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When Do You Reject the Null Hypothesis? (3 Examples)

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

We always use the following steps to perform a hypothesis test:

Step 1: State the null and alternative hypotheses.

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

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

2. Determine a significance level to use.

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

3. Calculate the test statistic and p-value.

Use the sample data to calculate a test statistic and a corresponding p-value .

4. Reject or fail to reject the null hypothesis.

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

If the p-value is not less than the significance level, then you fail to reject the null hypothesis.

You can use the following clever line to remember this rule:

“If the p is low, the null must go.”

In other words, if the p-value is low enough then we must reject the null hypothesis.

The following examples show when to reject (or fail to reject) the null hypothesis for the most common types of hypothesis tests.

Example 1: One Sample t-test

A  one sample t-test  is used to test whether or not the mean of a population is equal to some value.

For example, suppose we want to know whether or not the mean weight of a certain species of turtle is equal to 310 pounds.

We go out and collect a simple random sample of 40 turtles with the following information:

• Sample size n = 40
• Sample mean weight  x  = 300
• Sample standard deviation s = 18.5

We can use the following steps to perform a one sample t-test:

Step 1: State the Null and Alternative Hypotheses

We will perform the one sample t-test with the following hypotheses:

• H 0 :  μ = 310 (population mean is equal to 310 pounds)
• H A :  μ ≠ 310 (population mean is not equal to 310 pounds)

We will choose to use a significance level of 0.05 .

We can plug in the numbers for the sample size, sample mean, and sample standard deviation into this One Sample t-test Calculator to calculate the test statistic and p-value:

• t test statistic: -3.4187
• two-tailed p-value: 0.0015

Since the p-value (0.0015) is less than the significance level (0.05) we reject the null hypothesis .

We conclude that there is sufficient evidence to say that the mean weight of turtles in this population is not equal to 310 pounds.

Example 2: Two Sample t-test

A  two sample t-test is used to test whether or not two population means are equal.

For example, suppose we want to know whether or not the mean weight between two different species of turtles is equal.

We go out and collect a simple random sample from each population with the following information:

• Sample size n 1 = 40
• Sample mean weight  x 1  = 300
• Sample standard deviation s 1 = 18.5
• Sample size n 2 = 38
• Sample mean weight  x 2  = 305
• Sample standard deviation s 2 = 16.7

We can use the following steps to perform a two sample t-test:

We will perform the two sample t-test with the following hypotheses:

• H 0 :  μ 1  = μ 2 (the two population means are equal)
• H 1 :  μ 1  ≠ μ 2 (the two population means are not equal)

We will choose to use a significance level of 0.10 .

We can plug in the numbers for the sample sizes, sample means, and sample standard deviations into this Two Sample t-test Calculator to calculate the test statistic and p-value:

• t test statistic: -1.2508
• two-tailed p-value: 0.2149

Since the p-value (0.2149) is not less than the significance level (0.10) we fail to reject the null hypothesis .

We do not have sufficient evidence to say that the mean weight of turtles between these two populations is different.

Example 3: Paired Samples t-test

A paired samples t-test is used to compare the means of two samples when each observation in one sample can be paired with an observation in the other sample.

For example, suppose we want to know whether or not a certain training program is able to increase the max vertical jump of college basketball players.

To test this, we may recruit a simple random sample of 20 college basketball players and measure each of their max vertical jumps. Then, we may have each player use the training program for one month and then measure their max vertical jump again at the end of the month:

We can use the following steps to perform a paired samples t-test:

We will perform the paired samples t-test with the following hypotheses:

• H 0 :  μ before = μ after (the two population means are equal)
• H 1 :  μ before ≠ μ after (the two population means are not equal)

We will choose to use a significance level of 0.01 .

We can plug in the raw data for each sample into this Paired Samples t-test Calculator to calculate the test statistic and p-value:

• t test statistic: -3.226
• two-tailed p-value: 0.0045

Since the p-value (0.0045) is less than the significance level (0.01) we reject the null hypothesis .

We have sufficient evidence to say that the mean vertical jump before and after participating in the training program is not equal.

Bonus: Decision Rule Calculator 

You can use this decision rule calculator to automatically determine whether you should reject or fail to reject a null hypothesis for a hypothesis test based on the value of the test statistic.

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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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• 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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Module 8: Inference for One Proportion

Hypothesis testing (5 of 5), learning outcomes.

• Recognize type I and type II errors.

What Can Go Wrong: Two Types of Errors

Statistical investigations involve making decisions in the face of uncertainty, so there is always some chance of making a wrong decision. In hypothesis testing, two types of wrong decisions can occur.

If the null hypothesis is true, but we reject it, the error is a type I error.

If the null hypothesis is false, but we fail to reject it, the error is a type II error.

The following table summarizes type I and II errors.

Type I and type II errors are not caused by mistakes. These errors are the result of random chance. The data provide evidence for a conclusion that is false. It’s no one’s fault!

Data Use on Smart Phones

In a previous example, we looked at a hypothesis test about data usage on smart phones. The researcher investigated the claim that the mean data usage for all teens is greater than 62 MBs. The sample mean was 75 MBs. The P-value was approximately 0.023. In this situation, the P-value is the probability that we will get a sample mean of 75 MBs or higher if the true mean is 62 MBs.

Notice that the result (75 MBs) isn’t impossible, only very unusual. The result is rare enough that we question whether the null hypothesis is true. This is why we reject the null hypothesis. But it is possible that the null hypothesis hypothesis is true and the researcher happened to get a very unusual sample mean. In this case, the result is just due to chance, and the data have led to a type I error: rejecting the null hypothesis when it is actually true.

White Male Support for Obama in 2012

In a previous example, we conducted a hypothesis test using poll results to determine if white male support for Obama in 2012 will be less than 40%. Our poll of white males showed 35% planning to vote for Obama in 2012. Based on the sampling distribution, we estimated the P-value as 0.078. In this situation, the P-value is the probability that we will get a sample proportion of 0.35 or less if 0.40 of the population of white males support Obama.

At the 5% level, the poll did not give strong enough evidence for us to conclude that less than 40% of white males will vote for Obama in 2012.

Which type of error is possible in this situation? If, in fact, it is true that less than 40% of this population support Obama, then the data led to a type II error: failing to reject a null hypothesis that is false. In other words, we failed to accept an alternative hypothesis that is true.

We definitely did not make a type I error here because a type I error requires that we reject the null hypothesis.

What Is the Probability That We Will Make a Type I Error?

If the significance level is 5% (α = 0.05), then 5% of the time we will reject the null hypothesis (when it is true!). Of course we will not know if the null is true. But if it is, the natural variability that we expect in random samples will produce rare results 5% of the time. This makes sense because we assume the null hypothesis is true when we create the sampling distribution. We look at the variability in random samples selected from the population described by the null hypothesis.

Similarly, if the significance level is 1%, then 1% of the time sample results will be rare enough for us to reject the null hypothesis hypothesis. So if the null hypothesis is actually true, then by chance alone, 1% of the time we will reject a true null hypothesis. The probability of a type I error is therefore 1%.

In general, the probability of a type I error is α.

What Is the Probability That We Will Make a Type II Error?

The probability of a type I error, if the null hypothesis is true, is equal to the significance level. The probability of a type II error is much more complicated to calculate. We can reduce the risk of a type I error by using a lower significance level. The best way to reduce the risk of a type II error is by increasing the sample size. In theory, we could also increase the significance level, but doing so would increase the likelihood of a type I error at the same time. We discuss these ideas further in a later module.

A Fair Coin

In the long run, a fair coin lands heads up half of the time. (For this reason, a weighted coin is not fair.) We conducted a simulation in which each sample consists of 40 flips of a fair coin. Here is a simulated sampling distribution for the proportion of heads in 2,000 samples. Results ranged from 0.25 to 0.75.

In general, if the null hypothesis is true, the significance level gives the probability of making a type I error. If we conduct a large number of hypothesis tests using the same null hypothesis, then, a type I error will occur in a predictable percentage (α) of the hypothesis tests. This is a problem! If we run one hypothesis test and the data is significant at the 5% level, we have reasonably good evidence that the alternative hypothesis is true. If we run 20 hypothesis tests and the data in one of the tests is significant at the 5% level, it doesn’t tell us anything! We expect 5% of the tests (1 in 20) to show significant results just due to chance.

Cell Phones and Brain Cancer

The following is an excerpt from a 1999 New York Times article titled “Cell phones: questions but no answers,” as referenced by David S. Moore in Basic Practice of Statistics (4th ed., New York: W. H. Freeman, 2007):

• A hospital study that compared brain cancer patients and a similar group without brain cancer found no statistically significant association between cell phone use and a group of brain cancers known as gliomas. But when 20 types of glioma were considered separately, an association was found between cell phone use and one rare form. Puzzlingly, however, this risk appeared to decrease rather than increase with greater mobile phone use.

This is an example of a probable type I error. Suppose we conducted 20 hypotheses tests with the null hypothesis “Cell phone use is not associated with cancer” at the 5% level. We expect 1 in 20 (5%) to give significant results by chance alone when there is no association between cell phone use and cancer. So the conclusion that this one type of cancer is related to cell phone use is probably just a result of random chance and not an indication of an association.

Click here to see a fun cartoon that illustrates this same idea.

How Many People Are Telepathic?

Telepathy is the ability to read minds. Researchers used Zener cards in the early 1900s for experimental research into telepathy.

In a telepathy experiment, the “sender” looks at 1 of 5 Zener cards while the “receiver” guesses the symbol. This is repeated 40 times, and the proportion of correct responses is recorded. Because there are 5 cards, we expect random guesses to be right 20% of the time (1 out of 5) in the long run. So in 40 tries, 8 correct guesses, a proportion of 0.20, is common. But of course there will be variability even when someone is just guessing. Thirteen or more correct in 40 tries, a proportion of 0.325, is statistically significant at the 5% level. When people perform this well on the telepathy test, we conclude their performance is not due to chance and take it as an indication of the ability to read minds.

In the next section, “Hypothesis Test for a Population Proportion,” we learn the details of hypothesis testing for claims about a population proportion. Before we get into the details, we want to step back and think more generally about hypothesis testing. We close our introduction to hypothesis testing with a helpful analogy.

Courtroom Analogy for Hypothesis Tests

When a defendant stands trial for a crime, he or she is innocent until proven guilty. It is the job of the prosecution to present evidence showing that the defendant is guilty beyond a reasonable doubt . It is the job of the defense to challenge this evidence to establish a reasonable doubt. The jury weighs the evidence and makes a decision.

When a jury makes a decision, it has only two possible verdicts:

• Guilty: The jury concludes that there is enough evidence to convict the defendant. The evidence is so strong that there is not a reasonable doubt that the defendant is guilty.
• Not Guilty: The jury concludes that there is not enough evidence to conclude beyond a reasonable doubt that the person is guilty. Notice that they do not conclude that the person is innocent. This verdict says only that there is not enough evidence to return a guilty verdict.

How is this example like a hypothesis test?

The null hypothesis is “The person is innocent.” The alternative hypothesis is “The person is guilty.” The evidence is the data. In a courtroom, the person is assumed innocent until proven guilty. In a hypothesis test, we assume the null hypothesis is true until the data proves otherwise.

The two possible verdicts are similar to the two conclusions that are possible in a hypothesis test.

Reject the null hypothesis: When we reject a null hypothesis, we accept the alternative hypothesis. This is like a guilty verdict. The evidence is strong enough for the jury to reject the assumption of innocence. In a hypothesis test, the data is strong enough for us to reject the assumption that the null hypothesis is true.

Fail to reject the null hypothesis: When we fail to reject the null hypothesis, we are delivering a “not guilty” verdict. The jury concludes that the evidence is not strong enough to reject the assumption of innocence, so the evidence is too weak to support a guilty verdict. We conclude the data is not strong enough to reject the null hypothesis, so the data is too weak to accept the alternative hypothesis.

How does the courtroom analogy relate to type I and type II errors?

Type I error: The jury convicts an innocent person. By analogy, we reject a true null hypothesis and accept a false alternative hypothesis.

Type II error: The jury says a person is not guilty when he or she really is. By analogy, we fail to reject a null hypothesis that is false. In other words, we do not accept an alternative hypothesis when it is really true.

Let’s Summarize

In this section, we introduced the four-step process of hypothesis testing:

Step 1: Determine the hypotheses.

• The hypotheses are claims about the population(s).
• The null hypothesis is a hypothesis that the parameter equals a specific value.
• The alternative hypothesis is the competing claim that the parameter is less than, greater than, or not equal to the parameter value in the null. The claim that drives the statistical investigation is usually found in the alternative hypothesis.

Step 2: Collect the data.

Because the hypothesis test is based on probability, random selection or assignment is essential in data production.

Step 3: Assess the evidence.

• Use the data to find a P-value.
• The P-value is a probability statement about how unlikely the data is if the null hypothesis is true.
• More specifically, the P-value gives the probability of sample results at least as extreme as the data if the null hypothesis is true.

Step 4: Give the conclusion.

• A small P-value says the data is unlikely to occur if the null hypothesis is true. We therefore conclude that the null hypothesis is probably not true and that the alternative hypothesis is true instead.
• We often choose a significance level as a benchmark for judging if the P-value is small enough. If the P-value is less than or equal to the significance level, we reject the null hypothesis and accept the alternative hypothesis instead.
• If the P-value is greater than the significance level, we say we “fail to reject” the null hypothesis. We never say that we “accept” the null hypothesis. We just say that we don’t have enough evidence to reject it. This is equivalent to saying we don’t have enough evidence to support the alternative hypothesis.
• Our conclusion will respond to the research question, so we often state the conclusion in terms of the alternative hypothesis.

Inference is based on probability, so there is always uncertainty. Although we may have strong evidence against it, the null hypothesis may still be true. If this is the case, we have a type I error. Similarly, even if we fail to reject the null hypothesis, it does not mean the alternative hypothesis is false. In this case, we have a type II error. These errors are not the result of a mistake in conducting the hypothesis test. They occur because of random chance.

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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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Keyboard Shortcuts

Hypothesis testing.

Key Topics:

• Basic approach
• Null and alternative hypothesis
• Decision making and the p -value
• Z-test & Nonparametric alternative

Basic approach to hypothesis testing

• State a model describing the relationship between the explanatory variables and the outcome variable(s) in the population and the nature of the variability. State all of your assumptions .
• Specify the null and alternative hypotheses in terms of the parameters of the model.
• Invent a test statistic that will tend to be different under the null and alternative hypotheses.
• Using the assumptions of step 1, find the theoretical sampling distribution of the statistic under the null hypothesis of step 2. Ideally the form of the sampling distribution should be one of the “standard distributions”(e.g. normal, t , binomial..)
• Calculate a p -value , as the area under the sampling distribution more extreme than your statistic. Depends on the form of the alternative hypothesis.
• Choose your acceptable type 1 error rate (alpha) and apply the decision rule : reject the null hypothesis if the p-value is less than alpha, otherwise do not reject.
• \(\frac{\bar{X}-\mu_0}{\sigma / \sqrt{n}}\)
• general form is: (estimate - value we are testing)/(st.dev of the estimate)
• z-statistic follows N(0,1) distribution
• 2 × the area above |z|, area above z,or area below z, or
• compare the statistic to a critical value, |z| ≥ z α/2 , z ≥ z α , or z ≤ - z α
• Choose the acceptable level of Alpha = 0.05, we conclude …. ?

Making the Decision

It is either likely or unlikely that we would collect the evidence we did given the initial assumption. (Note: “likely” or “unlikely” is measured by calculating a probability!)

If it is likely , then we “ do not reject ” our initial assumption. There is not enough evidence to do otherwise.

If it is unlikely , then:

• either our initial assumption is correct and we experienced an unusual event or,
• our initial assumption is incorrect

In statistics, if it is unlikely, we decide to “ reject ” our initial assumption.

Example: Criminal Trial Analogy

First, state 2 hypotheses, the null hypothesis (“H 0 ”) and the alternative hypothesis (“H A ”)

• H 0 : Defendant is not guilty.
• H A : Defendant is guilty.

Usually the H 0 is a statement of “no effect”, or “no change”, or “chance only” about a population parameter.

While the H A , depending on the situation, is that there is a difference, trend, effect, or a relationship with respect to a population parameter.

• It can one-sided and two-sided.
• In two-sided we only care there is a difference, but not the direction of it. In one-sided we care about a particular direction of the relationship. We want to know if the value is strictly larger or smaller.

Then, collect evidence, such as finger prints, blood spots, hair samples, carpet fibers, shoe prints, ransom notes, handwriting samples, etc. (In statistics, the data are the evidence.)

Next, you make your initial assumption.

• Defendant is innocent until proven guilty.

In statistics, we always assume the null hypothesis is true .

Then, make a decision based on the available evidence.

• If there is sufficient evidence (“beyond a reasonable doubt”), reject the null hypothesis . (Behave as if defendant is guilty.)
• If there is not enough evidence, do not reject the null hypothesis . (Behave as if defendant is not guilty.)

If the observed outcome, e.g., a sample statistic, is surprising under the assumption that the null hypothesis is true, but more probable if the alternative is true, then this outcome is evidence against H 0 and in favor of H A .

An observed effect so large that it would rarely occur by chance is called statistically significant (i.e., not likely to happen by chance).

Using the p -value to make the decision

The p -value represents how likely we would be to observe such an extreme sample if the null hypothesis were true. The p -value is a probability computed assuming the null hypothesis is true, that the test statistic would take a value as extreme or more extreme than that actually observed. Since it's a probability, it is a number between 0 and 1. The closer the number is to 0 means the event is “unlikely.” So if p -value is “small,” (typically, less than 0.05), we can then reject the null hypothesis.

Significance level and p -value

Significance level, α, is a decisive value for p -value. In this context, significant does not mean “important”, but it means “not likely to happened just by chance”.

α is the maximum probability of rejecting the null hypothesis when the null hypothesis is true. If α = 1 we always reject the null, if α = 0 we never reject the null hypothesis. In articles, journals, etc… you may read: “The results were significant ( p <0.05).” So if p =0.03, it's significant at the level of α = 0.05 but not at the level of α = 0.01. If we reject the H 0 at the level of α = 0.05 (which corresponds to 95% CI), we are saying that if H 0 is true, the observed phenomenon would happen no more than 5% of the time (that is 1 in 20). If we choose to compare the p -value to α = 0.01, we are insisting on a stronger evidence!

So, what kind of error could we make? No matter what decision we make, there is always a chance we made an error.

Errors in Criminal Trial:

Errors in Hypothesis Testing

Type I error (False positive): The null hypothesis is rejected when it is true.

• α is the maximum probability of making a Type I error.

Type II error (False negative): The null hypothesis is not rejected when it is false.

• β is the probability of making a Type II error

There is always a chance of making one of these errors. But, a good scientific study will minimize the chance of doing so!

The power of a statistical test is its probability of rejecting the null hypothesis if the null hypothesis is false. That is, power is the ability to correctly reject H 0 and detect a significant effect. In other words, power is one minus the type II error risk.

\(\text{Power }=1-\beta = P\left(\text{reject} H_0 | H_0 \text{is false } \right)\)

Which error is worse?

Type I = you are innocent, yet accused of cheating on the test. Type II = you cheated on the test, but you are found innocent.

This depends on the context of the problem too. But in most cases scientists are trying to be “conservative”; it's worse to make a spurious discovery than to fail to make a good one. Our goal it to increase the power of the test that is to minimize the length of the CI.

We need to keep in mind:

• the effect of the sample size,
• the correctness of the underlying assumptions about the population,
• statistical vs. practical significance, etc…

(see the handout). To study the tradeoffs between the sample size, α, and Type II error we can use power and operating characteristic curves.

What type of error might we have made?

Type I error is claiming that average student height is not 65 inches, when it really is. Type II error is failing to claim that the average student height is not 65in when it is.

We rejected the null hypothesis, i.e., claimed that the height is not 65, thus making potentially a Type I error. But sometimes the p -value is too low because of the large sample size, and we may have statistical significance but not really practical significance! That's why most statisticians are much more comfortable with using CI than tests.

There is a need for a further generalization. What if we can't assume that σ is known? In this case we would use s (the sample standard deviation) to estimate σ.

If the sample is very large, we can treat σ as known by assuming that σ = s . According to the law of large numbers, this is not too bad a thing to do. But if the sample is small, the fact that we have to estimate both the standard deviation and the mean adds extra uncertainty to our inference. In practice this means that we need a larger multiplier for the standard error.

We need one-sample t -test.

One sample t -test

• Assume data are independently sampled from a normal distribution with unknown mean μ and variance σ 2 . Make an initial assumption, μ 0 .
• t-statistic: \(\frac{\bar{X}-\mu_0}{s / \sqrt{n}}\) where s is a sample st.dev.
• t-statistic follows t -distribution with df = n - 1
• Alpha = 0.05, we conclude ….

Testing for the population proportion

Let's go back to our CNN poll. Assume we have a SRS of 1,017 adults.

We are interested in testing the following hypothesis: H 0 : p = 0.50 vs. p > 0.50

What is the test statistic?

If alpha = 0.05, what do we conclude?

We will see more details in the next lesson on proportions, then distributions, and possible tests.

What 'Fail to Reject' Means in a Hypothesis Test

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In statistics , scientists can perform a number of different significance tests to determine if there is a relationship between two phenomena. One of the first they usually perform is a null hypothesis test. In short, the null hypothesis states that there is no meaningful relationship between two measured phenomena. After a performing a test, scientists can:

• Reject the null hypothesis (meaning there is a definite, consequential relationship between the two phenomena), or
• Fail to reject the null hypothesis (meaning the test has not identified a consequential relationship between the two phenomena)

Key Takeaways: The Null Hypothesis

• In a test of significance, the null hypothesis states that there is no meaningful relationship between two measured phenomena.

• By comparing the null hypothesis to an alternative hypothesis, scientists can either reject or fail to reject the null hypothesis.

• The null hypothesis cannot be positively proven. Rather, all that scientists can determine from a test of significance is that the evidence collected does or does not disprove the null hypothesis.

It is important to note that a failure to reject does not mean that the null hypothesis is true—only that the test did not prove it to be false. In some cases, depending on the experiment, a relationship may exist between two phenomena that is not identified by the experiment. In such cases, new experiments must be designed to rule out alternative hypotheses.

Null vs. Alternative Hypothesis

The null hypothesis is considered the default in a scientific experiment . In contrast, an alternative hypothesis is one that claims that there is a meaningful relationship between two phenomena. These two competing hypotheses can be compared by performing a statistical hypothesis test, which determines whether there is a statistically significant relationship between the data.

For example, scientists studying the water quality of a stream may wish to determine whether a certain chemical affects the acidity of the water. The null hypothesis—that the chemical has no effect on the water quality—can be tested by measuring the pH level of two water samples, one of which contains some of the chemical and one of which has been left untouched. If the sample with the added chemical is measurably more or less acidic—as determined through statistical analysis—it is a reason to reject the null hypothesis. If the sample's acidity is unchanged, it is a reason to not reject the null hypothesis.

When scientists design experiments, they attempt to find evidence for the alternative hypothesis. They do not try to prove that the null hypothesis is true. The null hypothesis is assumed to be an accurate statement until contrary evidence proves otherwise. As a result, a test of significance does not produce any evidence pertaining to the truth of the null hypothesis.

Failing to Reject vs. Accept

In an experiment, the null hypothesis and the alternative hypothesis should be carefully formulated such that one and only one of these statements is true. If the collected data supports the alternative hypothesis, then the null hypothesis can be rejected as false. However, if the data does not support the alternative hypothesis, this does not mean that the null hypothesis is true. All it means is that the null hypothesis has not been disproven—hence the term "failure to reject." A "failure to reject" a hypothesis should not be confused with acceptance.

In mathematics, negations are typically formed by simply placing the word “not” in the correct place. Using this convention, tests of significance allow scientists to either reject or not reject the null hypothesis. It sometimes takes a moment to realize that “not rejecting” is not the same as "accepting."

Null Hypothesis Example

In many ways, the philosophy behind a test of significance is similar to that of a trial. At the beginning of the proceedings, when the defendant enters a plea of “not guilty,” it is analogous to the statement of the null hypothesis. While the defendant may indeed be innocent, there is no plea of “innocent” to be formally made in court. The alternative hypothesis of “guilty” is what the prosecutor attempts to demonstrate.

The presumption at the outset of the trial is that the defendant is innocent. In theory, there is no need for the defendant to prove that he or she is innocent. The burden of proof is on the prosecuting attorney, who must marshal enough evidence to convince the jury that the defendant is guilty beyond a reasonable doubt. Likewise, in a test of significance, a scientist can only reject the null hypothesis by providing evidence for the alternative hypothesis.

If there is not enough evidence in a trial to demonstrate guilt, then the defendant is declared “not guilty.” This claim has nothing to do with innocence; it merely reflects the fact that the prosecution failed to provide enough evidence of guilt. In a similar way, a failure to reject the null hypothesis in a significance test does not mean that the null hypothesis is true. It only means that the scientist was unable to provide enough evidence for the alternative hypothesis.

For example, scientists testing the effects of a certain pesticide on crop yields might design an experiment in which some crops are left untreated and others are treated with varying amounts of pesticide. Any result in which the crop yields varied based on pesticide exposure—assuming all other variables are equal—would provide strong evidence for the alternative hypothesis (that the pesticide does affect crop yields). As a result, the scientists would have reason to reject the null hypothesis.

• Hypothesis Test for the Difference of Two Population Proportions
• Type I and Type II Errors in Statistics
• Null Hypothesis and Alternative Hypothesis
• Null Hypothesis Examples
• How to Conduct a Hypothesis Test
• An Example of a Hypothesis Test
• What Is a P-Value?
• The Difference Between Type I and Type II Errors in Hypothesis Testing
• What Is a Hypothesis? (Science)
• Null Hypothesis Definition and Examples
• Hypothesis Test Example
• The Runs Test for Random Sequences
• How to Do Hypothesis Tests With the Z.TEST Function in Excel
• Scientific Method Vocabulary Terms
• What Is the Difference Between Alpha and P-Values?
• Chi-Square Goodness of Fit Test

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Methodology

• How to Write a Strong Hypothesis | Steps & Examples

How to Write a Strong Hypothesis | Steps & Examples

Published on May 6, 2022 by Shona McCombes . Revised on November 20, 2023.

A hypothesis is a statement that can be tested by scientific research. If you want to test a relationship between two or more variables, you need to write hypotheses before you start your experiment or data collection .

Example: Hypothesis

Daily apple consumption leads to fewer doctor’s visits.

Table of contents

What is a hypothesis, developing a hypothesis (with example), hypothesis examples, other interesting articles, frequently asked questions about writing hypotheses.

A hypothesis states your predictions about what your research will find. It is a tentative answer to your research question that has not yet been tested. For some research projects, you might have to write several hypotheses that address different aspects of your research question.

A hypothesis is not just a guess – it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations and statistical analysis of data).

Variables in hypotheses

Hypotheses propose a relationship between two or more types of variables .

• An independent variable is something the researcher changes or controls.
• A dependent variable is something the researcher observes and measures.

If there are any control variables , extraneous variables , or confounding variables , be sure to jot those down as you go to minimize the chances that research bias  will affect your results.

In this example, the independent variable is exposure to the sun – the assumed cause . The dependent variable is the level of happiness – the assumed effect .

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Step 1. ask a question.

Writing a hypothesis begins with a research question that you want to answer. The question should be focused, specific, and researchable within the constraints of your project.

Step 2. Do some preliminary research

Your initial answer to the question should be based on what is already known about the topic. Look for theories and previous studies to help you form educated assumptions about what your research will find.

At this stage, you might construct a conceptual framework to ensure that you’re embarking on a relevant topic . This can also help you identify which variables you will study and what you think the relationships are between them. Sometimes, you’ll have to operationalize more complex constructs.

Step 3. Formulate your hypothesis

Now you should have some idea of what you expect to find. Write your initial answer to the question in a clear, concise sentence.

4. Refine your hypothesis

You need to make sure your hypothesis is specific and testable. There are various ways of phrasing a hypothesis, but all the terms you use should have clear definitions, and the hypothesis should contain:

• The relevant variables
• The specific group being studied
• The predicted outcome of the experiment or analysis

5. Phrase your hypothesis in three ways

To identify the variables, you can write a simple prediction in  if…then form. The first part of the sentence states the independent variable and the second part states the dependent variable.

In academic research, hypotheses are more commonly phrased in terms of correlations or effects, where you directly state the predicted relationship between variables.

If you are comparing two groups, the hypothesis can state what difference you expect to find between them.

6. Write a null hypothesis

If your research involves statistical hypothesis testing , you will also have to write a null hypothesis . The null hypothesis is the default position that there is no association between the variables. The null hypothesis is written as H 0 , while the alternative hypothesis is H 1 or H a .

• H 0 : The number of lectures attended by first-year students has no effect on their final exam scores.
• H 1 : The number of lectures attended by first-year students has a positive effect on their final exam scores.

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

• Sampling methods
• Simple random sampling
• Stratified sampling
• Cluster sampling
• Likert scales
• Reproducibility

 Statistics

• Null hypothesis
• Statistical power
• Probability distribution
• Effect size
• Poisson distribution

Research bias

• Optimism bias
• Cognitive bias
• Implicit bias
• Hawthorne effect
• Anchoring bias
• Explicit bias

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A hypothesis is not just a guess — it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations and statistical analysis of data).

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.

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.

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McCombes, S. (2023, November 20). How to Write a Strong Hypothesis | Steps & Examples. Scribbr. Retrieved April 11, 2024, from https://www.scribbr.com/methodology/hypothesis/

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Chapter 2: Summarizing and Visualizing Data

Chapter 3: measure of central tendency, chapter 4: measures of variation, chapter 5: measures of relative standing, chapter 6: probability distributions, chapter 7: estimates, chapter 8: distributions, chapter 9: hypothesis testing, chapter 10: analysis of variance, chapter 11: correlation and regression, chapter 12: statistics in practice.

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In an experiment, a farm with infected plants is subjected to a widely applicable insecticide.

This insecticide is expected to increase the number of healthy plants after its application. However, at the end of the experiment, the proportion of healthy and infected plants remained the same.

Here, the null hypothesis that the insecticide has no effect seems to hold, but should one accept the hypothesis or fail to reject it?

Accepting this hypothesis would mean that the insecticide is ineffective and cannot improve the plants' health.

This decision actually overlooks the other plausible explanations for the observed results.

In this case, using an unprescribed amount or concentration of insecticide might have resulted in no effect.

There is a possibility of plants being infected by something that the insecticide cannot target.

Failing to reject a null hypothesis means there is no sufficient evidence for the expected or the observed effect.

Today, if scientists had accepted null hypotheses, the discovery of plant viruses or the rediscovery of many extinct species would not have been possible.

9.8: Hypothesis: Accept or Fail to Reject?

The outcome of any hypothesis testing leads to rejecting or not rejecting the null hypothesis. This decision is taken based on the analysis of the data, an appropriate test statistic, an appropriate confidence level, the critical values, and P -values. However, when the evidence suggests that the null hypothesis cannot be rejected, is it right to say, 'Accept' the null hypothesis?

There are two ways to indicate that the null hypothesis is not rejected. 'Accept' the null hypothesis and 'fail to reject' the null hypothesis. Superficially, both these phrases mean the same, but in statistics, the meanings are somewhat different. The phrase 'accept the null hypothesis' implies that the null hypothesis is by nature true, and it is proved. But a hypothesis test simply provides information that there is no sufficient evidence in support of the alternative hypothesis, and therefore the null hypothesis cannot be rejected. The null hypothesis cannot be proven, although the hypothesis test begins with an assumption that the hypothesis is true, and the final result indicates the failure of the rejection of the null hypothesis. Thus, it is always advisable to state 'fail to reject the null hypothesis' instead of 'accept the null hypothesis.'

'Accepting' a hypothesis may also imply that the given hypothesis is now proven, so there is no need to study it further. Nevertheless, that is never the case, as newer scientific evidence often challenges the existing studies. Discovery of viruses and fossils, rediscovery of presumed extinct species, criminal trials, and novel drug tests follow the same principles of testing hypotheses. In those cases, 'accepting' a hypothesis may lead to severe consequences.

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11.3: Hypotheses in ANOVA

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So far we have seen what ANOVA is used for, why we use it, and how we use it. Now we can turn to the formal hypotheses we will be testing. As with before, we have a null and a research hypothesis to lay out.

Research Hypotheses

Our research hypothesis for ANOVA is more complex with more than two groups. Let’s take a look at it and then dive deeper into what it means.

What the ANOVA tests is whether there is a difference between any one set of means, but usually we still have expected directions of what means we think will be bigger than what other means. Let's work out an example. Let's say that my IV is mindset, and the three groups (levels) are:

• Growth Mindset
• Mixed Mindset (some Growth ideas and some Fixed ideas)
• Fixed Mindset

If we are measuring passing rates in math, we could write this all out in one sentence and one line of symbols:

• Research Hypothesis: Students with Growth Mindset with have higher average passing rates in math than students with either a mixed mindset or Fixed Mindset, but Fixed Mindset will have similar average passing rates to students with mixed mindset.
• Symbols: \( \overline{X}_{G} > \overline{X}_{M} = \overline{X}_{F} \)

But it ends up being easier to write out each pair of means:

• Research Hypothesis: Students with Growth Mindset with have higher average passing rates in math than students with a mixed mindset. Students with Growth Mindset with have higher average passing rates in math than students with a Fixed Mindset. Students with a Fixed Mindset will have similar average passing rates to students with mixed mindset.
• \( \overline{X}_{G} > \overline{X}_{M} \)
• \( \overline{X}_{G} > \overline{X}_{F} \)
• \( \overline{X}_{M} = \overline{X}_{F} \)

What you might notice is that one of these looks like a null hypothesis (no difference between the means)! And that is okay, as long as the research hypothesis predicts that at least one mean will differ from at least one other mean. It doesn't matter what order you list these means in; it helps to match the research hypothesis, but it's really to help you conceptualize the relationships that you are predicting so put it in the order that makes the most sense to you!

Why is it better to list out each pair of means? Well, look at this research hypothesis:

• Research Hypothesis: Students with Growth Mindset with have a similar average passing rate in math as students with a mixed mindset. Students with Growth Mindset with have higher average passing rates in math than students with a Fixed Mindset. Students with a Fixed Mindset will have similar average passing rates to students with mixed mindset.
• \( \overline{X}_{G} = \overline{X}_{M} \)

If you try to write that out in one line of symbols, it'll get confusing because you won't be able to easily show all three predictions. And if you have more than three groups, many research hypotheses won't be able to be represented in one line.

Another reason that this makes more sense is that each mean will be statistically compared with each other mean if the ANOVA results end up rejecting the null hypothesis. If you set up your research hypotheses this way in the first place (in pairs of means), then these pairwise comparisons make more sense later.

Null Hypotheses

Our null hypothesis is still the idea of “no difference” in our data. Because we have multiple group means, we simply list them out as equal to each other:

• Null Hypothesis: Students with Growth Mindset, mixed mindset, and Fixed Mindset will have similar average passing rates in math .
• Symbols: \( \overline{X}_{G} = \overline{X}_{M} = \overline{X}_{F} \)

You can list them all out, as well, but it's less necessary with a null hypothesis:

• Research Hypothesis: Students with Growth Mindset with have a similar average passing rate in math as students with a mixed mindset. Students with Growth Mindset with have a similar average passing rates in math than students with a Fixed Mindset. Students with a Fixed Mindset will have similar average passing rates to students with mixed mindset.
• \( \overline{X}_{G} = \overline{X}_{F} \)

Null Hypothesis Significance Testing

In our studies so far, when we've calculated an inferential test statistics, like a t-score, what do we do next? Compare it to a critical value in a table! And that's the same thing that we do with our calculated F-value. We compare our calculated value to our critical value to determine if we retain or reject the null hypothesis that all of the means are similar.

(Critical \(<\) Calculated) \(=\) Reject null \(=\) At least one mean is different from at least one other mean. \(= p<.05\)

(Critical \(>\) Calculated) \(=\) Retain null \(=\) All of the means are similar. \(= p>.05\)

What does Rejecting the Null Hypothesis Mean for a Research Hypothesis with Three or More Groups?

Remember when we rejected the null hypothesis when comparing two means with a t-test that we didn't have to do any additional comparisons; rejecting the null hypothesis with a t-test tells us that the two means are statistically significantly different, which means that the bigger mean was statistically significantly bigger. All we had to do was make sure that the means were in the direction that the research hypothesis predicted.

Unfortunately, with three or more group means, we do have to do additional statistical comparisons to test which means are statistically significantly different from which other means. The ANOVA only tells us that at least one mean is different from one other mean. So, rejecting the null hypothesis doesn't really tell us whether our research hypothesis is (fully) supported, partially supported, or not supported. When the null hypothesis is rejected, we will know that a difference exists somewhere, but we will not know where that difference is. Is Growth Mindset different from mixed mindset and Fixed Mindset, but mixed and Fixed are the same? Is Growth Mindset different from both mixed and Fixed Mindset? Are all three of them different from each other? And even if the means are different, are they different in the hypothesized direction? Does Growth Mindset always have a higher mean? We will come back to this issue later and see how to find out specific differences. For now, just remember that an ANOVA tests for any difference in group means, and it does not matter where that difference occurs. We must follow-up with any significant ANOVA to see which means are different from each other, and whether those mean differences (fully) support, partially support, or do not support the research hypothesis.