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• 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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Null Hypothesis

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In formal hypothesis testing, the null hypothesis ( H 0 ) is the hypothesis assumed to be true in the population and which gives rise to the sampling distribution of the test statistic in question (Hays 1994 ). The critical feature of the null hypothesis across hypothesis testing frameworks is that it is stated with enough precision that it can be tested.

Introduction

A hypothesis is a statement or explanation about the nature or causes of some phenomena of interest. In the process of scientific study, we can distinguish two forms of hypotheses. A research hypothesis poses the question of interest, and if well stated, will include the variables under study and the expected relationship between them. A statistical hypothesis translates the research hypothesis into a mathematically precise, statistically testable statement concerning the assumed value of a parameter of interest in the population. The null hypothesis is an example of a statistical hypothesis.

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Fisher, R. (1925). Statistical methods for research workers (1st ed.). Edinburgh: Oliver and Boyd.

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Gigerenzer, G. (2004). Mindless statistics. The Journal of Socio-Economics, 33 , 587–606.

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Hays, W. L. (1994). Statistics (5th ed.). Belmont: Wadsworth.

Neyman, J., & Pearson, E. S. (1933). On the problem of the most efficient tests of statistical hypotheses. Philosophical Transactions of the Royal Society of London, Series A, 231 , 289–337.

Szucs, D., & Ioannidis, J. P. A. (2016). When null hypothesis significance testing is unsuitable for research: A reassessment. bioRxiv . https://doi.org/10.1101/095570 .

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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: a statement of no difference between sample means or proportions or no difference between a sample mean or proportion and a population mean or proportion. In other words, the difference equals 0.

H a —, the alternative hypothesis: a claim about the population that is contradictory to H 0 and what we conclude when we reject H 0 .

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 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 percent of the registered voters in Santa Clara County voted in the primary election. p ≤ 30 H a : More than 30 percent 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 percent. 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 the following: 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 fewer than five years to graduate from college, on the average. The null and alternative hypotheses are the following: 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

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

On a state driver’s test, about 40 percent pass the test on the first try. We want to test if more than 40 percent 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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Chapter 13: Inferential Statistics

13.1 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 favor 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 favor 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). 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 “How Relationship Strength and Sample Size Combine to Determine Whether a Result Is Statistically Significant” 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”

Table 13.1 How Relationship Strength and Sample Size Combine to Determine Whether a Result Is Statistically Significant

Although Table 13.1 “How Relationship Strength and Sample Size Combine to Determine Whether a Result Is Statistically Significant” 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 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 “How Relationship Strength and Sample Size Combine to Determine Whether a Result Is Statistically Significant” 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). 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 favor 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.

Practice: Use Table 13.1 “How Relationship Strength and Sample Size Combine to Determine Whether a Result Is Statistically Significant” to decide whether each of the following results is statistically significant.

• 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.

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.

• Research Methods in Psychology. Provided by : University of Minnesota Libraries Publishing. Located at : http://open.lib.umn.edu/psychologyresearchmethods . License : CC BY-NC-SA: Attribution-NonCommercial-ShareAlike

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• Idea behind hypothesis testing

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Video transcript

What is Null Hypothesis? What Is Its Importance in Research?

Scientists begin their research with a hypothesis that a relationship of some kind exists between variables. The null hypothesis is the opposite stating that no such relationship exists. Null hypothesis may seem unexciting, but it is a very important aspect of research. In this article, we discuss what null hypothesis is, how to make use of it, and why you should use it to improve your statistical analyses.

What is the Null Hypothesis?

The null hypothesis can be tested using statistical analysis  and is often written as H 0 (read as “H-naught”). Once you determine how likely the sample relationship would be if the H 0   were true, you can run your analysis. Researchers use a significance test to determine the likelihood that the results supporting the H 0 are not due to chance.

The null hypothesis is not the same as an alternative hypothesis. An alternative hypothesis states, that there is a relationship between two variables, while H 0 posits the opposite. Let us consider the following example.

A researcher wants to discover the relationship between exercise frequency and appetite. She asks:

Q: Does increased exercise frequency lead to increased appetite? Alternative hypothesis: Increased exercise frequency leads to increased appetite. H 0 assumes that there is no relationship between the two variables: Increased exercise frequency does not lead to increased appetite.

Let us look at another example of how to state the null hypothesis:

Q: Does insufficient sleep lead to an increased risk of heart attack among men over age 50? H 0 : The amount of sleep men over age 50 get does not increase their risk of heart attack.

Why is Null Hypothesis Important?

Many scientists often neglect null hypothesis in their testing. As shown in the above examples, H 0 is often assumed to be the opposite of the hypothesis being tested. However, it is good practice to include H 0 and ensure it is carefully worded. To understand why, let us return to our previous example. In this case,

Alternative hypothesis: Getting too little sleep leads to an increased risk of heart attack among men over age 50.

H 0 : The amount of sleep men over age 50 get has no effect on their risk of heart attack.

Note that this H 0 is different than the one in our first example. What if we were to conduct this experiment and find that neither H 0 nor the alternative hypothesis was supported? The experiment would be considered invalid . Take our original H 0 in this case, “the amount of sleep men over age 50 get, does not increase their risk of heart attack”. If this H 0 is found to be untrue, and so is the alternative, we can still consider a third hypothesis. Perhaps getting insufficient sleep actually decreases the risk of a heart attack among men over age 50. Because we have tested H 0 , we have more information that we would not have if we had neglected it.

Do I Really Need to Test It?

The biggest problem with the null hypothesis is that many scientists see accepting it as a failure of the experiment. They consider that they have not proven anything of value. However, as we have learned from the replication crisis , negative results are just as important as positive ones. While they may seem less appealing to publishers, they can tell the scientific community important information about correlations that do or do not exist. In this way, they can drive science forward and prevent the wastage of resources.

Do you test for the null hypothesis? Why or why not? Let us know your thoughts in the comments below.

The following null hypotheses were formulated for this study: Ho1. There are no significant differences in the factors that influence urban gardening when respondents are grouped according to age, sex, household size, social status and average combined monthly income.

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null hypothesis

Usually a statement asserting that there is no difference or no association between variables. The null hypothesis is a tool that makes it possible to use certain statistical tests to figure out if another hypothesis of interest is likely to be accurate or not.

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3.3.4: Hypothesis Test for Simple Linear Regression

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We will now describe a hypothesis test to determine if the regression model is meaningful; in other words, does the value of \(X\) in any way help predict the expected value of \(Y\)?

Simple Linear Regression ANOVA Hypothesis Test

Model Assumptions

• The residual errors are random and are normally distributed.
• The standard deviation of the residual error does not depend on \(X\)
• A linear relationship exists between \(X\) and \(Y\)
• The samples are randomly selected

Test Hypotheses

\(H_o\):  \(X\) and \(Y\) are not correlated   

\(H_a\):  \(X\) and \(Y\) are correlated   

\(H_o\):  \(\beta_1\) (slope) = 0   

\(H_a\):  \(\beta_1\) (slope) ≠ 0

Test Statistic

\(F=\dfrac{M S_{\text {Regression }}}{M S_{\text {Error }}}\)

\(d f_{\text {num }}=1\)

\(d f_{\text {den }}=n-2\)

Sum of Squares

\(S S_{\text {Total }}=\sum(Y-\bar{Y})^{2}\)

\(S S_{\text {Error }}=\sum(Y-\hat{Y})^{2}\)

\(S S_{\text {Regression }}=S S_{\text {Total }}-S S_{\text {Error }}\)

In simple linear regression, this is equivalent to saying “Are X an Y correlated?”

In reviewing the model, \(Y=\beta_{0}+\beta_{1} X+\varepsilon\), as long as the slope (\(\beta_{1}\)) has any non‐zero value, \(X\) will add value in helping predict the expected value of \(Y\). However, if there is no correlation between X and Y, the value of the slope (\(\beta_{1}\)) will be zero. The model we can use is very similar to One Factor ANOVA.

The Results of the test can be summarized in a special ANOVA table:

Example: Rainfall and sales of sunglasses

Design : Is there a significant correlation between rainfall and sales of sunglasses?

Research Hypothese s:

\(H_o\):  Sales and Rainfall are not correlated      \(H_o\):  1 (slope) = 0

\(H_a\):  Sales and Rainfall are correlated      \(H_a\):  1 (slope) ≠ 0

Type I error would be to reject the Null Hypothesis and \(t\) claim that rainfall is correlated with sales of sunglasses, when they are not correlated. The test will be run at a level of significance (\(\alpha\)) of 5%.

The test statistic from the table will be \(\mathrm{F}=\dfrac{\text { MSRegression }}{\text { MSError }}\). The degrees of freedom for the numerator will be 1, and the degrees of freedom for denominator will be 5‐2=3.  

Critical Value for \(F\) at  \(\alpha\)of 5% with \(df_{num}=1\) and \(df_{den}=3} is 10.13.  Reject \(H_o\) if \(F >10.13\). We will also run this test using the \(p\)‐value method with statistical software, such as Minitab.  

Data/Results

\(F=341.422 / 12.859=26.551\), which is more than the critical value of 10.13, so Reject \(H_o\). Also, the \(p\)‐value = 0.0142 < 0.05 which also supports rejecting \(H_o\).  

Sales of Sunglasses and Rainfall are negatively correlated.

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General Principles of Preclinical Study Design

Wenlong huang.

Institute of Medical Sciences, School of Medicine, Medical Sciences and Nutrition, University of Aberdeen, Aberdeen, UK

Nathalie Percie du Sert

NC3Rs, London, UK

Preclinical studies using animals to study the potential of a therapeutic drug or strategy are important steps before translation to clinical trials. However, evidence has shown that poor quality in the design and conduct of these studies has not only impeded clinical translation but also led to significant waste of valuable research resources. It is clear that experimental biases are related to the poor quality seen with preclinical studies. In this chapter, we will focus on hypothesis testing type of preclinical studies and explain general concepts and principles in relation to the design of in vivo experiments, provide definitions of experimental biases and how to avoid them, and discuss major sources contributing to experimental biases and how to mitigate these sources. We will also explore the differences between confirmatory and exploratory studies, and discuss available guidelines on preclinical studies and how to use them. This chapter, together with relevant information in other chapters in the handbook, provides a powerful tool to enhance scientific rigour for preclinical studies without restricting creativity.

This chapter will give an overview of some generic concepts pertinent to the design of preclinical research. The emphasis is on the requirements of in vivo experiments which use experimental animals to discover and validate new clinical therapeutic approaches. However, these general principles are, by and large, generically relevant to all areas of preclinical research. The overarching requirement should be that preclinical research should only be conducted to answer an important question for which a robust scrutiny of the available evidence demonstrates that the answer is not already known. Furthermore, such experiments must be designed, conducted, analysed and reported to the highest levels of rigour and transparency. Assessments of research outputs should focus more on these factors and less on any apparent “novelty”.

1. An Overview

Broadly, preclinical research can be classified into two distinct categories depending on the aim and purpose of the experiment, namely, “hypothesis generating” (exploratory) and “hypothesis testing” (confirmatory) research ( Fig. 1 ). Hypothesis generating studies are often scientifically-informed, curiosity and intuition-driven explorations which may generate testable theories regarding the pathophysiology of disease and potential drug targets. The freedom of researchers to explore such innovative ideas is the lifeblood of preclinical science and should not be stifled by excessive constraints in terms of experimental design and conduct. Nevertheless, in order to subsequently assess the veracity of hypotheses generated in this way, and certainly to justify clinical development of a therapeutic target, hypothesis testing studies which seek to show reproducible intervention effects in relevant animal models must be designed, conducted, analysed and reported to the highest possible levels of rigour and transparency. This will also contribute to reducing research “waste” ( Ioannidis et al. 2014 ; Macleod et al. 2014 ). Chapter “Resolving the Tension Between Exploration and Confirmation in Preclinical Biomedical Research” of the handbook will deal with exploratory and confirmatory studies in details. This chapter will only focus on general design principles for hypothesis testing studies. We will address the issue of design principles for hypothesis-generating studies at the end of this chapter. We advise that when researchers design and conduct hypothesis testing in vivo studies, they should conform to the general principles for the major domains that are outlined in Sect. 4 of the chapter and incorporate these principles into a protocol that can be registered and published. The purpose of using these principles is to enhance scientific rigour without restricting creativity. It is advisable that sometimes there can be exploratory elements within the same hypothesis testing studies; therefore, extra care in terms of applying these principles to reduce experimental biases would be needed before the start of the studies. This chapter will not cover reporting, which will be detailed in chapters “Minimum Information and Quality Standards for Conducting, Reporting, and Organizing In Vitro Research”, “Minimum Information in In Vivo Research”, and “Quality Governance in Biomedical Research” of the handbook.

Comparison of exploratory (hypothesis generating) and confirmatory (hypothesis testing) preclinical studies. Descriptive statistics describes data and provides descriptions of the population, using numerical calculations, graphs, and tables. In contrast, inferential statistics predicts and infers about a population using a sample of data from the population, therefore one can take data from samples and make generalisation about a population

We would recommend that researchers who conduct hypothesis testing in vivo studies should prepare clear protocols, which include a statistical analysis plan, detailing how they are going to set up measures to address the major domains of experimental biases before the experiments start. Ideally, these protocols should be preregistered and/or published, so that the methods which will be used to reduce the impact of bias are documented in an a priori fashion. The process of peer review of a protocol prior to initiating experiments of course is a valuable opportunity for refinement and improvement. Registering protocols encourages rigour and transparency, even if the protocol is not peer-reviewed. Some journals are open to submissions of these types of protocols, such as BMJ Open Science, and many journals offer the Registered Reports format. In addition, there are online resources that allow researchers to preregister their experimental protocols, such as preclinical. eu and osf.io/registries.

2. General Scientific Methods for Designing In Vivo Experiments

Designing an in vivo experiment involves taking a number of decisions on different aspects of the experimental plan. Typically, a comparative experiment can be broken into several component parts.

2.1. Hypotheses and Effect Size

The objective is usually to test a hypothesis. On some occasions, two hypotheses may be postulated: the null hypothesis and the alternative hypothesis. The alternative hypothesis refers to the presumption that the experimental manipulation has an effect on the response measured; the null hypothesis is the hypothesis of no change, or no effect. In a statistical test, the p-value reports the probability of observing an effect as large or larger than the one being observed if the null hypothesis was true; the smaller the p -value, the least likely it is that the null hypothesis is true. The null hypothesis cannot be accepted or proven true. This also defines the effect of interest, i.e. the outcome that will be measured to test the hypothesis. The minimum effect size is the smallest effect the researcher designs the experiment to be able to detect and should be declared in the protocol; it is set up as the minimum difference which would be of biological relevance. The effect size is then used in the sample size calculation to ensure that the experiment is powered to detect only meaningful effects and does not generate statistically significant results that are not biologically relevant. In many cases, it will be hard to determine the minimum difference of biological relevance as for early stage experiments it might be completely unknown, or translatability between clinical relevance and experimental detection thresholds will be complex. There is no simple and easy answer to this question, but in general, a minimum effect size should be set so one can assume to have a beneficial effect for individuals rather than large cohorts, the difference must be experimentally testable and reasonable to achieve, and should have a rationale for translation into patients in the long run.

2.2. Groups, Experimental Unit and Sample Size

In comparative experiments, animals are split into groups, and each group is subjected to different interventions, such as a drug or vehicle injection, or a surgical procedure. The sample size is the number of experimental units per group; identifying the experimental unit underpins the reliability of the experiment, but it is often incorrectly identified ( Lazic et al. 2018 ). The experimental unit is the entity subjected to an intervention independently of all other units; it must be possible to assign any two experimental units to different comparison groups. For example, if the treatment is applied to individual mice by injection, the experimental unit may be the animal, in which case the number of experimental units per group and the number of animals per group is the same. However, if there is any contamination between mice within a cage, the treatment given to one mouse might influence other mice in that cage, and it would be more appropriate to subject all mice in one cage to the same treatment and treat the cage as the experimental unit. In another example, if the treatment is added to the water in a fish tank, two fish in the same tank cannot receive different treatments; thus the experimental unit is the tank, and the sample size is the number of tanks per group. Once identified, experimental units are allocated to the different comparison groups of the desired sample size; this is done using an appropriate method of randomisation to prevent selection bias (see Sect. 3 ). Each comparison group will be subjected to different interventions, at least one of which will be a control. The purpose of the control group is to allow the researcher to investigate the effect of a treatment and distinguish it from other confounding experimental effects. It is therefore crucial that any control group is treated exactly in the same way as the other comparison groups. Types of control group to consider include negative control, vehicle control, positive control, sham control, comparative control and naïve control ( Bate and Clark 2014 ).

2.3. Measurements and Outcome Measures

Measurements are taken to assess the results; these are recorded as outcome measures (also known as dependent variable). A number of outcome measures can be recorded in a single experiment, for example, if burrowing behaviour is measured, the outcome measure might be the weight of gravel displaced, or if neuronal density is measured from histological brain slides, the outcome measure might be the neuron count. The primary outcome measure should be identified in the planning stage of the experiment and stated in the protocol; it is the outcome of greatest importance, which will answer the main experimental question. The number of animals in the experiment is determined by the power needed to detect a difference in the primary outcome measure. A hypothesis testing experiment may also include additional outcome measures, i.e. secondary outcome measures, which can be used to generate hypotheses for follow-up experiments. Secondary outcome measures cannot be used to draw conclusions about the experiment if the experiment was not powered to detect a minimum difference for these outcome measures.

For the purpose of the statistical analysis, outcome measures fall into two broad categories: continuous or categorical. Continuous measures are sometimes referred to as quantitative data and are measured on a numerical scale. Continuous measures include truly continuous data but also discrete data. Examples of true continuous data include bodyweight, body temperature, blood/CSF concentration or time to event, while examples of discrete data include litter size, number of correct response or clinical score. Categorical responses are measured on a nonnumerical scale; they can be ordinal (e.g. severity score, mild/moderate/severe), nominal (e.g. behavioural response, left/middle/right arm maze) or binary (e.g. disease state, present/absent). Continuous responses may take longer to measure, but they contain more information. If possible, it is preferable to measure a continuous rather than categorical response because continuous data can be analysed using the parametric analyses, which have higher power; this reduces the sample size needed ( Bate and Clark 2014 ).

2.4. Independent Variables and Analysis

There are many ways to analyse data from in vivo experiments; the first step in devising the analysis plan is to identify the independent variables. There can be two broad types: independent variables of interest which the researcher specifically manipulates to test the hypothesis, for example, a drug with different doses, and nuisance variables, which are other sources of variability that may impact on the outcome measure, but are not of direct interest to the researcher. Examples of nuisance variables could be the day of the experiment, if animals used on different days, or baseline body weight or locomotor activity. Every experiment has nuisance variables. Identifying them at the protocol stage and accounting for them in the design and the analysis, for example, as blocking factors, or co-variables, increase the sensitivity of the experiment to detect changes induced by the independent variable(s) of interest. The analysis plan should be established before the experiment starts and any data is collected; it should also be included in the protocol. Additional analyses can be performed on the data, but if an analysis was not planned before the data was collected, it should be clearly reported as a post hoc or exploratory analysis. Exploratory analyses are at greater risk of yielding false positive results.

3. Experimental Biases: Definitions and Methods to Reduce Them

For any researcher who intends to carry out preclinical in vivo studies, it is important to understand what experimental biases are. First, we need to know the definition of bias. It is the inadequacies in the design, conduct, analysis or reporting of an experiment that cause systematic distortion of the estimated intervention effect away from the “truth” ( Altman et al. 2001 ; van der Worp et al. 2010 ), and it will significantly confound in vivo studies and reduce their internal validity. Sources of bias are multiple and in many cases context dependant. In this overview chapter, it is not possible to give an exhaustive list of potential sources of bias, and it behoves the researcher to systematically identify all potential significant sources of bias for the particular experiment being in planned and to design appropriate mitigation tactics into the protocol. Major known types of biases include selection bias, performance bias, detection bias, and attrition bias. Table 1 gives the definition of each type of bias and describe the methods to reduce them.

Researchers who conduct hypothesis testing in vivo animal work should understand the importance of limiting the impact of experimental biases in the design, conduct, analysis and reporting of in vivo experiments. Experimental biases can cause significant weakness in the design, conduct and analysis of in vivo animal studies, which can produce misleading results and waste valuable resources. In biomedical research, many effects of interventions are fairly small, and small effects therefore are difficult to distinguish from experimental biases ( Ioannidis et al. 2014 ). Evidence (1960–2012 from PubMed) shows that adequate steps to reduce biases, e.g. blinded assessment of outcome and randomisation, have not been taken in more than 20% and 50% of biomedical studies, respectively, leading to inflated estimates of effectiveness, e.g. in the fields of preclinical stroke, multiple sclerosis, Parkinson’s disease, bone cancer pain and myocardial infarction research ( Currie et al. 2013 ; Macleod et al. 2008 ; Rooke et al. 2011 ; Sena et al. 2007 ; van Hout et al. 2016 ; Vesterinen et al. 2010 ) and consequently significant research waste ( Ioannidis et al. 2014 ; Macleod et al. 2014 , 2015 ). Therefore, it is imperative that biomedical researchers should spend efforts on improvements in the quality of their studies using the methods described in this chapter to reduce experimental biases which will lead to increased effect-to-bias ratio.

However, it is worth pointing out that the notion that experimental biases could significantly impact on in vivo animal studies is often assumed because they are believed to be important in clinical research. Therefore, such an assumption may be flawed, as the body of evidence showing the importance of bias-reducing methods such as randomisation, blinding, etc. for animal studies is still limited and most of the evidence is indirect. Furthermore, there may also be sources of bias which impact on preclinical studies which are currently unknown. Thus, systematic review and metaanalysis of in vivo studies have shown that papers that do not report bias-reducing methods report larger effect sizes ( Vesterinen et al. 2010 ). However, these studies are based on reported data alone, and therefore there might be a difference between what researchers do and what they report in their publications ( Reichlin et al. 2016 ). Reporting of the precise details of bias reduction methods is often scanty, and therefore accurate assessment of the precise method and rigour of such procedures is challenging. Moreover, those papers that do not report one bias-reducing method, e.g. randomisation, also tend to not report other bias-reducing methods, e.g. blinding and sample size calculation, suggesting that there could be interactions between these methods.

4. Experimental Biases: Major Domains and General Principles

In this section, we will describe the major domains, in other words, sources that could contribute to experimental bias if not carefully considered and if mitigating tactics are not included in the design of hypothesis testing experiments before data collection starts. These include sample size estimation, randomisation, allocation concealment, blinding, primary and secondary outcome measures and inclusion/exclusion criteria. General descriptions for these domains ( Macleod et al. 2009 ; Rice et al. 2008 ; Rice 2010 ; van der Worp et al. 2010 ) are shown in the following Table 2 . It is important to note that these domains are key things to be included in a protocol as mentioned in Sect. 1 .

General principles to reduce experimental bias in each of the above-mentioned domains ( Andrews et al. 2016 ; Knopp et al. 2015 ) are outlined in the following Table 3 .

5. Existing Guidelines and How to Use Them

There are resources to assist investigators in designing rigorous protocols and identify sources of bias. Cross-referencing to experimental reporting guidelines and checklists (e.g. ARRIVE (NC3Rs 2018a) , the NIH guidelines ( NIH 2018a ) and the Nature reporting of animal studies checklist ( Nature 2013 )) can be informative and helpful when planning an experimental protocol. However, it is important to bear in mind that these are primarily designed for reporting purposes and are not specifically designed for use in assisting with experimental design. There are more comprehensive planning guidelines specifically aiming at early experimental design stage. Henderson et al. identified 26 guidelines for in vivo experiments in animals in 2012 ( Henderson et al. 2013 ) (and a few more have been published since, like PREPARE ( Smith et al. 2018 ), developed by the NORECEPA (Norway’s National Consensus Platform for the advancement of the 3Rs), and PPRECISE for the field of pain research ( Andrews et al. 2016 )). Most of them have been developed for a specific research field but carry ideas and principles that can be transferred to all forms of in vivo experiments. Notable are, for example, the very detailed Lambeth Conventions ( Curtis et al. 2013 ) (developed for cardiac arrhythmia research), from Alzheimer’s research recommendations by Shineman et al. (2011) and generally applicable call by Landis et al. (2012) .

The authors of many of these guidelines state that their list might need adaption to the specific experiment. This is pointing out the general shortcoming that a fixed-item list can hardly foresee and account for any possible experimental situation and a blind ticking of boxes ticking of boxes is unlikely to improve experimental design. Such guidelines rather serve an educational purpose of making researchers aware of possible pitfalls and biases before the experimental conduct.

Two examples for a more adaptive and reactive way to serve a similar purpose should be stated: the NIH pages on rigour and reproducibility ( NIH 2018b ) provide in-depth information and collect important publications and workshop updates on these topics and have a funding scheme specifically for rigour and reproducibility. Second, using the Experimental Design Assistant (EDA) ( NC3Rs 2018b ; Percie du Sert et al. 2017 ) developed by the UK’s National Centre for the 3Rs (NC3Rs), a free to use online platform guiding researchers through experimental planning will give researchers the opportunity to adopt guideline and rigour principles precisely to their needs. The researcher creates a flow diagram of their experimental set-up grouped in three domains: the experiment (general questions on hypotheses and aims, animals used, animal strains, etc.), the practical steps (experimental conduct, assessment, etc.) and the analysis stage (e.g. outcome measures, statistical methods, data processing). Unlike a fixed checklist, the EDA checks the specific design as presented by the experimenter within the tool using logic algorithms. The user is then faced with the flaws the EDA identified and can adjust their design accordingly. This process can go through multiple rounds, by that forming a dynamic feedback loop educating the researcher and providing more nuanced assistance than a static checklist can.

While this process, however valid, might take time, the following steps of the EDA actively guide researchers through crucial and complex questions of the experiment, by suggesting fitting methods of statistical analyses of the experiment and subsequently carrying out sample size calculations. The EDA can then also generate a randomization sequence or compile a report of the planned experiment that can, e.g. be part of a preregistration of the experimental protocol.

6. Exploratory and Confirmatory Research

It is necessary to understand that there are in general two types of preclinical research, namely, exploratory and confirmatory research, respectively. Figure 1 shows that exploratory studies mainly aim to produce theories regarding the pathophysiology of disease (hypothesis generating), while confirmatory studies seek to reproduce exploratory findings as clearly defined intervention effects in relevant animal models (hypothesis testing). The next chapter will deal with exploratory and confirmatory studies in details. Similar standards of rigour are advisable for both forms of studies; this may be achieved by conforming to the general principles for the major domains that are outlined in Table 2 and incorporating these principles into a protocol that can be registered and published. It is important to note that both exploratory and confirmatory research can be closely linked: sometimes there can be exploratory and confirmatory components within the same studies. For example, a newly generated knockout mouse model is used to examine the effect of knockout on one specific phenotype (hypothesis testing–confirmatory) but may also describe a variety of other phenotypic characteristics as well (hypothesis generating–exploratory). Therefore, extra care in terms of applying these principles to reduce experimental bias would be needed before the commence of the studies. It also worth noting that sometimes it might not be compulsory or necessary to use some of the principles during exploratory studies such as sample size estimation and blinding which are albeit of highest importance in confirmatory research.

However, it is necessary to recognise how hypothesis confirming and hypothesis generating research relate to each other: while confirmatory research can turn into exploratory (e.g. if the findings are contrary to the hypothesis, this can lead to a new hypothesis that can be tested in a separate experiment), under no circumstances exploratory findings should be disseminated as the result of hypothesis confirming research by fitting a hypothesis to your results, i.e. to your p -values (often called HARKing = hypothesising after results are known or p -hacking = sifting through a multitude of p -values to find one below 0.05).

In conclusion, this chapter provides general concepts and principles that are important for the design and conduct of preclinical in vivo experiments, including experimental biases and how to reduce these biases in order to achieve the highest levels of rigour for hypothesis generating research using animals. The chapter should be used in conjunction with other relevant chapters in the handbook such as chapters “Blinding and Randomization”, “Minimum Information and Quality Standards for Conducting, Reporting, and Organizing In Vitro Research”, “Minimum Information in In Vivo Research”, “A Reckless Guide to P -Values: Local Evidence, Global Errors”, and “Quality Governance in Biomedical Research”.

Contributor Information

Wenlong Huang, Institute of Medical Sciences, School of Medicine, Medical Sciences and Nutrition, University of Aberdeen, Aberdeen, UK.

Nathalie Percie du Sert, NC3Rs, London, UK.

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Null Hypothesis Examples

ThoughtCo / Hilary Allison

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• Ph.D., Biomedical Sciences, University of Tennessee at Knoxville
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The null hypothesis —which assumes that there is no meaningful relationship between two variables—may be the most valuable hypothesis for the scientific method because it is the easiest to test using a statistical analysis. This means you can support your hypothesis with a high level of confidence. Testing the null hypothesis can tell you whether your results are due to the effect of manipulating ​ the dependent variable or due to chance.

What Is the Null Hypothesis?

The null hypothesis states there is no relationship between the measured phenomenon (the dependent variable) and the independent variable . You do not​ need to believe that the null hypothesis is true to test it. On the contrary, you will likely suspect that there is a relationship between a set of variables. One way to prove that this is the case is to reject the null hypothesis. Rejecting a hypothesis does not mean an experiment was "bad" or that it didn't produce results. In fact, it is often one of the first steps toward further inquiry.

To distinguish it from other hypotheses, the null hypothesis is written as ​ H 0  (which is read as “H-nought,” "H-null," or "H-zero"). A significance test is used to determine the likelihood that the results supporting the null hypothesis are not due to chance. A confidence level of 95 percent or 99 percent is common. Keep in mind, even if the confidence level is high, there is still a small chance the null hypothesis is not true, perhaps because the experimenter did not account for a critical factor or because of chance. This is one reason why it's important to repeat experiments.

Examples of the Null Hypothesis

To write a null hypothesis, first start by asking a question. Rephrase that question in a form that assumes no relationship between the variables. In other words, assume a treatment has no effect. Write your hypothesis in a way that reflects this.

• What Is a Hypothesis? (Science)
• What 'Fail to Reject' Means in a Hypothesis Test
• What Are the Elements of a Good Hypothesis?
• Scientific Method Vocabulary Terms
• Null Hypothesis Definition and Examples
• Definition of a Hypothesis
• Six Steps of the Scientific Method
• Hypothesis Test for the Difference of Two Population Proportions
• Understanding Simple vs Controlled Experiments
• What Is the Difference Between Alpha and P-Values?
• Null Hypothesis and Alternative Hypothesis
• What Are Examples of a Hypothesis?
• What It Means When a Variable Is Spurious
• Hypothesis Test Example
• How to Conduct a Hypothesis Test
• What Is a P-Value?