certain level of confidence).

A null hypothesis is rejected if the measured data is significantly unlikely to have occurred and a null hypothesis is accepted if the observed outcome is consistent with the position held by the null hypothesis.

Rejecting the null hypothesis sets the stage for further experimentation to see if a relationship between two variables exists. 

Hypothesis testing is a critical part of the scientific method as it helps decide whether the results of a research study support a particular theory about a given population. Hypothesis testing is a systematic way of backing up researchers’ predictions with statistical analysis.

It helps provide sufficient statistical evidence that either favors or rejects a certain hypothesis about the population parameter. 

Purpose of a Null Hypothesis 

• The primary purpose of the null hypothesis is to disprove an assumption. 
• Whether rejected or accepted, the null hypothesis can help further progress a theory in many scientific cases.
• A null hypothesis can be used to ascertain how consistent the outcomes of multiple studies are.

Do you always need both a Null Hypothesis and an Alternative Hypothesis?

The null (H0) and alternative (Ha or H1) hypotheses are two competing claims that describe the effect of the independent variable on the dependent variable. They are mutually exclusive, which means that only one of the two hypotheses can be true. 

While the null hypothesis states that there is no effect in the population, an alternative hypothesis states that there is statistical significance between two variables. 

The goal of hypothesis testing is to make inferences about a population based on a sample. In order to undertake hypothesis testing, you must express your research hypothesis as a null and alternative hypothesis. Both hypotheses are required to cover every possible outcome of the study. 

What is the difference between a null hypothesis and an alternative hypothesis?

The alternative hypothesis is the complement to the null hypothesis. The null hypothesis states that there is no effect or no relationship between variables, while the alternative hypothesis claims that there is an effect or relationship in the population.

It is the claim that you expect or hope will be true. The null hypothesis and the alternative hypothesis are always mutually exclusive, meaning that only one can be true at a time.

What are some problems with the null hypothesis?

One major problem with the null hypothesis is that researchers typically will assume that accepting the null is a failure of the experiment. However, accepting or rejecting any hypothesis is a positive result. Even if the null is not refuted, the researchers will still learn something new.

Why can a null hypothesis not be accepted?

We can either reject or fail to reject a null hypothesis, but never accept it. If your test fails to detect an effect, this is not proof that the effect doesn’t exist. It just means that your sample did not have enough evidence to conclude that it exists.

We can’t accept a null hypothesis because a lack of evidence does not prove something that does not exist. Instead, we fail to reject it.

Failing to reject the null indicates that the sample did not provide sufficient enough evidence to conclude that an effect exists.

If the p-value is greater than the significance level, then you fail to reject the null hypothesis.

Is a null hypothesis directional or non-directional?

A hypothesis test can either contain an alternative directional hypothesis or a non-directional alternative hypothesis. A directional hypothesis is one that contains the less than (“<“) or greater than (“>”) sign.

A nondirectional hypothesis contains the not equal sign (“≠”).  However, a null hypothesis is neither directional nor non-directional.

A null hypothesis is a prediction that there will be no change, relationship, or difference between two variables.

The directional hypothesis or nondirectional hypothesis would then be considered alternative hypotheses to the null hypothesis.

Gill, J. (1999). The insignificance of null hypothesis significance testing.  Political research quarterly ,  52 (3), 647-674.

Krueger, J. (2001). Null hypothesis significance testing: On the survival of a flawed method.  American Psychologist ,  56 (1), 16.

Masson, M. E. (2011). A tutorial on a practical Bayesian alternative to null-hypothesis significance testing.  Behavior research methods ,  43 , 679-690.

Nickerson, R. S. (2000). Null hypothesis significance testing: a review of an old and continuing controversy.  Psychological methods ,  5 (2), 241.

Rozeboom, W. W. (1960). The fallacy of the null-hypothesis significance test.  Psychological bulletin ,  57 (5), 416.

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6a.1 - introduction to hypothesis testing, basic terms section  .

The first step in hypothesis testing is to set up two competing hypotheses. The hypotheses are the most important aspect. If the hypotheses are incorrect, your conclusion will also be incorrect.

The two hypotheses are named the null hypothesis and the alternative hypothesis.

The goal of hypothesis testing is to see if there is enough evidence against the null hypothesis. In other words, to see if there is enough evidence to reject the null hypothesis. If there is not enough evidence, then we fail to reject the null hypothesis.

Consider the following example where we set up these hypotheses.

Example 6-1 Section  

A man, Mr. Orangejuice, goes to trial and is tried for the murder of his ex-wife. He is either guilty or innocent. Set up the null and alternative hypotheses for this example.

Putting this in a hypothesis testing framework, the hypotheses being tested are:

• The man is guilty
• The man is innocent

Let's set up the null and alternative hypotheses.

\(H_0\colon \) Mr. Orangejuice is innocent

\(H_a\colon \) Mr. Orangejuice is guilty

Remember that we assume the null hypothesis is true and try to see if we have evidence against the null. Therefore, it makes sense in this example to assume the man is innocent and test to see if there is evidence that he is guilty.

The Logic of Hypothesis Testing Section  

We want to know the answer to a research question. We determine our null and alternative hypotheses. Now it is time to make a decision.

The decision is either going to be...

• reject the null hypothesis or...
• fail to reject the null hypothesis.

Consider the following table. The table shows the decision/conclusion of the hypothesis test and the unknown "reality", or truth. We do not know if the null is true or if it is false. If the null is false and we reject it, then we made the correct decision. If the null hypothesis is true and we fail to reject it, then we made the correct decision.

So what happens when we do not make the correct decision?

When doing hypothesis testing, two types of mistakes may be made and we call them Type I error and Type II error. If we reject the null hypothesis when it is true, then we made a type I error. If the null hypothesis is false and we failed to reject it, we made another error called a Type II error.

Types of errors

The “reality”, or truth, about the null hypothesis is unknown and therefore we do not know if we have made the correct decision or if we committed an error. We can, however, define the likelihood of these events.

\(\alpha\) and \(\beta\) are probabilities of committing an error so we want these values to be low. However, we cannot decrease both. As \(\alpha\) decreases, \(\beta\) increases.

Example 6-1 Cont'd... Section  

A man, Mr. Orangejuice, goes to trial and is tried for the murder of his ex-wife. He is either guilty or not guilty. We found before that...

• \( H_0\colon \) Mr. Orangejuice is innocent
• \( H_a\colon \) Mr. Orangejuice is guilty

Interpret Type I error, \(\alpha \), Type II error, \(\beta \).

As you can see here, the Type I error (putting an innocent man in jail) is the more serious error. Ethically, it is more serious to put an innocent man in jail than to let a guilty man go free. So to minimize the probability of a type I error we would choose a smaller significance level.

Try it! Section  

An inspector has to choose between certifying a building as safe or saying that the building is not safe. There are two hypotheses:

• Building is safe
• Building is not safe

Set up the null and alternative hypotheses. Interpret Type I and Type II error.

\( H_0\colon\) Building is not safe vs \(H_a\colon \) Building is safe

Power and \(\beta \) are complements of each other. Therefore, they have an inverse relationship, i.e. as one increases, the other decreases.

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• Knowledge Base
• Null and Alternative Hypotheses | Definitions & Examples

Null and Alternative Hypotheses | Definitions & Examples

Published on 5 October 2022 by Shaun Turney . Revised on 6 December 2022.

The null and alternative hypotheses are two competing claims that researchers weigh evidence for and against using a statistical test :

• Null hypothesis (H 0 ): There’s no effect in the population .
• Alternative hypothesis (H A ): There’s an effect in the population.

The effect is usually the effect of the independent variable on the dependent variable .

Table of contents

Answering your research question with hypotheses, what is a null hypothesis, what is an alternative hypothesis, differences between null and alternative hypotheses, how to write null and alternative hypotheses, frequently asked questions about null and alternative hypotheses.

The null and alternative hypotheses offer competing answers to your research question . When the research question asks “Does the independent variable affect the dependent variable?”, the null hypothesis (H 0 ) answers “No, there’s no effect in the population.” On the other hand, the alternative hypothesis (H A ) answers “Yes, there is an effect in the population.”

The null and alternative are always claims about the population. That’s because the goal of hypothesis testing is to make inferences about a population based on a sample . Often, we infer whether there’s an effect in the population by looking at differences between groups or relationships between variables in the sample.

You can use a statistical test to decide whether the evidence favors the null or alternative hypothesis. Each type of statistical test comes with a specific way of phrasing the null and alternative hypothesis. However, the hypotheses can also be phrased in a general way that applies to any test.

The null hypothesis is the claim that there’s no effect in the population.

If the sample provides enough evidence against the claim that there’s no effect in the population ( p ≤ α), then we can reject the null hypothesis . Otherwise, we fail to reject the null hypothesis.

Although “fail to reject” may sound awkward, it’s the only wording that statisticians accept. Be careful not to say you “prove” or “accept” the null hypothesis.

Null hypotheses often include phrases such as “no effect”, “no difference”, or “no relationship”. When written in mathematical terms, they always include an equality (usually =, but sometimes ≥ or ≤).

Examples of null hypotheses

The table below gives examples of research questions and null hypotheses. There’s always more than one way to answer a research question, but these null hypotheses can help you get started.

*Note that some researchers prefer to always write the null hypothesis in terms of “no effect” and “=”. It would be fine to say that daily meditation has no effect on the incidence of depression and p 1 = p 2 .

The alternative hypothesis (H A ) is the other answer to your research question . It claims that there’s an effect in the population.

Often, your alternative hypothesis is the same as your research hypothesis. In other words, it’s the claim that you expect or hope will be true.

The alternative hypothesis is the complement to the null hypothesis. Null and alternative hypotheses are exhaustive, meaning that together they cover every possible outcome. They are also mutually exclusive, meaning that only one can be true at a time.

Alternative hypotheses often include phrases such as “an effect”, “a difference”, or “a relationship”. When alternative hypotheses are written in mathematical terms, they always include an inequality (usually ≠, but sometimes > or <). As with null hypotheses, there are many acceptable ways to phrase an alternative hypothesis.

Examples of alternative hypotheses

The table below gives examples of research questions and alternative hypotheses to help you get started with formulating your own.

Null and alternative hypotheses are similar in some ways:

• They’re both answers to the research question
• They both make claims about the population
• They’re both evaluated by statistical tests.

However, there are important differences between the two types of hypotheses, summarized in the following table.

To help you write your hypotheses, you can use the template sentences below. If you know which statistical test you’re going to use, you can use the test-specific template sentences. Otherwise, you can use the general template sentences.

The only thing you need to know to use these general template sentences are your dependent and independent variables. To write your research question, null hypothesis, and alternative hypothesis, fill in the following sentences with your variables:

Does independent variable affect dependent variable ?

• Null hypothesis (H 0 ): Independent variable does not affect dependent variable .
• Alternative hypothesis (H A ): Independent variable affects dependent variable .

Test-specific

Once you know the statistical test you’ll be using, you can write your hypotheses in a more precise and mathematical way specific to the test you chose. The table below provides template sentences for common statistical tests.

Note: The template sentences above assume that you’re performing one-tailed tests . One-tailed tests are appropriate for most studies.

The null hypothesis is often abbreviated as H 0 . When the null hypothesis is written using mathematical symbols, it always includes an equality symbol (usually =, but sometimes ≥ or ≤).

The alternative hypothesis is often abbreviated as H a or H 1 . When the alternative hypothesis is written using mathematical symbols, it always includes an inequality symbol (usually ≠, but sometimes < or >).

A research hypothesis is your proposed answer to your research question. The research hypothesis usually includes an explanation (‘ x affects y because …’).

A statistical hypothesis, on the other hand, is a mathematical statement about a population parameter. Statistical hypotheses always come in pairs: the null and alternative hypotheses. In a well-designed study , the statistical hypotheses correspond logically to the research hypothesis.

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Hypothesis Testing (cont...)

Hypothesis testing, the null and alternative hypothesis.

In order to undertake hypothesis testing you need to express your research hypothesis as a null and alternative hypothesis. The null hypothesis and alternative hypothesis are statements regarding the differences or effects that occur in the population. You will use your sample to test which statement (i.e., the null hypothesis or alternative hypothesis) is most likely (although technically, you test the evidence against the null hypothesis). So, with respect to our teaching example, the null and alternative hypothesis will reflect statements about all statistics students on graduate management courses.

The null hypothesis is essentially the "devil's advocate" position. That is, it assumes that whatever you are trying to prove did not happen ( hint: it usually states that something equals zero). For example, the two different teaching methods did not result in different exam performances (i.e., zero difference). Another example might be that there is no relationship between anxiety and athletic performance (i.e., the slope is zero). The alternative hypothesis states the opposite and is usually the hypothesis you are trying to prove (e.g., the two different teaching methods did result in different exam performances). Initially, you can state these hypotheses in more general terms (e.g., using terms like "effect", "relationship", etc.), as shown below for the teaching methods example:

Depending on how you want to "summarize" the exam performances will determine how you might want to write a more specific null and alternative hypothesis. For example, you could compare the mean exam performance of each group (i.e., the "seminar" group and the "lectures-only" group). This is what we will demonstrate here, but other options include comparing the distributions , medians , amongst other things. As such, we can state:

Now that you have identified the null and alternative hypotheses, you need to find evidence and develop a strategy for declaring your "support" for either the null or alternative hypothesis. We can do this using some statistical theory and some arbitrary cut-off points. Both these issues are dealt with next.

Significance levels

The level of statistical significance is often expressed as the so-called p -value . Depending on the statistical test you have chosen, you will calculate a probability (i.e., the p -value) of observing your sample results (or more extreme) given that the null hypothesis is true . Another way of phrasing this is to consider the probability that a difference in a mean score (or other statistic) could have arisen based on the assumption that there really is no difference. Let us consider this statement with respect to our example where we are interested in the difference in mean exam performance between two different teaching methods. If there really is no difference between the two teaching methods in the population (i.e., given that the null hypothesis is true), how likely would it be to see a difference in the mean exam performance between the two teaching methods as large as (or larger than) that which has been observed in your sample?

So, you might get a p -value such as 0.03 (i.e., p = .03). This means that there is a 3% chance of finding a difference as large as (or larger than) the one in your study given that the null hypothesis is true. However, you want to know whether this is "statistically significant". Typically, if there was a 5% or less chance (5 times in 100 or less) that the difference in the mean exam performance between the two teaching methods (or whatever statistic you are using) is as different as observed given the null hypothesis is true, you would reject the null hypothesis and accept the alternative hypothesis. Alternately, if the chance was greater than 5% (5 times in 100 or more), you would fail to reject the null hypothesis and would not accept the alternative hypothesis. As such, in this example where p = .03, we would reject the null hypothesis and accept the alternative hypothesis. We reject it because at a significance level of 0.03 (i.e., less than a 5% chance), the result we obtained could happen too frequently for us to be confident that it was the two teaching methods that had an effect on exam performance.

Whilst there is relatively little justification why a significance level of 0.05 is used rather than 0.01 or 0.10, for example, it is widely used in academic research. However, if you want to be particularly confident in your results, you can set a more stringent level of 0.01 (a 1% chance or less; 1 in 100 chance or less).

One- and two-tailed predictions

When considering whether we reject the null hypothesis and accept the alternative hypothesis, we need to consider the direction of the alternative hypothesis statement. For example, the alternative hypothesis that was stated earlier is:

The alternative hypothesis tells us two things. First, what predictions did we make about the effect of the independent variable(s) on the dependent variable(s)? Second, what was the predicted direction of this effect? Let's use our example to highlight these two points.

Sarah predicted that her teaching method (independent variable: teaching method), whereby she not only required her students to attend lectures, but also seminars, would have a positive effect (that is, increased) students' performance (dependent variable: exam marks). If an alternative hypothesis has a direction (and this is how you want to test it), the hypothesis is one-tailed. That is, it predicts direction of the effect. If the alternative hypothesis has stated that the effect was expected to be negative, this is also a one-tailed hypothesis.

Alternatively, a two-tailed prediction means that we do not make a choice over the direction that the effect of the experiment takes. Rather, it simply implies that the effect could be negative or positive. If Sarah had made a two-tailed prediction, the alternative hypothesis might have been:

In other words, we simply take out the word "positive", which implies the direction of our effect. In our example, making a two-tailed prediction may seem strange. After all, it would be logical to expect that "extra" tuition (going to seminar classes as well as lectures) would either have a positive effect on students' performance or no effect at all, but certainly not a negative effect. However, this is just our opinion (and hope) and certainly does not mean that we will get the effect we expect. Generally speaking, making a one-tail prediction (i.e., and testing for it this way) is frowned upon as it usually reflects the hope of a researcher rather than any certainty that it will happen. Notable exceptions to this rule are when there is only one possible way in which a change could occur. This can happen, for example, when biological activity/presence in measured. That is, a protein might be "dormant" and the stimulus you are using can only possibly "wake it up" (i.e., it cannot possibly reduce the activity of a "dormant" protein). In addition, for some statistical tests, one-tailed tests are not possible.

Rejecting or failing to reject the null hypothesis

Let's return finally to the question of whether we reject or fail to reject the null hypothesis.

If our statistical analysis shows that the significance level is below the cut-off value we have set (e.g., either 0.05 or 0.01), we reject the null hypothesis and accept the alternative hypothesis. Alternatively, if the significance level is above the cut-off value, we fail to reject the null hypothesis and cannot accept the alternative hypothesis. You should note that you cannot accept the null hypothesis, but only find evidence against it.

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Chapter 13: Inferential Statistics

Understanding Null Hypothesis Testing

Learning Objectives

• Explain the purpose of null hypothesis testing, including the role of sampling error.
• Describe the basic logic of null hypothesis testing.
• Describe the role of relationship strength and sample size in determining statistical significance and make reasonable judgments about statistical significance based on these two factors.

The Purpose of Null Hypothesis Testing

As we have seen, psychological research typically involves measuring one or more variables for a sample and computing descriptive statistics for that sample. In general, however, the researcher’s goal is not to draw conclusions about that sample but to draw conclusions about the population that the sample was selected from. Thus researchers must use sample statistics to draw conclusions about the corresponding values in the population. These corresponding values in the population are called  parameters . Imagine, for example, that a researcher measures the number of depressive symptoms exhibited by each of 50 clinically depressed adults and computes the mean number of symptoms. The researcher probably wants to use this sample statistic (the mean number of symptoms for the sample) to draw conclusions about the corresponding population parameter (the mean number of symptoms for clinically depressed adults).

Unfortunately, sample statistics are not perfect estimates of their corresponding population parameters. This is because there is a certain amount of random variability in any statistic from sample to sample. The mean number of depressive symptoms might be 8.73 in one sample of clinically depressed adults, 6.45 in a second sample, and 9.44 in a third—even though these samples are selected randomly from the same population. Similarly, the correlation (Pearson’s  r ) between two variables might be +.24 in one sample, −.04 in a second sample, and +.15 in a third—again, even though these samples are selected randomly from the same population. This random variability in a statistic from sample to sample is called  sampling error . (Note that the term error  here refers to random variability and does not imply that anyone has made a mistake. No one “commits a sampling error.”)

One implication of this is that when there is a statistical relationship in a sample, it is not always clear that there is a statistical relationship in the population. A small difference between two group means in a sample might indicate that there is a small difference between the two group means in the population. But it could also be that there is no difference between the means in the population and that the difference in the sample is just a matter of sampling error. Similarly, a Pearson’s  r  value of −.29 in a sample might mean that there is a negative relationship in the population. But it could also be that there is no relationship in the population and that the relationship in the sample is just a matter of sampling error.

In fact, any statistical relationship in a sample can be interpreted in two ways:

• There is a relationship in the population, and the relationship in the sample reflects this.
• There is no relationship in the population, and the relationship in the sample reflects only sampling error.

The purpose of null hypothesis testing is simply to help researchers decide between these two interpretations.

The Logic of Null Hypothesis Testing

Null hypothesis testing  is a formal approach to deciding between two interpretations of a statistical relationship in a sample. One interpretation is called the   null hypothesis  (often symbolized  H 0  and read as “H-naught”). This is the idea that there is no relationship in the population and that the relationship in the sample reflects only sampling error. Informally, the null hypothesis is that the sample relationship “occurred by chance.” The other interpretation is called the  alternative hypothesis  (often symbolized as  H 1 ). This is the idea that there is a relationship in the population and that the relationship in the sample reflects this relationship in the population.

Again, every statistical relationship in a sample can be interpreted in either of these two ways: It might have occurred by chance, or it might reflect a relationship in the population. So researchers need a way to decide between them. Although there are many specific null hypothesis testing techniques, they are all based on the same general logic. The steps are as follows:

• Assume for the moment that the null hypothesis is true. There is no relationship between the variables in the population.
• Determine how likely the sample relationship would be if the null hypothesis were true.
• If the sample relationship would be extremely unlikely, then reject the null hypothesis  in favour of the alternative hypothesis. If it would not be extremely unlikely, then  retain the null hypothesis .

Following this logic, we can begin to understand why Mehl and his colleagues concluded that there is no difference in talkativeness between women and men in the population. In essence, they asked the following question: “If there were no difference in the population, how likely is it that we would find a small difference of  d  = 0.06 in our sample?” Their answer to this question was that this sample relationship would be fairly likely if the null hypothesis were true. Therefore, they retained the null hypothesis—concluding that there is no evidence of a sex difference in the population. We can also see why Kanner and his colleagues concluded that there is a correlation between hassles and symptoms in the population. They asked, “If the null hypothesis were true, how likely is it that we would find a strong correlation of +.60 in our sample?” Their answer to this question was that this sample relationship would be fairly unlikely if the null hypothesis were true. Therefore, they rejected the null hypothesis in favour of the alternative hypothesis—concluding that there is a positive correlation between these variables in the population.

A crucial step in null hypothesis testing is finding the likelihood of the sample result if the null hypothesis were true. This probability is called the  p value . A low  p  value means that the sample result would be unlikely if the null hypothesis were true and leads to the rejection of the null hypothesis. A high  p  value means that the sample result would be likely if the null hypothesis were true and leads to the retention of the null hypothesis. But how low must the  p  value be before the sample result is considered unlikely enough to reject the null hypothesis? In null hypothesis testing, this criterion is called  α (alpha)  and is almost always set to .05. If there is less than a 5% chance of a result as extreme as the sample result if the null hypothesis were true, then the null hypothesis is rejected. When this happens, the result is said to be  statistically significant . If there is greater than a 5% chance of a result as extreme as the sample result when the null hypothesis is true, then the null hypothesis is retained. This does not necessarily mean that the researcher accepts the null hypothesis as true—only that there is not currently enough evidence to conclude that it is true. Researchers often use the expression “fail to reject the null hypothesis” rather than “retain the null hypothesis,” but they never use the expression “accept the null hypothesis.”

The Misunderstood  p  Value

The  p  value is one of the most misunderstood quantities in psychological research (Cohen, 1994) [1] . Even professional researchers misinterpret it, and it is not unusual for such misinterpretations to appear in statistics textbooks!

The most common misinterpretation is that the  p  value is the probability that the null hypothesis is true—that the sample result occurred by chance. For example, a misguided researcher might say that because the  p  value is .02, there is only a 2% chance that the result is due to chance and a 98% chance that it reflects a real relationship in the population. But this is incorrect . The  p  value is really the probability of a result at least as extreme as the sample result  if  the null hypothesis  were  true. So a  p  value of .02 means that if the null hypothesis were true, a sample result this extreme would occur only 2% of the time.

You can avoid this misunderstanding by remembering that the  p  value is not the probability that any particular  hypothesis  is true or false. Instead, it is the probability of obtaining the  sample result  if the null hypothesis were true.

Role of Sample Size and Relationship Strength

Recall that null hypothesis testing involves answering the question, “If the null hypothesis were true, what is the probability of a sample result as extreme as this one?” In other words, “What is the  p  value?” It can be helpful to see that the answer to this question depends on just two considerations: the strength of the relationship and the size of the sample. Specifically, the stronger the sample relationship and the larger the sample, the less likely the result would be if the null hypothesis were true. That is, the lower the  p  value. This should make sense. Imagine a study in which a sample of 500 women is compared with a sample of 500 men in terms of some psychological characteristic, and Cohen’s  d  is a strong 0.50. If there were really no sex difference in the population, then a result this strong based on such a large sample should seem highly unlikely. Now imagine a similar study in which a sample of three women is compared with a sample of three men, and Cohen’s  d  is a weak 0.10. If there were no sex difference in the population, then a relationship this weak based on such a small sample should seem likely. And this is precisely why the null hypothesis would be rejected in the first example and retained in the second.

Of course, sometimes the result can be weak and the sample large, or the result can be strong and the sample small. In these cases, the two considerations trade off against each other so that a weak result can be statistically significant if the sample is large enough and a strong relationship can be statistically significant even if the sample is small. Table 13.1 shows roughly how relationship strength and sample size combine to determine whether a sample result is statistically significant. The columns of the table represent the three levels of relationship strength: weak, medium, and strong. The rows represent four sample sizes that can be considered small, medium, large, and extra large in the context of psychological research. Thus each cell in the table represents a combination of relationship strength and sample size. If a cell contains the word  Yes , then this combination would be statistically significant for both Cohen’s  d  and Pearson’s  r . If it contains the word  No , then it would not be statistically significant for either. There is one cell where the decision for  d  and  r  would be different and another where it might be different depending on some additional considerations, which are discussed in Section 13.2 “Some Basic Null Hypothesis Tests”

Although Table 13.1 provides only a rough guideline, it shows very clearly that weak relationships based on medium or small samples are never statistically significant and that strong relationships based on medium or larger samples are always statistically significant. If you keep this lesson in mind, you will often know whether a result is statistically significant based on the descriptive statistics alone. It is extremely useful to be able to develop this kind of intuitive judgment. One reason is that it allows you to develop expectations about how your formal null hypothesis tests are going to come out, which in turn allows you to detect problems in your analyses. For example, if your sample relationship is strong and your sample is medium, then you would expect to reject the null hypothesis. If for some reason your formal null hypothesis test indicates otherwise, then you need to double-check your computations and interpretations. A second reason is that the ability to make this kind of intuitive judgment is an indication that you understand the basic logic of this approach in addition to being able to do the computations.

Statistical Significance Versus Practical Significance

Table 13.1 illustrates another extremely important point. A statistically significant result is not necessarily a strong one. Even a very weak result can be statistically significant if it is based on a large enough sample. This is closely related to Janet Shibley Hyde’s argument about sex differences (Hyde, 2007) [2] . The differences between women and men in mathematical problem solving and leadership ability are statistically significant. But the word  significant  can cause people to interpret these differences as strong and important—perhaps even important enough to influence the college courses they take or even who they vote for. As we have seen, however, these statistically significant differences are actually quite weak—perhaps even “trivial.”

This is why it is important to distinguish between the  statistical  significance of a result and the  practical  significance of that result.  Practical significance refers to the importance or usefulness of the result in some real-world context. Many sex differences are statistically significant—and may even be interesting for purely scientific reasons—but they are not practically significant. In clinical practice, this same concept is often referred to as “clinical significance.” For example, a study on a new treatment for social phobia might show that it produces a statistically significant positive effect. Yet this effect still might not be strong enough to justify the time, effort, and other costs of putting it into practice—especially if easier and cheaper treatments that work almost as well already exist. Although statistically significant, this result would be said to lack practical or clinical significance.

Key Takeaways

• Null hypothesis testing is a formal approach to deciding whether a statistical relationship in a sample reflects a real relationship in the population or is just due to chance.
• The logic of null hypothesis testing involves assuming that the null hypothesis is true, finding how likely the sample result would be if this assumption were correct, and then making a decision. If the sample result would be unlikely if the null hypothesis were true, then it is rejected in favour of the alternative hypothesis. If it would not be unlikely, then the null hypothesis is retained.
• The probability of obtaining the sample result if the null hypothesis were true (the  p  value) is based on two considerations: relationship strength and sample size. Reasonable judgments about whether a sample relationship is statistically significant can often be made by quickly considering these two factors.
• Statistical significance is not the same as relationship strength or importance. Even weak relationships can be statistically significant if the sample size is large enough. It is important to consider relationship strength and the practical significance of a result in addition to its statistical significance.
• Discussion: Imagine a study showing that people who eat more broccoli tend to be happier. Explain for someone who knows nothing about statistics why the researchers would conduct a null hypothesis test.
• The correlation between two variables is  r  = −.78 based on a sample size of 137.
• The mean score on a psychological characteristic for women is 25 ( SD  = 5) and the mean score for men is 24 ( SD  = 5). There were 12 women and 10 men in this study.
• In a memory experiment, the mean number of items recalled by the 40 participants in Condition A was 0.50 standard deviations greater than the mean number recalled by the 40 participants in Condition B.
• In another memory experiment, the mean scores for participants in Condition A and Condition B came out exactly the same!
• A student finds a correlation of  r  = .04 between the number of units the students in his research methods class are taking and the students’ level of stress.

Long Descriptions

“Null Hypothesis” long description: A comic depicting a man and a woman talking in the foreground. In the background is a child working at a desk. The man says to the woman, “I can’t believe schools are still teaching kids about the null hypothesis. I remember reading a big study that conclusively disproved it years ago.” [Return to “Null Hypothesis”]

“Conditional Risk” long description: A comic depicting two hikers beside a tree during a thunderstorm. A bolt of lightning goes “crack” in the dark sky as thunder booms. One of the hikers says, “Whoa! We should get inside!” The other hiker says, “It’s okay! Lightning only kills about 45 Americans a year, so the chances of dying are only one in 7,000,000. Let’s go on!” The comic’s caption says, “The annual death rate among people who know that statistic is one in six.” [Return to “Conditional Risk”]

Media Attributions

• Null Hypothesis by XKCD  CC BY-NC (Attribution NonCommercial)
• Conditional Risk by XKCD  CC BY-NC (Attribution NonCommercial)
• Cohen, J. (1994). The world is round: p < .05. American Psychologist, 49 , 997–1003. ↵
• Hyde, J. S. (2007). New directions in the study of gender similarities and differences. Current Directions in Psychological Science, 16 , 259–263. ↵

Values in a population that correspond to variables measured in a study.

The random variability in a statistic from sample to sample.

A formal approach to deciding between two interpretations of a statistical relationship in a sample.

The idea that there is no relationship in the population and that the relationship in the sample reflects only sampling error.

The idea that there is a relationship in the population and that the relationship in the sample reflects this relationship in the population.

When the relationship found in the sample would be extremely unlikely, the idea that the relationship occurred “by chance” is rejected.

When the relationship found in the sample is likely to have occurred by chance, the null hypothesis is not rejected.

The probability that, if the null hypothesis were true, the result found in the sample would occur.

How low the p value must be before the sample result is considered unlikely in null hypothesis testing.

When there is less than a 5% chance of a result as extreme as the sample result occurring and the null hypothesis is rejected.

Research Methods in Psychology - 2nd Canadian Edition Copyright © 2015 by Paul C. Price, Rajiv Jhangiani, & I-Chant A. Chiang is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

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Support or Reject Null Hypothesis in Easy Steps

What does it mean to reject the null hypothesis.

• General Situations: P Value
• P Value Guidelines
• A Proportion
• A Proportion (second example)

In many statistical tests, you’ll want to either reject or support the null hypothesis . For elementary statistics students, the term can be a tricky term to grasp, partly because the name “null hypothesis” doesn’t make it clear about what the null hypothesis actually is!

The null hypothesis can be thought of as a nullifiable hypothesis. That means you can nullify it, or reject it. What happens if you reject the null hypothesis? It gets replaced with the alternate hypothesis, which is what you think might actually be true about a situation. For example, let’s say you think that a certain drug might be responsible for a spate of recent heart attacks. The drug company thinks the drug is safe. The null hypothesis is always the accepted hypothesis; in this example, the drug is on the market, people are using it, and it’s generally accepted to be safe. Therefore, the null hypothesis is that the drug is safe. The alternate hypothesis — the one you want to replace the null hypothesis, is that the drug isn’t safe. Rejecting the null hypothesis in this case means that you will have to prove that the drug is not safe.

To reject the null hypothesis, perform the following steps:

Step 1: State the null hypothesis. When you state the null hypothesis, you also have to state the alternate hypothesis. Sometimes it is easier to state the alternate hypothesis first, because that’s the researcher’s thoughts about the experiment. How to state the null hypothesis (opens in a new window).

Step 2: Support or reject the null hypothesis . Several methods exist, depending on what kind of sample data you have. For example, you can use the P-value method. For a rundown on all methods, see: Support or reject the null hypothesis.

If you are able to reject the null hypothesis in Step 2, you can replace it with the alternate hypothesis.

That’s it!

When to Reject the Null hypothesis

Basically, you reject the null hypothesis when your test value falls into the rejection region . There are four main ways you’ll compute test values and either support or reject your null hypothesis. Which method you choose depends mainly on if you have a proportion or a p-value .

Support or Reject the Null Hypothesis: Steps

Click the link the skip to the situation you need to support or reject null hypothesis for: General Situations: P Value P Value Guidelines A Proportion A Proportion (second example)

Support or Reject Null Hypothesis with a P Value

If you have a P-value , or are asked to find a p-value, follow these instructions to support or reject the null hypothesis. This method works if you are given an alpha level and if you are not given an alpha level. If you are given a confidence level , just subtract from 1 to get the alpha level. See: How to calculate an alpha level .

Step 1: State the null hypothesis and the alternate hypothesis (“the claim”). If you aren’t sure how to do this, follow this link for How To State the Null and Alternate Hypothesis .

Step 2: Find the critical value . We’re dealing with a normally distributed population, so the critical value is a z-score . Use the following formula to find the z-score .

Click here if you want easy, step-by-step instructions for solving this formula.

Step 4: Find the P-Value by looking up your answer from step 3 in the z-table . To get the p-value, subtract the area from 1. For example, if your area is .990 then your p-value is 1-.9950 = 0.005. Note: for a two-tailed test , you’ll need to halve this amount to get the p-value in one tail.

Step 5: Compare your answer from step 4 with the α value given in the question. Should you support or reject the null hypothesis? If step 7 is less than or equal to α, reject the null hypothesis, otherwise do not reject it.

P-Value Guidelines

Use these general guidelines to decide if you should reject or keep the null:

If p value > .10 → “not significant ” If p value ≤ .10 → “marginally significant” If p value ≤ .05 → “significant” If p value ≤ .01 → “highly significant.”

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Support or Reject Null Hypothesis for a Proportion

Sometimes, you’ll be given a proportion of the population or a percentage and asked to support or reject null hypothesis. In this case you can’t compute a test value by calculating a z-score (you need actual numbers for that), so we use a slightly different technique.

Example question: A researcher claims that Democrats will win the next election. 4300 voters were polled; 2200 said they would vote Democrat. Decide if you should support or reject null hypothesis. Is there enough evidence at α=0.05 to support this claim?

Step 1: State the null hypothesis and the alternate hypothesis (“the claim”) . H o :p ≤ 0.5 H 1 :p > .5

Step 3: Use the following formula to calculate your test value.

Where: Phat is calculated in Step 2 P the null hypothesis p value (.05) Q is 1 – p

The z-score is: .512 – .5 / √(.5(.5) / 4300)) = 1.57

Step 4: Look up Step 3 in the z-table to get .9418.

Step 5: Calculate your p-value by subtracting Step 4 from 1. 1-.9418 = .0582

Step 6: Compare your answer from step 5 with the α value given in the question . Support or reject the null hypothesis? If step 5 is less than α, reject the null hypothesis, otherwise do not reject it. In this case, .582 (5.82%) is not less than our α, so we do not reject the null hypothesis.

Support or Reject Null Hypothesis for a Proportion: Second example

Example question: A researcher claims that more than 23% of community members go to church regularly. In a recent survey, 126 out of 420 people stated they went to church regularly. Is there enough evidence at α = 0.05 to support this claim? Use the P-Value method to support or reject null hypothesis.

Step 1: State the null hypothesis and the alternate hypothesis (“the claim”) . H o :p ≤ 0.23; H 1 :p > 0.23 (claim)

Step 3: Find ‘p’ by converting the stated claim to a decimal: 23% = 0.23. Also, find ‘q’ by subtracting ‘p’ from 1: 1 – 0.23 = 0.77.

Step 4: Use the following formula to calculate your test value.

If formulas confuse you, this is asking you to:

• Multiply p and q together, then divide by the number in the random sample. (0.23 x 0.77) / 420 = 0.00042
• Take the square root of your answer to 2 . √( 0.1771) = 0. 0205
• Divide your answer to 1. by your answer in 3. 0.07 / 0. 0205 = 3.41

Step 5: Find the P-Value by looking up your answer from step 5 in the z-table . The z-score for 3.41 is .4997. Subtract from 0.500: 0.500-.4997 = 0.003.

Step 6: Compare your P-value to α . Support or reject null hypothesis? If the P-value is less, reject the null hypothesis. If the P-value is more, keep the null hypothesis. 0.003 < 0.05, so we have enough evidence to reject the null hypothesis and accept the claim.

Note: In Step 5, I’m using the z-table on this site to solve this problem. Most textbooks have the right of z-table . If you’re seeing .9997 as an answer in your textbook table, then your textbook has a “whole z” table, in which case don’t subtract from .5, subtract from 1. 1-.9997 = 0.003.

Check out our Youtube channel for video tips!

Everitt, B. S.; Skrondal, A. (2010), The Cambridge Dictionary of Statistics , Cambridge University Press. Gonick, L. (1993). The Cartoon Guide to Statistics . HarperPerennial.

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Statistics By Jim

Making statistics intuitive

Failing to Reject the Null Hypothesis

By Jim Frost 69 Comments

Failing to reject the null hypothesis is an odd way to state that the results of your hypothesis test are not statistically significant. Why the peculiar phrasing? “Fail to reject” sounds like one of those double negatives that writing classes taught you to avoid. What does it mean exactly? There’s an excellent reason for the odd wording!

In this post, learn what it means when you fail to reject the null hypothesis and why that’s the correct wording. While accepting the null hypothesis sounds more straightforward, it is not statistically correct!

Before proceeding, let’s recap some necessary information. In all statistical hypothesis tests, you have the following two hypotheses:

• The null hypothesis states that there is no effect or relationship between the variables.
• The alternative hypothesis states the effect or relationship exists.

We assume that the null hypothesis is correct until we have enough evidence to suggest otherwise.

After you perform a hypothesis test, there are only two possible outcomes.

• When your p-value is greater than your significance level, you fail to reject the null hypothesis. Your results are not significant. You’ll learn more about interpreting this outcome later in this post.

Related posts : Hypothesis Testing Overview and The Null Hypothesis

Why Don’t Statisticians Accept the Null Hypothesis?

To understand why we don’t accept the null, consider the fact that you can’t prove a negative. A lack of evidence only means that you haven’t proven that something exists. It does not prove that something doesn’t exist. It might exist, but your study missed it. That’s a huge difference and it is the reason for the convoluted wording. Let’s look at several analogies.

Species Presumed to be Extinct

Lack of proof doesn’t represent proof that something doesn’t exist!

Criminal Trials

Perhaps the prosecutor conducted a shoddy investigation and missed clues? Or, the defendant successfully covered his tracks? Consequently, the verdict in these cases is “not guilty.” That judgment doesn’t say the defendant is proven innocent, just that there wasn’t enough evidence to move the jury from the default assumption of innocence.

Hypothesis Tests

The hypothesis test assesses the evidence in your sample. If your test fails to detect an effect, it’s not proof that the effect doesn’t exist. It just means your sample contained an insufficient amount of evidence to conclude that it exists. Like the species that were presumed extinct, or the prosecutor who missed clues, the effect might exist in the overall population but not in your particular sample. Consequently, the test results fail to reject the null hypothesis, which is analogous to a “not guilty” verdict in a trial. There just wasn’t enough evidence to move the hypothesis test from the default position that the null is true.

The critical point across these analogies is that a lack of evidence does not prove something does not exist—just that you didn’t find it in your specific investigation. Hence, you never accept the null hypothesis.

Related post : The Significance Level as an Evidentiary Standard

What Does Fail to Reject the Null Hypothesis Mean?

Accepting the null hypothesis would indicate that you’ve proven an effect doesn’t exist. As you’ve seen, that’s not the case at all. You can’t prove a negative! Instead, the strength of your evidence falls short of being able to reject the null. Consequently, we fail to reject it.

Failing to reject the null indicates that our sample did not provide sufficient evidence to conclude that the effect exists. However, at the same time, that lack of evidence doesn’t prove that the effect does not exist. Capturing all that information leads to the convoluted wording!

What are the possible implications of failing to reject the null hypothesis? Let’s work through them.

First, it is possible that the effect truly doesn’t exist in the population, which is why your hypothesis test didn’t detect it in the sample. Makes sense, right? While that is one possibility, it doesn’t end there.

Another possibility is that the effect exists in the population, but the test didn’t detect it for a variety of reasons. These reasons include the following:

• The sample size was too small to detect the effect.
• The variability in the data was too high. The effect exists, but the noise in your data swamped the signal (effect).
• By chance, you collected a fluky sample. When dealing with random samples, chance always plays a role in the results. The luck of the draw might have caused your sample not to reflect an effect that exists in the population.

Notice how studies that collect a small amount of data or low-quality data are likely to miss an effect that exists? These studies had inadequate statistical power to detect the effect. We certainly don’t want to take results from low-quality studies as proof that something doesn’t exist!

However, failing to detect an effect does not necessarily mean a study is low-quality. Random chance in the sampling process can work against even the best research projects!

If you’re learning about hypothesis testing and like the approach I use in my blog, check out my eBook!

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Reader Interactions

May 8, 2024 at 9:08 am

Thank you very much for explaining the topic. It brings clarity and makes statistics very simple and interesting. Its helping me in the field of Medical Research.

February 26, 2024 at 7:54 pm

Hi Jim, My question is that can I reverse Null hyposthesis and start with Null: µ1 ≠ µ2 ? Then, if I can reject Null, I will end up with µ1=µ2 for mean comparison and this what I am looking for. But isn’t this cheating?

February 26, 2024 at 11:41 pm

That can be done but it requires you to revamp the entire test. Keep in mind that the reason you normally start out with the null equating to no relationship is because the researchers typically want to prove that a relationship or effect exists. This format forces the researchers to collect a substantial amount of high quality data to have a chance at demonstrating that an effect exists. If they collect a small sample and/or poor quality (e.g., noisy or imprecise), then the results default back to the null stating that no effect exists. So, they have to collect good data and work hard to get findings that suggest the effect exists.

There are tests that flip it around as you suggest where the null states that a relationship does exist. For example, researchers perform an equivalency test when they want to show that there is no difference. That the groups are equal. The test is designed such that it requires a good sample size and high quality data to have a chance at proving equivalency. If they have a small sample size and/or poor quality data, the results default back to the groups being unequal, which is not what they want to show.

So, choose the null hypothesis and corresponding analysis based on what you hope to find. Choose the null hypothesis that forces you to work hard to reject it and get the results that you want. It forces you to collect better evidence to make your case and the results default back to what you don’t want if you do a poor job.

I hope that makes sense!

October 13, 2023 at 5:10 am

Really appreciate how you have been able to explain something difficult in very simple terms. Also covering why you can’t accept a null hypothesis – something which I think is frequently missed. Thank you, Jim.

February 22, 2022 at 11:18 am

Hi Jim, I really appreciate your blog, making difficult things sound simple is a great gift.

I have a doubt about the p-value. You said there are two options when it comes to hypothesis tests results . Reject or failing to reject the null, depending on the p-value and your significant level.

But… a P-value of 0,001 means a stronger evidence than a P-value of 0,01? ( both with a significant level of 5%. Or It doesn`t matter, and just every p-Value under your significant level means the same burden of evidence against the null?

I hope I made my point clear. Thanks a lot for your time.

February 23, 2022 at 9:06 pm

There are different schools of thought about this question. The traditional approach is clear cut. Your results are statistically significance when your p-value is less than or equal to your significance level. When the p-value is greater than the significance level, your results are not significant.

However, as you point out, lower p-values indicate stronger evidence against the null hypothesis. I write about this aspect of p-values in several articles, interpreting p-values (near the end) and p-values and reproducibility .

Personally, I consider both aspects. P-values near 0.05 provide weak evidence. Consequently, I’d be willing to say that p-values less than or equal to 0.05 are statistically significant, but when they’re near 0.05, I’d consider it as a preliminary result that requires more research. However, if the p-value is less 0.01, or even better 0.001, then that’s much stronger evidence and I’ll give those results more weight in my evaluation.

If you read those two articles, I think you’ll see what I mean.

January 1, 2022 at 6:00 pm

HI, I have a quick question that you may be able to help me with. I am using SPSS and carrying out a Mann W U Test it says to retain the null hypothesis. The hypothesis is that males are faster than women at completing a task. So is that saying that they are or are not

January 1, 2022 at 8:17 pm

In that case, your sample data provides insufficient evidence to conclude that males are faster. The results do not prove that males and females are the same speed. You just don’t have enough evidence to say males are faster. In this post, I cover the reasons why you can’t prove the null is true.

November 23, 2021 at 5:36 pm

What if I have to prove in my hypothesis that there shouldn’t be any affect of treatment on patients? Can I say that if my null hypothesis is accepted i have got my results (no effect)? I am confused what to do in this situation. As for null hypothesis we always have to write it with some type of equality. What if I want my result to be what i have stated in null hypothesis i.e. no effect? How to write statements in this case? I am using non parametric test, Mann whitney u test

November 27, 2021 at 4:56 pm

You need to perform an equivalence test, which is a special type of procedure when you want to prove that the results are equal. The problem with a regular hypothesis test is that when you fail to reject the null, you’re not proving that they the outcomes are equal. You can fail to reject the null thanks to a small sample size, noisy data, or a small effect size even when the outcomes are truly different at the population level. An equivalence test sets things up so you need strong evidence to really show that two outcomes are equal.

Unfortunately, I don’t have any content for equivalence testing at this point, but you can read an article about it at Wikipedia: Equivalence Test .

August 13, 2021 at 9:41 pm

Great explanation and great analogies! Thanks.

August 11, 2021 at 2:02 am

I got problems with analysis. I did wound healing experiments with drugs treatment (total 9 groups). When I do the 2-way ANOVA in excel, I got the significant results in sample (Drug Treatment) and columns (Day, Timeline) . But I did not get the significantly results in interactions. Can I still reject the null hypothesis and continue the post-hoc test?

Thank you very much.

June 13, 2021 at 4:51 am

Hi Jim, There are so many books covering maths/programming related to statistics/DS, but may be hardly any book to develop an intuitive understanding. Thanks to you for filling up that gap. After statistics, hypothesis-testing, regression, will it be possible for you to write such books on more topics in DS such as trees, deep-learning etc.

I recently started with reading your book on hypothesis testing (just finished the first chapter). I have a question w.r.t the fuel cost example (from first chapter), where a random sample of 25 families (with sample mean 330.6) is taken. To do the hypothesis testing here, we are taking a sampling distribution with a mean of 260. Then based on the p-value and significance level, we find whether to reject or accept the null hypothesis. The entire decision (to accept or reject the null hypothesis) is based on the sampling distribution about which i have the following questions : a) we are assuming that the sampling distribution is normally distributed. what if it has some other distribution, how can we find that ? b) We have assumed that the sampling distribution is normally distributed and then further assumed that its mean is 260 (as required for the hypothesis testing). But we need the standard deviation as well to define the normal distribution, can you please let me know how do we find the standard deviation for the sampling distribution ? Thanks.

April 24, 2021 at 2:25 pm

Maybe its the idea of “Innocent until proven guilty”? Your Null assume the person is not guilty, and your alternative assumes the person is guilty, only when you have enough evidence (finding statistical significance P0.05 you have failed to reject null hypothesis, null stands,implying the person is not guilty. Or, the person remain innocent.. Correct me if you think it’s wrong but this is the way I interpreted.

April 25, 2021 at 5:10 pm

I used the courtroom/trial analogy within this post. Read that for more details. I’d agree with your general take on the issue except when you have enough evidence you actually reject the null, which in the trial means the defendant is found guilty.

April 17, 2021 at 6:10 am

Can regression analysis be done using 5 companies variables for predicting working capital management and profitability positive/negative relationship?

Also, does null hypothesis rejecting means whatsoever is stated in null hypothesis that is false proved through regression analysis?

I have very less knowledge about regression analysis. Please help me, Sir. As I have my project report due on next week. Thanks in advance!

April 18, 2021 at 10:48 pm

Hi Ahmed, yes, regression analysis can be used for the scenario you describe as long as you have the required data.

For more about the null hypothesis in relation to regression analysis, read my post about regression coefficients and their p-values . I describe the null hypothesis in it.

January 26, 2021 at 7:32 pm

With regards to the legal example above. While your explanation makes sense when simplified to this statistical level, from a legal perspective it is not correct. The presumption of innocence means one does not need to be proven innocent. They are innocent. The onus of proof lies with proving they are guilty. So if you can’t prove someones guilt then in fact you must accept the null hypothesis that they are innocent. It’s not a statistical test so a little bit misleading using it an example, although I see why you would.

If it were a statistical test, then we would probably be rather paranoid that everyone is a murderer but they just haven’t been proven to be one yet.

Great article though, a nice simple and thoughtout explanation.

January 26, 2021 at 9:11 pm

It seems like you misread my post. The hypothesis testing/legal analogy is very strong both in making the case and in the result.

In hypothesis testing, the data have to show beyond a reasonable doubt that the alternative hypothesis is true. In a court case, the prosecutor has to present sufficient evidence to show beyond a reasonable doubt that the defendant is guilty.

In terms of the test/case results. When the evidence (data) is insufficient, you fail to reject the null hypothesis but you do not conclude that the data proves the null is true. In a legal case that has insufficient evidence, the jury finds the defendant to be “not guilty” but they do not say that s/he is proven innocent. To your point specifically, it is not accurate to say that “not guilty” is the same as “proven innocent.”

It’s a very strong parallel.

January 9, 2021 at 11:45 am

Just a question, in my research on hypotheses for an assignment, I am finding it difficult to find an exact definition for a hypothesis itself. I know the defintion, but I’m looking for a citable explanation, any ideas?

January 10, 2021 at 1:37 am

To be clear, do you need to come up with a statistical hypothesis? That’s one where you’ll use a particular statistical hypothesis test. If so, I’ll need to know more about what you’re studying, your variables, and the type of hypothesis test you plan to use.

There are also scientific hypotheses that you’ll state in your proposals, study papers, etc. Those are different from statistical hypotheses (although related). However, those are very study area specific and I don’t cover those types on this blog because this is a statistical blog. But, if it’s a statistical hypothesis for a hypothesis test, then let me know the information I mention above and I can help you out!

November 7, 2020 at 8:33 am

Hi, good read, I’m kind of a novice here, so I’m trying to write a research paper, and I’m trying to make a hypothesis. however looking at the literature, there are contradicting results.

researcher A found that there is relationship between X and Y

however, researcher B found that there is no relationship between X and Y

therefore, what is the null hypothesis between X and y? do we choose what we assumed to be correct for our study? or is is somehow related to the alternative hypothesis? I’m confused.

thank you very much for the help.

November 8, 2020 at 12:07 am

Hypotheses for a statistical test are different than a researcher’s hypothesis. When you’re constructing the statistical hypothesis, you don’t need to consider what other researchers have found. Instead, you construct them so that the test only produces statistically significant results (rejecting the null) when your data provides strong evidence. I talk about that process in this post.

Typically, researchers are hoping to establish that an effect or relationship exists. Consequently, the null and alternative hypotheses are typically the following:

Null: The effect or relationship doesn’t not exist. Alternative: The effect or relationship does exist.

However, if you’re hoping to prove that there is no effect or no relationship, you then need to flip those hypotheses and use a special test, such as an equivalences test.

So, there’s no need to consider what researchers have found but instead what you’re looking for. In most cases, you are looking for an effect/relationship, so you’d go with the hypotheses as I show them above.

I hope that helps!

October 22, 2020 at 6:13 pm

Great, deep detailed answer. Appreciated!

September 16, 2020 at 12:03 pm

Thank you for explaining it too clearly. I have the following situation with a Box Bohnken design of three levels and three factors for multiple responses. F-value for second order model is not significant (failing to reject null hypothesis, p-value > 0.05) but, lack of fit of the model is not significant. What can you suggest me about statistical analysis?

September 17, 2020 at 2:42 am

Are your first order effects significant?

You want the lack of fit to be nonsignificant. If it’s significant, that means the model doesn’t fit the data well. So, you’re good there! 🙂

September 14, 2020 at 5:18 pm

thank you for all the explicit explanation on the subject.

However, i still got a question about “accepting the null hypothesis”. from textbook, the p-value is the probability that a statistic would take a value that is as extreme as or more extreme than that actually observed.

so, that’s why when p<0.01 we reject the null hypothesis, because it's too rare (p0.05, i can understand that for most cases we cannot accept the null, for example, if p=0.5, it means that the probability to get a statistic from the distribution is 0.5, which is totally random.

But how about when the p is very close to 1, like p=0.95, or p=0.99999999, can’t we say that the probability that the statistic is not from this distribution is less than 0.05, | or in another way, the probability that the statistic is from the distribution is almost 1. can’t we accept the null in such circumstance?

September 11, 2020 at 12:14 pm

Wow! This is beautifully explained. “Lack of proof doesn’t represent proof that something doesn’t exist!”. This kinda, hit me with such force. Can I then, use the same analogy for many other things in life? LOL! 🙂

H0 = God does not exist; H1 = God does exist; WE fail to reject H0 as there is no evidence.

Thank you sir, this has answered many of my questions, statistically speaking! No pun intended with the above.

September 11, 2020 at 4:58 pm

Hi, LOL, I’m glad it had such meaning for you! I’ll leave the determination about the existence of god up to each person, but in general, yes, I think statistical thinking can be helpful when applied to real life. It is important to realize that lack of proof truly is not proof that something doesn’t exist. But, I also consider other statistical concepts, such as confounders and sampling methodology, to be useful keeping in mind when I’m considering everyday life stuff–even when I’m not statistically analyzing it. Those concepts are generally helpful when trying to figure out what is going on in your life! Are there other alternative explanations? Is what you’re perceiving likely to be biased by something that’s affecting the “data” you can observe? Am I drawing a conclusion based on a large or small sample? How strong is the evidence?

A lot of those concepts are great considerations even when you’re just informally assessing and draw conclusions about things happening in your daily life.

August 13, 2020 at 12:04 am

Dear Jim, thanks for clarifying. absolutely, now it makes sense. the topic is murky but it is good to have your guidance, and be clear. I have not come across an instructor as clear in explaining as you do. Appreciate your direction. Thanks a lot, Geetanjali

August 15, 2020 at 3:48 pm

Hi Geetanjali,

I’m glad my website is helpful! That makes my day hearing that. Thanks so much for writing!

August 12, 2020 at 9:37 am

Hi Jim. I am doing data analyis for my masters thesis and my hypothesis testings were insignificant. And I am ok with that. But there is something bothering me. It is the low reliabilities of the 4-Items sub-scales (.55, .68, .75), though the overall alpha is good (.85). I just wonder if it is affecting my hypothesis testings.

August 11, 2020 at 9:23 pm

Thank you sir for replying, yes sir we it’s a RCT study.. where we did within and between the groups analysis and found p>0.05 in between the groups using Mann Whitney U test. So in such cases if the results comes like this we need to Mention that we failed reject the null hypothesis? Is that correct? Whether it tells that the study is inefficient as we couldn’t accept the alternative hypothesis. Thanks is advance.

August 11, 2020 at 9:43 pm

Hi Saumya, ah, this becomes clearer. When ask statistical questions, please be sure to include all relevant information because the details are extremely important. I didn’t know it was an RCT with a treatment and control group. Yes, given that your p-value is greater than your significance level, you fail to reject the null hypothesis. The results are not significant. The experiment provides insufficient evidence to conclude that the outcome in the treatment group is different than the control group.

By the way, you never accept the alternative hypothesis (or the null). The two options are to either reject the null or fail to reject the null. In your case, you fail to reject the null hypothesis.

I hope this helps!

August 11, 2020 at 9:41 am

Sir, p value is0.05, by which we interpret that both the groups are equally effective. In this case I had to reject the alternative hypothesis/ failed to reject null hypothessis.

August 11, 2020 at 12:37 am

sir, within the group analysis the p value for both the groups is significant (p0.05, by which we interpret that though both the treatments are effective, there in no difference between the efficacy of one over the other.. in other words.. no intervention is superior and both are equally effective.

August 11, 2020 at 2:45 pm

Thanks for the additional details. If I understand correctly, there were separate analyses before that determined each treatment had a statistically significance effect. However, when you compare the two treatments, there difference between them is not statistically significant.

If that’s the case, the interpretation is fairly straightforward. You have evidence that suggests that both treatments are effective. However, you don’t have evidence to conclude that one is better than the other.

August 10, 2020 at 9:26 am

Hi thank you for a wonderful explanation. I have a doubt: My Null hypothesis says: no significant difference between the effect fo A and B treatment Alternative hypothesis: there will be significant difference between the effect of A and B treatment. and my results show that i fail to reject null hypothesis.. Both the treatments were effective, but not significant difference.. how do I interpret this?

August 10, 2020 at 1:32 pm

First, I need to ask you a question. If your p-value is not significant, and so you fail to reject the null, why do you say that the treatment is effective? I can answer you question better after knowing the reason you say that. Thanks!

August 9, 2020 at 9:40 am

Dear Jim, thanks for making stats much more understandable and answering all question so painstakingly. I understand the following on p value and null. If our sample yields a p value of .01, it means that that there is a 1% probability that our kind of sample exists in the population. that is a rare event. So why shouldn’t we accept the HO as the probability of our event was v rare. Pls can you correct me. Thanks, G

August 10, 2020 at 1:53 pm

That’s a great question! They key thing to remember is that p-values are a conditional probability. P-value calculations assume that the null hypothesis is true. So, a p-value of 0.01 indicates that there is a 1% probability of observing your sample results, or more extreme, *IF* the null hypothesis is true.

The kicker is that we don’t whether the null is true or not. But, using this process does limit the likelihood of a false positive to your significance level (alpha). But, we don’t know whether the null is true and you had an unusual sample or whether the null is false. Usually, with a p-value of 0.01, we’d reject the null and conclude it is false.

I hope that answered your question. This topic can be murky and I wasn’t quite clear which part you needed clarification.

August 4, 2020 at 11:16 pm

Thank you for the wonderful explanation. However, I was just curious to know that what if in a particular test, we get a p-value less than the level of significance, leading to evidence against null hypothesis. Is there any possibility that our interpretation of population effect might be wrong due to randomness of samples? Also, how do we conclude whether the evidence is enough for our alternate hypothesis?

August 4, 2020 at 11:55 pm

Hi Abhilash,

Yes, unfortunately, when you’re working with samples, there’s always the possibility that random chance will cause your sample to not represent the population. For information about these errors, read my post about the types of errors in hypothesis testing .

In hypothesis testing, you determine whether your evidence is strong enough to reject the null. You don’t accept the alternative hypothesis. I cover that in my post about interpreting p-values .

August 1, 2020 at 3:50 pm

Hi, I am trying to interpret this phenomenon after my research. The null hypothesis states that “The use of combined drugs A and B does not lower blood pressure when compared to if drug A or B is used singularly”

The alternate hypothesis states: The use of combined drugs A and B lower blood pressure compared to if drug A or B is used singularly.

At the end of the study, majority of the people did not actually combine drugs A and B, rather indicated they either used drug A or drug B but not a combination. I am finding it very difficult to explain this outcome more so that it is a descriptive research. Please how do I go about this? Thanks a lot

June 22, 2020 at 10:01 am

What confuses me is how we set/determine the null hypothesis? For example stating that two sets of data are either no different or have no relationship will give completely different outcomes, so which is correct? Is the null that they are different or the same?

June 22, 2020 at 2:16 pm

Typically, the null states there is no effect/no relationship. That’s true for 99% of hypothesis tests. However, there are some equivalence tests where you are trying to prove that the groups are equal. In that case, the null hypothesis states that groups are not equal.

The null hypothesis is typically what you *don’t* want to find. You have to work hard, design a good experiment, collect good data, and end up with sufficient evidence to favor the alternative hypothesis. Usually in an experiment you want to find an effect. So, usually the null states there is no effect and you have get good evidence to reject that notion.

However, there are a few tests where you actually want to prove something is equal, so you need the null to state that they’re not equal in those cases and then do all the hard work and gather good data to suggest that they are equal. Basically, set up the hypothesis so it takes a good experiment and solid evidence to be able to reject the null and favor the hypothesis that you’re hoping is true.

June 5, 2020 at 11:54 am

Thank you for the explanation. I have one question that. If Null hypothesis is failed to reject than is possible to interpret the analysis further?

June 5, 2020 at 7:36 pm

Hi Mottakin,

Typically, if your result is that you fail to reject the null hypothesis there’s not much further interpretation. You don’t want to be in a situation where you’re endlessly trying new things on a quest for obtaining significant results. That’s data mining.

May 25, 2020 at 7:55 am

I hope all is well. I am enjoying your blog. I am not a statistician, however, I use statistical formulae to provide insight on the direction in which data is going. I have used both the regression analysis and a T-Test. I know that both use a null hypothesis and an alternative hypothesis. Could you please clarity the difference between a regression analysis and a T-Test? Are there conditions where one is a better option than the other?

May 26, 2020 at 9:18 pm

t-Tests compare the means of one or two groups. Regression analysis typically describes the relationships between a set of independent variables and the dependent variables. Interestingly, you can actually use regression analysis to perform a t-test. However, that would be overkill. If you just want to compare the means of one or two groups, use a t-test. Read my post about performing t-tests in Excel to see what they can do. If you have a more complex model than just comparing one or two means, regression might be the way to go. Read my post about when to use regression analysis .

May 12, 2020 at 5:45 pm

This article is really enlightening but there is still some darkness looming around. I see that low p-values mean strong evidence against null hypothesis and finding such a sample is highly unlikely when null hypothesis is true. So , is it OK to say that when p-value is 0.01 , it was very unlikely to have found such a sample but we still found it and hence finding such a sample has not occurred just by chance which leads towards rejection of null hypothesis.

May 12, 2020 at 11:16 pm

That’s mostly correct. I wouldn’t say, “has not occurred by chance.” So, when you get a very low p-value it does mean that you are unlikely to obtain that sample if the null is true. However, once you obtain that result, you don’t know for sure which of the two occurred:

• The effect exists in the population.
• Random chance gave you an unusual sample (i.e., Type I error).

You really don’t know for sure. However, by the decision making results you set about the strength of evidence required to reject the null, you conclude that the effect exists. Just always be aware that it could be a false positive.

That’s all a long way of saying that your sample was unlikely to occur by chance if the null is true.

April 29, 2020 at 11:59 am

Why do we consult the statistical tables to find out the critical values of our test statistics?

April 30, 2020 at 5:05 pm

Statistical tables started back in the “olden days” when computers didn’t exist. You’d calculate the test statistic value for your sample. Then, you’d look in the appropriate table and using the degrees of freedom for your design and find the critical values for the test statistic. If the value of your test statistics exceeded the critical value, your results were statistically significant.

With powerful and readily available computers, researchers could analyze their data and calculate the p-values and compare them directly to the significance level.

I hope that answers your question!

April 15, 2020 at 10:12 am

If we are not able to reject the null hypothesis. What could be the solution?

April 16, 2020 at 11:13 pm

Hi Shazzad,

The first thing to recognize is that failing to reject the null hypothesis might not be an error. If the null hypothesis is false, then the correct outcome is failing to reject the null.

However, if the null hypothesis is false and you fail to reject, it is a type II error, or a false negative. Read my post about types of errors in hypothesis tests for more information.

This type of error can occur for a variety of reasons, including the following:

• Fluky sample. When working with random samples, random error can cause anomalous results purely by chance.
• Sample is too small. Perhaps the sample was too small, which means the test didn’t have enough statistical power to detect the difference.
• Problematic data or sampling methodology. There could be a problem with how you collected the data or your sampling methodology.

There are various other possibilities, but those are several common problems.

April 14, 2020 at 12:19 pm

Thank you so much for this article! I am taking my first Statistics class in college and I have one question about this.

I understand that the default position is that the null is correct, and you explained that (just like a court case), the sample evidence must EXCEED the “evidentiary standard” (which is the significance level) to conclude that an effect/relationship exists. And, if an effect/relationship exists, that means that it’s the alternative hypothesis that “wins” (not sure if that’s the correct way of wording it, but I’m trying to make this as simple as possible in my head!).

But what I don’t understand is that if the P-value is GREATER than the significance value, we fail to reject the null….because shouldn’t a higher P-value, mean that our sample evidence EXCEEDS the evidentiary standard (aka the significance level), and therefore an effect/relationship exists? In my mind it would make more sense to reject the null, because our P-value is higher and therefore we have enough evidence to reject the null.

I hope I worded this in a way that makes sense. Thank you in advance!

April 14, 2020 at 10:42 pm

That’s a great question. The key thing to remember is that higher p-values correspond to weaker evidence against the null hypothesis. A high p-value indicates that your sample is likely (high probability = high p-value) if the null hypothesis is true. Conversely, low p-values represent stronger evidence against the null. You were unlikely (low probability = low p-value) to have collect a sample with the measured characteristics if the null is true.

So, there is negative correlation between p-values and strength of evidence against the null hypothesis. Low p-values indicate stronger evidence. Higher p-value represent weaker evidence.

In a nutshell, you reject the null hypothesis with a low p-value because it indicates your sample data are unusual if the null is true. When it’s unusual enough, you reject the null.

March 5, 2020 at 11:10 am

There is something I am confused about. If our significance level is .05 and our resulting p-value is .02 (thus the strength of our evidence is strong enough to reject the null hypothesis), do we state that we reject the null hypothesis with 95% confidence or 98% confidence?

My guess is our confidence level is 95% since or alpha was .05. But if the strength of our evidence is 98%, why wouldn’t we use that as our stated confidence in our results?

March 5, 2020 at 4:19 pm

Hi Michael,

You’d state that you can reject the null at a significance level of 5% or conversely at the 95% confidence level. A key reason is to avoid cherry picking your results. In other words, you don’t want to choose the significance level based on your results.

Consequently, set the significance level/confidence level before performing your analysis. Then, use those preset levels to determine statistical significance. I always recommend including the exact p-value when you report on statistical significance. Exact p-values do provide information about the strength of evidence against the null.

March 5, 2020 at 9:58 am

Thank you for sharing this knowledge , it is very appropriate in explaining some observations in the study of forest biodiversity.

March 4, 2020 at 2:01 am

Thank you so much. This provides for my research

March 3, 2020 at 7:28 pm

If one couples this with what they call estimated monetary value of risk in risk management, one can take better decisions.

March 3, 2020 at 3:12 pm

Thank you for providing this clear insight.

March 3, 2020 at 3:29 am

Nice article Jim. The risk of such failure obviously reduces when a lower significance level is specified.One benefits most by reading this article in conjunction with your other article “Understanding Significance Levels in Statistics”.

March 3, 2020 at 2:43 am

That’s fine. My question is why doesn’t the numerical value of type 1 error coincide with the significance level in the backdrop that the type 1 error and the significance level are both the same ? I hope you got my question.

March 3, 2020 at 3:30 am

Hi, they are equal. As I indicated, the significance level equals the type I error rate.

March 3, 2020 at 1:27 am

Kindly elighten me on one confusion. We set out our significance level before setting our hypothesis. When we calculate the type 1 error, which happens to be a significance level, the numerical value doesn’t equals (either undermining value comes out or an exceeding value comescout ) our significance level that was preassigned. Why is this so ?

March 3, 2020 at 2:24 am

Hi Ratnadeep,

You’re correct. The significance level (alpha) is the same as the type I error rate. However, you compare the p-value to the significance level. It’s the p-value that can be greater than or less than the significance level.

The significance level is the evidentiary standard. How strong does the evidence in your sample need to be before you can reject the null? The p-value indicates the strength of the evidence that is present in your sample. By comparing the p-value to the significance level, you’re comparing the actual strength of the sample evidence to the evidentiary standard to determine whether your sample evidence is strong enough to conclude that the effect exists in the population.

I write about this in my post about the understanding significance levels . I think that will help answer your questions!

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Null Hypothesis and Alternative Hypothesis

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Hypothesis testing involves the careful construction of two statements: the null hypothesis and the alternative hypothesis. These hypotheses can look very similar but are actually different.

How do we know which hypothesis is the null and which one is the alternative? We will see that there are a few ways to tell the difference.

The Null Hypothesis

The null hypothesis reflects that there will be no observed effect in our experiment. In a mathematical formulation of the null hypothesis, there will typically be an equal sign. This hypothesis is denoted by H 0 .

The null hypothesis is what we attempt to find evidence against in our hypothesis test. We hope to obtain a small enough p-value that it is lower than our level of significance alpha and we are justified in rejecting the null hypothesis. If our p-value is greater than alpha, then we fail to reject the null hypothesis.

If the null hypothesis is not rejected, then we must be careful to say what this means. The thinking on this is similar to a legal verdict. Just because a person has been declared "not guilty", it does not mean that he is innocent. In the same way, just because we failed to reject a null hypothesis it does not mean that the statement is true.

For example, we may want to investigate the claim that despite what convention has told us, the mean adult body temperature is not the accepted value of 98.6 degrees Fahrenheit . The null hypothesis for an experiment to investigate this is “The mean adult body temperature for healthy individuals is 98.6 degrees Fahrenheit.” If we fail to reject the null hypothesis, then our working hypothesis remains that the average adult who is healthy has a temperature of 98.6 degrees. We do not prove that this is true.

If we are studying a new treatment, the null hypothesis is that our treatment will not change our subjects in any meaningful way. In other words, the treatment will not produce any effect in our subjects.

The Alternative Hypothesis

The alternative or experimental hypothesis reflects that there will be an observed effect for our experiment. In a mathematical formulation of the alternative hypothesis, there will typically be an inequality, or not equal to symbol. This hypothesis is denoted by either H a or by H 1 .

The alternative hypothesis is what we are attempting to demonstrate in an indirect way by the use of our hypothesis test. If the null hypothesis is rejected, then we accept the alternative hypothesis. If the null hypothesis is not rejected, then we do not accept the alternative hypothesis. Going back to the above example of mean human body temperature, the alternative hypothesis is “The average adult human body temperature is not 98.6 degrees Fahrenheit.”

If we are studying a new treatment, then the alternative hypothesis is that our treatment does, in fact, change our subjects in a meaningful and measurable way.

The following set of negations may help when you are forming your null and alternative hypotheses. Most technical papers rely on just the first formulation, even though you may see some of the others in a statistics textbook.

• Null hypothesis: “ x is equal to y .” Alternative hypothesis “ x is not equal to y .”
• Null hypothesis: “ x is at least y .” Alternative hypothesis “ x is less than y .”
• Null hypothesis: “ x is at most y .” Alternative hypothesis “ x is greater than y .”
• What 'Fail to Reject' Means in a Hypothesis Test
• Type I and Type II Errors in Statistics
• An Example of a Hypothesis Test
• The Runs Test for Random Sequences
• An Example of Chi-Square Test for a Multinomial Experiment
• The Difference Between Type I and Type II Errors in Hypothesis Testing
• What Level of Alpha Determines Statistical Significance?
• What Is the Difference Between Alpha and P-Values?
• What Is ANOVA?
• How to Find Critical Values with a Chi-Square Table
• Example of a Permutation Test
• Degrees of Freedom for Independence of Variables in Two-Way Table
• Example of an ANOVA Calculation
• How to Find Degrees of Freedom in Statistics
• How to Construct a Confidence Interval for a Population Proportion
• Degrees of Freedom in Statistics and Mathematics

9.1 Null and Alternative Hypotheses

The actual test begins by considering two hypotheses . They are called the null hypothesis and the alternative hypothesis . These hypotheses contain opposing viewpoints.

H 0 : The null hypothesis: It is a statement of no difference between the variables—they are not related. This can often be considered the status quo and as a result if you cannot accept the null it requires some action.

H a : The alternative hypothesis: It is a claim about the population that is contradictory to H 0 and what we conclude when we reject H 0 . This is usually what the researcher is trying to prove.

Since the null and alternative hypotheses are contradictory, you must examine evidence to decide if you have enough evidence to reject the null hypothesis or not. The evidence is in the form of sample data.

After you have determined which hypothesis the sample supports, you make a decision. There are two options for a decision. They are "reject H 0 " if the sample information favors the alternative hypothesis or "do not reject H 0 " or "decline to reject H 0 " if the sample information is insufficient to reject the null hypothesis.

Mathematical Symbols Used in H 0 and H a :

H 0 always has a symbol with an equal in it. H a never has a symbol with an equal in it. The choice of symbol depends on the wording of the hypothesis test. However, be aware that many researchers (including one of the co-authors in research work) use = in the null hypothesis, even with > or < as the symbol in the alternative hypothesis. This practice is acceptable because we only make the decision to reject or not reject the null hypothesis.

Example 9.1

H 0 : No more than 30% of the registered voters in Santa Clara County voted in the primary election. p ≤ .30 H a : More than 30% of the registered voters in Santa Clara County voted in the primary election. p > 30

A medical trial is conducted to test whether or not a new medicine reduces cholesterol by 25%. State the null and alternative hypotheses.

Example 9.2

We want to test whether the mean GPA of students in American colleges is different from 2.0 (out of 4.0). The null and alternative hypotheses are: H 0 : μ = 2.0 H a : μ ≠ 2.0

We want to test whether the mean height of eighth graders is 66 inches. State the null and alternative hypotheses. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

• H 0 : μ __ 66
• H a : μ __ 66

Example 9.3

We want to test if college students take less than five years to graduate from college, on the average. The null and alternative hypotheses are: H 0 : μ ≥ 5 H a : μ < 5

We want to test if it takes fewer than 45 minutes to teach a lesson plan. State the null and alternative hypotheses. Fill in the correct symbol ( =, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

• H 0 : μ __ 45
• H a : μ __ 45

Example 9.4

In an issue of U. S. News and World Report , an article on school standards stated that about half of all students in France, Germany, and Israel take advanced placement exams and a third pass. The same article stated that 6.6% of U.S. students take advanced placement exams and 4.4% pass. Test if the percentage of U.S. students who take advanced placement exams is more than 6.6%. State the null and alternative hypotheses. H 0 : p ≤ 0.066 H a : p > 0.066

On a state driver’s test, about 40% pass the test on the first try. We want to test if more than 40% pass on the first try. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

• H 0 : p __ 0.40
• H a : p __ 0.40

Collaborative Exercise

Bring to class a newspaper, some news magazines, and some Internet articles . In groups, find articles from which your group can write null and alternative hypotheses. Discuss your hypotheses with the rest of the class.

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What Is a Null Hypothesis?

The alternative hypothesis.

• Additional Examples
• Null Hypothesis and Investments

The Bottom Line

• Corporate Finance
• Financial Ratios

Null Hypothesis: What Is It, and How Is It Used in Investing?

Adam Hayes, Ph.D., CFA, is a financial writer with 15+ years Wall Street experience as a derivatives trader. Besides his extensive derivative trading expertise, Adam is an expert in economics and behavioral finance. Adam received his master's in economics from The New School for Social Research and his Ph.D. from the University of Wisconsin-Madison in sociology. He is a CFA charterholder as well as holding FINRA Series 7, 55 & 63 licenses. He currently researches and teaches economic sociology and the social studies of finance at the Hebrew University in Jerusalem.

A null hypothesis is a type of statistical hypothesis that proposes that no statistical significance exists in a set of given observations. Hypothesis testing is used to assess the credibility of a hypothesis by using sample data. Sometimes referred to simply as the “null,” it is represented as H 0 .

The null hypothesis, also known as “the conjecture,” is used in quantitative analysis to test theories about markets, investing strategies, and economies to decide if an idea is true or false.

Key Takeaways

• A null hypothesis is a type of conjecture in statistics that proposes that there is no difference between certain characteristics of a population or data-generating process.
• The alternative hypothesis proposes that there is a difference.
• Hypothesis testing provides a method to reject a null hypothesis within a certain confidence level.
• If you can reject the null hypothesis, it provides support for the alternative hypothesis.
• Null hypothesis testing is the basis of the principle of falsification in science.

Alex Dos Diaz / Investopedia

Understanding a Null Hypothesis

A gambler may be interested in whether a game of chance is fair. If it is, then the expected earnings per play come to zero for both players. If it is not, then the expected earnings are positive for one player and negative for the other.

To test whether the game is fair, the gambler collects earnings data from many repetitions of the game, calculates the average earnings from these data, then tests the null hypothesis that the expected earnings are not different from zero.

If the average earnings from the sample data are sufficiently far from zero, then the gambler will reject the null hypothesis and conclude the alternative hypothesis—namely, that the expected earnings per play are different from zero. If the average earnings from the sample data are near zero, then the gambler will not reject the null hypothesis, concluding instead that the difference between the average from the data and zero is explainable by chance alone.

A null hypothesis can only be rejected, not proven.

The null hypothesis assumes that any kind of difference between the chosen characteristics that you see in a set of data is due to chance. For example, if the expected earnings for the gambling game are truly equal to zero, then any difference between the average earnings in the data and zero is due to chance.

Analysts look to reject   the null hypothesis because doing so is a strong conclusion. This requires evidence in the form of an observed difference that is too large to be explained solely by chance. Failing to reject the null hypothesis—that the results are explainable by chance alone—is a weak conclusion because it allows that while factors other than chance may be at work, they may not be strong enough for the statistical test to detect them.

An important point to note is that we are testing the null hypothesis because there is an element of doubt about its validity. Whatever information that is against the stated null hypothesis is captured in the alternative (alternate) hypothesis (H 1 ).

For the examples below, the alternative hypothesis would be:

• Students score an average that is not equal to seven.
• The mean annual return of a mutual fund is not equal to 8% per year.

In other words, the alternative hypothesis is a direct contradiction of the null hypothesis.

Null Hypothesis Examples

Here is a simple example: A school principal claims that students in her school score an average of seven out of 10 in exams. The null hypothesis is that the population mean is not 7.0. To test this null hypothesis, we record marks of, say, 30 students ( sample ) from the entire student population of the school (say, 300) and calculate the mean of that sample.

We can then compare the (calculated) sample mean to the (hypothesized) population mean of 7.0 and attempt to reject the null hypothesis. (The null hypothesis here—that the population mean is not 7.0—cannot be proved using the sample data. It can only be rejected.)

Take another example: The annual return of a particular  mutual fund  is claimed to be 8%. Assume that the mutual fund has been in existence for 20 years. The null hypothesis is that the mean return is not 8% for the mutual fund. We take a random sample of annual returns of the mutual fund for, say, five years (sample) and calculate the sample mean. We then compare the (calculated) sample mean to the (claimed) population mean (8%) to test the null hypothesis.

For the above examples, null hypotheses are:

• Example A: Students in the school don’t score an average of seven out of 10 in exams.
• Example B: The mean annual return of the mutual fund is not 8% per year.

For the purposes of determining whether to reject the null hypothesis (abbreviated H0), said hypothesis is assumed, for the sake of argument, to be true. Then the likely range of possible values of the calculated statistic (e.g., the average score on 30 students’ tests) is determined under this presumption (e.g., the range of plausible averages might range from 6.2 to 7.8 if the population mean is 7.0).

If the sample average is outside of this range, the null hypothesis is rejected. Otherwise, the difference is said to be “explainable by chance alone,” being within the range that is determined by chance alone.

How Null Hypothesis Testing Is Used in Investments

As an example related to financial markets, assume Alice sees that her investment strategy produces higher average returns than simply buying and holding a stock . The null hypothesis states that there is no difference between the two average returns, and Alice is inclined to believe this until she can conclude contradictory results.

Refuting the null hypothesis would require showing statistical significance, which can be found by a variety of tests. The alternative hypothesis would state that the investment strategy has a higher average return than a traditional buy-and-hold strategy.

One tool that can determine the statistical significance of the results is the p-value. A p-value represents the probability that a difference as large or larger than the observed difference between the two average returns could occur solely by chance.

A p-value that is less than or equal to 0.05 often indicates whether there is evidence against the null hypothesis. If Alice conducts one of these tests, such as a test using the normal model, resulting in a significant difference between her returns and the buy-and-hold returns (the p-value is less than or equal to 0.05), she can then reject the null hypothesis and conclude the alternative hypothesis.

How Is the Null Hypothesis Identified?

The analyst or researcher establishes a null hypothesis based on the research question or problem they are trying to answer. Depending on the question, the null may be identified differently. For example, if the question is simply whether an effect exists (e.g., does X influence Y?), the null hypothesis could be H 0 : X = 0. If the question is instead, is X the same as Y, the H 0 would be X = Y. If it is that the effect of X on Y is positive, H 0 would be X > 0. If the resulting analysis shows an effect that is statistically significantly different from zero, the null can be rejected.

How Is Null Hypothesis Used in Finance?

In finance , a null hypothesis is used in quantitative analysis. It tests the premise of an investing strategy, the markets, or an economy to determine if it is true or false.

For instance, an analyst may want to see if two stocks, ABC and XYZ, are closely correlated. The null hypothesis would be ABC ≠ XYZ.

How Are Statistical Hypotheses Tested?

Statistical hypotheses are tested by a four-step process . The first is for the analyst to state the two hypotheses so that only one can be right. The second is to formulate an analysis plan, which outlines how the data will be evaluated. The third is to carry out the plan and physically analyze the sample data. The fourth and final step is to analyze the results and either reject the null hypothesis or claim that the observed differences are explainable by chance alone.

What Is an Alternative Hypothesis?

An alternative hypothesis is a direct contradiction of a null hypothesis. This means that if one of the two hypotheses is true, the other is false.

A null hypothesis states there is no difference between groups or relationship between variables. It is a type of statistical hypothesis and proposes that no statistical significance exists in a set of given observations. “Null” means nothing.

The null hypothesis is used in quantitative analysis to test theories about economies, investing strategies, and markets to decide if an idea is true or false. Hypothesis testing assesses the credibility of a hypothesis by using sample data. It is represented as H 0 and is sometimes simply known as “the null.”

Correction—July 23, 2024:  This article was corrected to state accurate examples of null hypothesis in the Null Hypothesis Examples section.

Sage Publishing. “ Chapter 8: Introduction to Hypothesis Testing ,” Page 4.

Sage Publishing. “ Chapter 8: Introduction to Hypothesis Testing ,” Pages 4 to 7.

Sage Publishing. “ Chapter 8: Introduction to Hypothesis Testing ,” Page 7.

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