the research hypothesis should be rejected

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

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

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

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

Consider the following example where we set up these hypotheses.

Example 6-1 Section  

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

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

• The man is guilty
• The man is innocent

Let's set up the null and alternative hypotheses.

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

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

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

The Logic of Hypothesis Testing Section  

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

The decision is either going to be...

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

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

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

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

Types of errors

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

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

Example 6-1 Cont'd... Section  

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

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

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

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

Try it! Section  

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

• Building is safe
• Building is not safe

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

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

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

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Chapter 13: Inferential Statistics

Understanding Null Hypothesis Testing

Learning Objectives

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

The Purpose of Null Hypothesis Testing

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

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

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

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

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

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

The Logic of Null Hypothesis Testing

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

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

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

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

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

The Misunderstood  p  Value

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

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

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

Role of Sample Size and Relationship Strength

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

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

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

Statistical Significance Versus Practical Significance

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

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

Key Takeaways

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

Long Descriptions

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

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

Media Attributions

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

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

The random variability in a statistic from sample to sample.

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

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

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

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

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

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

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

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

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

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Type 1 and Type 2 Errors in Statistics

Saul Mcleod, PhD

Editor-in-Chief for Simply Psychology

BSc (Hons) Psychology, MRes, PhD, University of Manchester

Saul Mcleod, PhD., is a qualified psychology teacher with over 18 years of experience in further and higher education. He has been published in peer-reviewed journals, including the Journal of Clinical Psychology.

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A statistically significant result cannot prove that a research hypothesis is correct (which implies 100% certainty). Because a p -value is based on probabilities, there is always a chance of making an incorrect conclusion regarding accepting or rejecting the null hypothesis ( H 0 ).

Anytime we make a decision using statistics, there are four possible outcomes, with two representing correct decisions and two representing errors.

The chances of committing these two types of errors are inversely proportional: that is, decreasing type I error rate increases type II error rate and vice versa.

As the significance level (α) increases, it becomes easier to reject the null hypothesis, decreasing the chance of missing a real effect (Type II error, β). If the significance level (α) goes down, it becomes harder to reject the null hypothesis , increasing the chance of missing an effect while reducing the risk of falsely finding one (Type I error).

Type I error 

A type 1 error is also known as a false positive and occurs when a researcher incorrectly rejects a true null hypothesis. Simply put, it’s a false alarm.

This means that you report that your findings are significant when they have occurred by chance.

The probability of making a type 1 error is represented by your alpha level (α), the p- value below which you reject the null hypothesis.

A p -value of 0.05 indicates that you are willing to accept a 5% chance of getting the observed data (or something more extreme) when the null hypothesis is true.

You can reduce your risk of committing a type 1 error by setting a lower alpha level (like α = 0.01). For example, a p-value of 0.01 would mean there is a 1% chance of committing a Type I error.

However, using a lower value for alpha means that you will be less likely to detect a true difference if one really exists (thus risking a type II error).

Scenario: Drug Efficacy Study

Imagine a pharmaceutical company is testing a new drug, named “MediCure”, to determine if it’s more effective than a placebo at reducing fever. They experimented with two groups: one receives MediCure, and the other received a placebo.

• Null Hypothesis (H0) : MediCure is no more effective at reducing fever than the placebo.
• Alternative Hypothesis (H1) : MediCure is more effective at reducing fever than the placebo.

After conducting the study and analyzing the results, the researchers found a p-value of 0.04.

If they use an alpha (α) level of 0.05, this p-value is considered statistically significant, leading them to reject the null hypothesis and conclude that MediCure is more effective than the placebo.

However, MediCure has no actual effect, and the observed difference was due to random variation or some other confounding factor. In this case, the researchers have incorrectly rejected a true null hypothesis.

Error : The researchers have made a Type 1 error by concluding that MediCure is more effective when it isn’t.

Implications

Resource Allocation : Making a Type I error can lead to wastage of resources. If a business believes a new strategy is effective when it’s not (based on a Type I error), they might allocate significant financial and human resources toward that ineffective strategy.

Unnecessary Interventions : In medical trials, a Type I error might lead to the belief that a new treatment is effective when it isn’t. As a result, patients might undergo unnecessary treatments, risking potential side effects without any benefit.

Reputation and Credibility : For researchers, making repeated Type I errors can harm their professional reputation. If they frequently claim groundbreaking results that are later refuted, their credibility in the scientific community might diminish.

Type II error

A type 2 error (or false negative) happens when you accept the null hypothesis when it should actually be rejected.

Here, a researcher concludes there is not a significant effect when actually there really is.

The probability of making a type II error is called Beta (β), which is related to the power of the statistical test (power = 1- β). You can decrease your risk of committing a type II error by ensuring your test has enough power.

You can do this by ensuring your sample size is large enough to detect a practical difference when one truly exists.

Scenario: Efficacy of a New Teaching Method

Educational psychologists are investigating the potential benefits of a new interactive teaching method, named “EduInteract”, which utilizes virtual reality (VR) technology to teach history to middle school students.

They hypothesize that this method will lead to better retention and understanding compared to the traditional textbook-based approach.

• Null Hypothesis (H0) : The EduInteract VR teaching method does not result in significantly better retention and understanding of history content than the traditional textbook method.
• Alternative Hypothesis (H1) : The EduInteract VR teaching method results in significantly better retention and understanding of history content than the traditional textbook method.

The researchers designed an experiment where one group of students learns a history module using the EduInteract VR method, while a control group learns the same module using a traditional textbook.

After a week, the student’s retention and understanding are tested using a standardized assessment.

Upon analyzing the results, the psychologists found a p-value of 0.06. Using an alpha (α) level of 0.05, this p-value isn’t statistically significant.

Therefore, they fail to reject the null hypothesis and conclude that the EduInteract VR method isn’t more effective than the traditional textbook approach.

However, let’s assume that in the real world, the EduInteract VR truly enhances retention and understanding, but the study failed to detect this benefit due to reasons like small sample size, variability in students’ prior knowledge, or perhaps the assessment wasn’t sensitive enough to detect the nuances of VR-based learning.

Error : By concluding that the EduInteract VR method isn’t more effective than the traditional method when it is, the researchers have made a Type 2 error.

This could prevent schools from adopting a potentially superior teaching method that might benefit students’ learning experiences.

Missed Opportunities : A Type II error can lead to missed opportunities for improvement or innovation. For example, in education, if a more effective teaching method is overlooked because of a Type II error, students might miss out on a better learning experience.

Potential Risks : In healthcare, a Type II error might mean overlooking a harmful side effect of a medication because the research didn’t detect its harmful impacts. As a result, patients might continue using a harmful treatment.

Stagnation : In the business world, making a Type II error can result in continued investment in outdated or less efficient methods. This can lead to stagnation and the inability to compete effectively in the marketplace.

How do Type I and Type II errors relate to psychological research and experiments?

Type I errors are like false alarms, while Type II errors are like missed opportunities. Both errors can impact the validity and reliability of psychological findings, so researchers strive to minimize them to draw accurate conclusions from their studies.

How does sample size influence the likelihood of Type I and Type II errors in psychological research?

Sample size in psychological research influences the likelihood of Type I and Type II errors. A larger sample size reduces the chances of Type I errors, which means researchers are less likely to mistakenly find a significant effect when there isn’t one.

A larger sample size also increases the chances of detecting true effects, reducing the likelihood of Type II errors.

Are there any ethical implications associated with Type I and Type II errors in psychological research?

Yes, there are ethical implications associated with Type I and Type II errors in psychological research.

Type I errors may lead to false positive findings, resulting in misleading conclusions and potentially wasting resources on ineffective interventions. This can harm individuals who are falsely diagnosed or receive unnecessary treatments.

Type II errors, on the other hand, may result in missed opportunities to identify important effects or relationships, leading to a lack of appropriate interventions or support. This can also have negative consequences for individuals who genuinely require assistance.

Therefore, minimizing these errors is crucial for ethical research and ensuring the well-being of participants.

Further Information

• Publication manual of the American Psychological Association
• Statistics for Psychology Book Download

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
• What Level of Alpha Determines Statistical Significance?
• 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?

What Is A Research (Scientific) Hypothesis? A plain-language explainer + examples

By:  Derek Jansen (MBA)  | Reviewed By: Dr Eunice Rautenbach | June 2020

If you’re new to the world of research, or it’s your first time writing a dissertation or thesis, you’re probably noticing that the words “research hypothesis” and “scientific hypothesis” are used quite a bit, and you’re wondering what they mean in a research context .

“Hypothesis” is one of those words that people use loosely, thinking they understand what it means. However, it has a very specific meaning within academic research. So, it’s important to understand the exact meaning before you start hypothesizing. 

Research Hypothesis 101

• What is a hypothesis ?
• What is a research hypothesis (scientific hypothesis)?
• Requirements for a research hypothesis
• Definition of a research hypothesis
• The null hypothesis

What is a hypothesis?

Let’s start with the general definition of a hypothesis (not a research hypothesis or scientific hypothesis), according to the Cambridge Dictionary:

Hypothesis: an idea or explanation for something that is based on known facts but has not yet been proved.

In other words, it’s a statement that provides an explanation for why or how something works, based on facts (or some reasonable assumptions), but that has not yet been specifically tested . For example, a hypothesis might look something like this:

Hypothesis: sleep impacts academic performance.

This statement predicts that academic performance will be influenced by the amount and/or quality of sleep a student engages in – sounds reasonable, right? It’s based on reasonable assumptions , underpinned by what we currently know about sleep and health (from the existing literature). So, loosely speaking, we could call it a hypothesis, at least by the dictionary definition.

But that’s not good enough…

Unfortunately, that’s not quite sophisticated enough to describe a research hypothesis (also sometimes called a scientific hypothesis), and it wouldn’t be acceptable in a dissertation, thesis or research paper . In the world of academic research, a statement needs a few more criteria to constitute a true research hypothesis .

What is a research hypothesis?

A research hypothesis (also called a scientific hypothesis) is a statement about the expected outcome of a study (for example, a dissertation or thesis). To constitute a quality hypothesis, the statement needs to have three attributes – specificity , clarity and testability .

Let’s take a look at these more closely.

Need a helping hand?

Hypothesis Essential #1: Specificity & Clarity

A good research hypothesis needs to be extremely clear and articulate about both what’ s being assessed (who or what variables are involved ) and the expected outcome (for example, a difference between groups, a relationship between variables, etc.).

Let’s stick with our sleepy students example and look at how this statement could be more specific and clear.

Hypothesis: Students who sleep at least 8 hours per night will, on average, achieve higher grades in standardised tests than students who sleep less than 8 hours a night.

As you can see, the statement is very specific as it identifies the variables involved (sleep hours and test grades), the parties involved (two groups of students), as well as the predicted relationship type (a positive relationship). There’s no ambiguity or uncertainty about who or what is involved in the statement, and the expected outcome is clear.

Contrast that to the original hypothesis we looked at – “Sleep impacts academic performance” – and you can see the difference. “Sleep” and “academic performance” are both comparatively vague , and there’s no indication of what the expected relationship direction is (more sleep or less sleep). As you can see, specificity and clarity are key.

Hypothesis Essential #2: Testability (Provability)

A statement must be testable to qualify as a research hypothesis. In other words, there needs to be a way to prove (or disprove) the statement. If it’s not testable, it’s not a hypothesis – simple as that.

For example, consider the hypothesis we mentioned earlier:

Hypothesis: Students who sleep at least 8 hours per night will, on average, achieve higher grades in standardised tests than students who sleep less than 8 hours a night.  

We could test this statement by undertaking a quantitative study involving two groups of students, one that gets 8 or more hours of sleep per night for a fixed period, and one that gets less. We could then compare the standardised test results for both groups to see if there’s a statistically significant difference. 

Again, if you compare this to the original hypothesis we looked at – “Sleep impacts academic performance” – you can see that it would be quite difficult to test that statement, primarily because it isn’t specific enough. How much sleep? By who? What type of academic performance?

So, remember the mantra – if you can’t test it, it’s not a hypothesis 🙂

Defining A Research Hypothesis

You’re still with us? Great! Let’s recap and pin down a clear definition of a hypothesis.

A research hypothesis (or scientific hypothesis) is a statement about an expected relationship between variables, or explanation of an occurrence, that is clear, specific and testable.

So, when you write up hypotheses for your dissertation or thesis, make sure that they meet all these criteria. If you do, you’ll not only have rock-solid hypotheses but you’ll also ensure a clear focus for your entire research project.

What about the null hypothesis?

You may have also heard the terms null hypothesis , alternative hypothesis, or H-zero thrown around. At a simple level, the null hypothesis is the counter-proposal to the original hypothesis.

For example, if the hypothesis predicts that there is a relationship between two variables (for example, sleep and academic performance), the null hypothesis would predict that there is no relationship between those variables.

At a more technical level, the null hypothesis proposes that no statistical significance exists in a set of given observations and that any differences are due to chance alone.

And there you have it – hypotheses in a nutshell. 

If you have any questions, be sure to leave a comment below and we’ll do our best to help you. If you need hands-on help developing and testing your hypotheses, consider our private coaching service , where we hold your hand through the research journey.

Psst... there’s more!

This post was based on one of our popular Research Bootcamps . If you're working on a research project, you'll definitely want to check this out ...

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16 Comments

Very useful information. I benefit more from getting more information in this regard.

Very great insight,educative and informative. Please give meet deep critics on many research data of public international Law like human rights, environment, natural resources, law of the sea etc

In a book I read a distinction is made between null, research, and alternative hypothesis. As far as I understand, alternative and research hypotheses are the same. Can you please elaborate? Best Afshin

This is a self explanatory, easy going site. I will recommend this to my friends and colleagues.

Very good definition. How can I cite your definition in my thesis? Thank you. Is nul hypothesis compulsory in a research?

It’s a counter-proposal to be proven as a rejection

Please what is the difference between alternate hypothesis and research hypothesis?

It is a very good explanation. However, it limits hypotheses to statistically tasteable ideas. What about for qualitative researches or other researches that involve quantitative data that don’t need statistical tests?

In qualitative research, one typically uses propositions, not hypotheses.

could you please elaborate it more

I’ve benefited greatly from these notes, thank you.

This is very helpful

well articulated ideas are presented here, thank you for being reliable sources of information

Excellent. Thanks for being clear and sound about the research methodology and hypothesis (quantitative research)

I have only a simple question regarding the null hypothesis. – Is the null hypothesis (Ho) known as the reversible hypothesis of the alternative hypothesis (H1? – How to test it in academic research?

this is very important note help me much more

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Statistics Made Easy

Understanding the Null Hypothesis for ANOVA Models

A one-way ANOVA is used to determine if there is a statistically significant difference between the mean of three or more independent groups.

A one-way ANOVA uses the following null and alternative hypotheses:

• H 0 :  μ 1  = μ 2  = μ 3  = … = μ k  (all of the group means are equal)
• H A : At least one group mean is different   from the rest

To decide if we should reject or fail to reject the null hypothesis, we must refer to the p-value in the output of the ANOVA table.

If the p-value is less than some significance level (e.g. 0.05) then we can reject the null hypothesis and conclude that not all group means are equal.

A two-way ANOVA is used to determine whether or not there is a statistically significant difference between the means of three or more independent groups that have been split on two variables (sometimes called “factors”).

A two-way ANOVA tests three null hypotheses at the same time:

• All group means are equal at each level of the first variable
• All group means are equal at each level of the second variable
• There is no interaction effect between the two variables

To decide if we should reject or fail to reject each null hypothesis, we must refer to the p-values in the output of the two-way ANOVA table.

The following examples show how to decide to reject or fail to reject the null hypothesis in both a one-way ANOVA and two-way ANOVA.

Example 1: One-Way ANOVA

Suppose we want to know whether or not three different exam prep programs lead to different mean scores on a certain exam. To test this, we recruit 30 students to participate in a study and split them into three groups.

The students in each group are randomly assigned to use one of the three exam prep programs for the next three weeks to prepare for an exam. At the end of the three weeks, all of the students take the same exam. 

The exam scores for each group are shown below:

When we enter these values into the One-Way ANOVA Calculator , we receive the following ANOVA table as the output:

Notice that the p-value is 0.11385 .

For this particular example, we would use the following null and alternative hypotheses:

• H 0 :  μ 1  = μ 2  = μ 3 (the mean exam score for each group is equal)

Since the p-value from the ANOVA table is not less than 0.05, we fail to reject the null hypothesis.

This means we don’t have sufficient evidence to say that there is a statistically significant difference between the mean exam scores of the three groups.

Example 2: Two-Way ANOVA

Suppose a botanist wants to know whether or not plant growth is influenced by sunlight exposure and watering frequency.

She plants 40 seeds and lets them grow for two months under different conditions for sunlight exposure and watering frequency. After two months, she records the height of each plant. The results are shown below:

In the table above, we see that there were five plants grown under each combination of conditions.

For example, there were five plants grown with daily watering and no sunlight and their heights after two months were 4.8 inches, 4.4 inches, 3.2 inches, 3.9 inches, and 4.4 inches:

She performs a two-way ANOVA in Excel and ends up with the following output:

We can see the following p-values in the output of the two-way ANOVA table:

• The p-value for watering frequency is 0.975975 . This is not statistically significant at a significance level of 0.05.
• The p-value for sunlight exposure is 3.9E-8 (0.000000039) . This is statistically significant at a significance level of 0.05.
• The p-value for the interaction between watering  frequency and sunlight exposure is 0.310898 . This is not statistically significant at a significance level of 0.05.

These results indicate that sunlight exposure is the only factor that has a statistically significant effect on plant height.

And because there is no interaction effect, the effect of sunlight exposure is consistent across each level of watering frequency.

That is, whether a plant is watered daily or weekly has no impact on how sunlight exposure affects a plant.

Additional Resources

The following tutorials provide additional information about ANOVA models:

How to Interpret the F-Value and P-Value in ANOVA How to Calculate Sum of Squares in ANOVA What Does a High F Value Mean in ANOVA?

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Research questions, hypotheses and objectives

Patricia farrugia.

* Michael G. DeGroote School of Medicine, the

Bradley A. Petrisor

† Division of Orthopaedic Surgery and the

Forough Farrokhyar

‡ Departments of Surgery and

§ Clinical Epidemiology and Biostatistics, McMaster University, Hamilton, Ont

Mohit Bhandari

There is an increasing familiarity with the principles of evidence-based medicine in the surgical community. As surgeons become more aware of the hierarchy of evidence, grades of recommendations and the principles of critical appraisal, they develop an increasing familiarity with research design. Surgeons and clinicians are looking more and more to the literature and clinical trials to guide their practice; as such, it is becoming a responsibility of the clinical research community to attempt to answer questions that are not only well thought out but also clinically relevant. The development of the research question, including a supportive hypothesis and objectives, is a necessary key step in producing clinically relevant results to be used in evidence-based practice. A well-defined and specific research question is more likely to help guide us in making decisions about study design and population and subsequently what data will be collected and analyzed. 1

Objectives of this article

In this article, we discuss important considerations in the development of a research question and hypothesis and in defining objectives for research. By the end of this article, the reader will be able to appreciate the significance of constructing a good research question and developing hypotheses and research objectives for the successful design of a research study. The following article is divided into 3 sections: research question, research hypothesis and research objectives.

Research question

Interest in a particular topic usually begins the research process, but it is the familiarity with the subject that helps define an appropriate research question for a study. 1 Questions then arise out of a perceived knowledge deficit within a subject area or field of study. 2 Indeed, Haynes suggests that it is important to know “where the boundary between current knowledge and ignorance lies.” 1 The challenge in developing an appropriate research question is in determining which clinical uncertainties could or should be studied and also rationalizing the need for their investigation.

Increasing one’s knowledge about the subject of interest can be accomplished in many ways. Appropriate methods include systematically searching the literature, in-depth interviews and focus groups with patients (and proxies) and interviews with experts in the field. In addition, awareness of current trends and technological advances can assist with the development of research questions. 2 It is imperative to understand what has been studied about a topic to date in order to further the knowledge that has been previously gathered on a topic. Indeed, some granting institutions (e.g., Canadian Institute for Health Research) encourage applicants to conduct a systematic review of the available evidence if a recent review does not already exist and preferably a pilot or feasibility study before applying for a grant for a full trial.

In-depth knowledge about a subject may generate a number of questions. It then becomes necessary to ask whether these questions can be answered through one study or if more than one study needed. 1 Additional research questions can be developed, but several basic principles should be taken into consideration. 1 All questions, primary and secondary, should be developed at the beginning and planning stages of a study. Any additional questions should never compromise the primary question because it is the primary research question that forms the basis of the hypothesis and study objectives. It must be kept in mind that within the scope of one study, the presence of a number of research questions will affect and potentially increase the complexity of both the study design and subsequent statistical analyses, not to mention the actual feasibility of answering every question. 1 A sensible strategy is to establish a single primary research question around which to focus the study plan. 3 In a study, the primary research question should be clearly stated at the end of the introduction of the grant proposal, and it usually specifies the population to be studied, the intervention to be implemented and other circumstantial factors. 4

Hulley and colleagues 2 have suggested the use of the FINER criteria in the development of a good research question ( Box 1 ). The FINER criteria highlight useful points that may increase the chances of developing a successful research project. A good research question should specify the population of interest, be of interest to the scientific community and potentially to the public, have clinical relevance and further current knowledge in the field (and of course be compliant with the standards of ethical boards and national research standards).

FINER criteria for a good research question

Adapted with permission from Wolters Kluwer Health. 2

Whereas the FINER criteria outline the important aspects of the question in general, a useful format to use in the development of a specific research question is the PICO format — consider the population (P) of interest, the intervention (I) being studied, the comparison (C) group (or to what is the intervention being compared) and the outcome of interest (O). 3 , 5 , 6 Often timing (T) is added to PICO ( Box 2 ) — that is, “Over what time frame will the study take place?” 1 The PICOT approach helps generate a question that aids in constructing the framework of the study and subsequently in protocol development by alluding to the inclusion and exclusion criteria and identifying the groups of patients to be included. Knowing the specific population of interest, intervention (and comparator) and outcome of interest may also help the researcher identify an appropriate outcome measurement tool. 7 The more defined the population of interest, and thus the more stringent the inclusion and exclusion criteria, the greater the effect on the interpretation and subsequent applicability and generalizability of the research findings. 1 , 2 A restricted study population (and exclusion criteria) may limit bias and increase the internal validity of the study; however, this approach will limit external validity of the study and, thus, the generalizability of the findings to the practical clinical setting. Conversely, a broadly defined study population and inclusion criteria may be representative of practical clinical practice but may increase bias and reduce the internal validity of the study.

PICOT criteria 1

A poorly devised research question may affect the choice of study design, potentially lead to futile situations and, thus, hamper the chance of determining anything of clinical significance, which will then affect the potential for publication. Without devoting appropriate resources to developing the research question, the quality of the study and subsequent results may be compromised. During the initial stages of any research study, it is therefore imperative to formulate a research question that is both clinically relevant and answerable.

Research hypothesis

The primary research question should be driven by the hypothesis rather than the data. 1 , 2 That is, the research question and hypothesis should be developed before the start of the study. This sounds intuitive; however, if we take, for example, a database of information, it is potentially possible to perform multiple statistical comparisons of groups within the database to find a statistically significant association. This could then lead one to work backward from the data and develop the “question.” This is counterintuitive to the process because the question is asked specifically to then find the answer, thus collecting data along the way (i.e., in a prospective manner). Multiple statistical testing of associations from data previously collected could potentially lead to spuriously positive findings of association through chance alone. 2 Therefore, a good hypothesis must be based on a good research question at the start of a trial and, indeed, drive data collection for the study.

The research or clinical hypothesis is developed from the research question and then the main elements of the study — sampling strategy, intervention (if applicable), comparison and outcome variables — are summarized in a form that establishes the basis for testing, statistical and ultimately clinical significance. 3 For example, in a research study comparing computer-assisted acetabular component insertion versus freehand acetabular component placement in patients in need of total hip arthroplasty, the experimental group would be computer-assisted insertion and the control/conventional group would be free-hand placement. The investigative team would first state a research hypothesis. This could be expressed as a single outcome (e.g., computer-assisted acetabular component placement leads to improved functional outcome) or potentially as a complex/composite outcome; that is, more than one outcome (e.g., computer-assisted acetabular component placement leads to both improved radiographic cup placement and improved functional outcome).

However, when formally testing statistical significance, the hypothesis should be stated as a “null” hypothesis. 2 The purpose of hypothesis testing is to make an inference about the population of interest on the basis of a random sample taken from that population. The null hypothesis for the preceding research hypothesis then would be that there is no difference in mean functional outcome between the computer-assisted insertion and free-hand placement techniques. After forming the null hypothesis, the researchers would form an alternate hypothesis stating the nature of the difference, if it should appear. The alternate hypothesis would be that there is a difference in mean functional outcome between these techniques. At the end of the study, the null hypothesis is then tested statistically. If the findings of the study are not statistically significant (i.e., there is no difference in functional outcome between the groups in a statistical sense), we cannot reject the null hypothesis, whereas if the findings were significant, we can reject the null hypothesis and accept the alternate hypothesis (i.e., there is a difference in mean functional outcome between the study groups), errors in testing notwithstanding. In other words, hypothesis testing confirms or refutes the statement that the observed findings did not occur by chance alone but rather occurred because there was a true difference in outcomes between these surgical procedures. The concept of statistical hypothesis testing is complex, and the details are beyond the scope of this article.

Another important concept inherent in hypothesis testing is whether the hypotheses will be 1-sided or 2-sided. A 2-sided hypothesis states that there is a difference between the experimental group and the control group, but it does not specify in advance the expected direction of the difference. For example, we asked whether there is there an improvement in outcomes with computer-assisted surgery or whether the outcomes worse with computer-assisted surgery. We presented a 2-sided test in the above example because we did not specify the direction of the difference. A 1-sided hypothesis states a specific direction (e.g., there is an improvement in outcomes with computer-assisted surgery). A 2-sided hypothesis should be used unless there is a good justification for using a 1-sided hypothesis. As Bland and Atlman 8 stated, “One-sided hypothesis testing should never be used as a device to make a conventionally nonsignificant difference significant.”

The research hypothesis should be stated at the beginning of the study to guide the objectives for research. Whereas the investigators may state the hypothesis as being 1-sided (there is an improvement with treatment), the study and investigators must adhere to the concept of clinical equipoise. According to this principle, a clinical (or surgical) trial is ethical only if the expert community is uncertain about the relative therapeutic merits of the experimental and control groups being evaluated. 9 It means there must exist an honest and professional disagreement among expert clinicians about the preferred treatment. 9

Designing a research hypothesis is supported by a good research question and will influence the type of research design for the study. Acting on the principles of appropriate hypothesis development, the study can then confidently proceed to the development of the research objective.

Research objective

The primary objective should be coupled with the hypothesis of the study. Study objectives define the specific aims of the study and should be clearly stated in the introduction of the research protocol. 7 From our previous example and using the investigative hypothesis that there is a difference in functional outcomes between computer-assisted acetabular component placement and free-hand placement, the primary objective can be stated as follows: this study will compare the functional outcomes of computer-assisted acetabular component insertion versus free-hand placement in patients undergoing total hip arthroplasty. Note that the study objective is an active statement about how the study is going to answer the specific research question. Objectives can (and often do) state exactly which outcome measures are going to be used within their statements. They are important because they not only help guide the development of the protocol and design of study but also play a role in sample size calculations and determining the power of the study. 7 These concepts will be discussed in other articles in this series.

From the surgeon’s point of view, it is important for the study objectives to be focused on outcomes that are important to patients and clinically relevant. For example, the most methodologically sound randomized controlled trial comparing 2 techniques of distal radial fixation would have little or no clinical impact if the primary objective was to determine the effect of treatment A as compared to treatment B on intraoperative fluoroscopy time. However, if the objective was to determine the effect of treatment A as compared to treatment B on patient functional outcome at 1 year, this would have a much more significant impact on clinical decision-making. Second, more meaningful surgeon–patient discussions could ensue, incorporating patient values and preferences with the results from this study. 6 , 7 It is the precise objective and what the investigator is trying to measure that is of clinical relevance in the practical setting.

The following is an example from the literature about the relation between the research question, hypothesis and study objectives:

Study: Warden SJ, Metcalf BR, Kiss ZS, et al. Low-intensity pulsed ultrasound for chronic patellar tendinopathy: a randomized, double-blind, placebo-controlled trial. Rheumatology 2008;47:467–71.

Research question: How does low-intensity pulsed ultrasound (LIPUS) compare with a placebo device in managing the symptoms of skeletally mature patients with patellar tendinopathy?

Research hypothesis: Pain levels are reduced in patients who receive daily active-LIPUS (treatment) for 12 weeks compared with individuals who receive inactive-LIPUS (placebo).

Objective: To investigate the clinical efficacy of LIPUS in the management of patellar tendinopathy symptoms.

The development of the research question is the most important aspect of a research project. A research project can fail if the objectives and hypothesis are poorly focused and underdeveloped. Useful tips for surgical researchers are provided in Box 3 . Designing and developing an appropriate and relevant research question, hypothesis and objectives can be a difficult task. The critical appraisal of the research question used in a study is vital to the application of the findings to clinical practice. Focusing resources, time and dedication to these 3 very important tasks will help to guide a successful research project, influence interpretation of the results and affect future publication efforts.

Tips for developing research questions, hypotheses and objectives for research studies

• Perform a systematic literature review (if one has not been done) to increase knowledge and familiarity with the topic and to assist with research development.
• Learn about current trends and technological advances on the topic.
• Seek careful input from experts, mentors, colleagues and collaborators to refine your research question as this will aid in developing the research question and guide the research study.
• Use the FINER criteria in the development of the research question.
• Ensure that the research question follows PICOT format.
• Develop a research hypothesis from the research question.
• Develop clear and well-defined primary and secondary (if needed) objectives.
• Ensure that the research question and objectives are answerable, feasible and clinically relevant.

FINER = feasible, interesting, novel, ethical, relevant; PICOT = population (patients), intervention (for intervention studies only), comparison group, outcome of interest, time.

Competing interests: No funding was received in preparation of this paper. Dr. Bhandari was funded, in part, by a Canada Research Chair, McMaster University.

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9.1: Null and Alternative Hypotheses

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

\(H_0\): The null hypothesis: It is a statement of no difference between the variables—they are not related. This can often be considered the status quo and as a result if you cannot accept the null it requires some action.

\(H_a\): The alternative hypothesis: It is a claim about the population that is contradictory to \(H_0\) and what we conclude when we reject \(H_0\). This is usually what the researcher is trying to prove.

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

After you have determined which hypothesis the sample supports, you make a decision. There are two options for a decision. They are "reject \(H_0\)" if the sample information favors the alternative hypothesis or "do not reject \(H_0\)" or "decline to reject \(H_0\)" if the sample information is insufficient to reject the null hypothesis.

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

Example \(\PageIndex{1}\)

• \(H_{0}\): No more than 30% of the registered voters in Santa Clara County voted in the primary election. \(p \leq 30\)
• \(H_{a}\): More than 30% of the registered voters in Santa Clara County voted in the primary election. \(p > 30\)

Exercise \(\PageIndex{1}\)

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

• \(H_{0}\): The drug reduces cholesterol by 25%. \(p = 0.25\)
• \(H_{a}\): The drug does not reduce cholesterol by 25%. \(p \neq 0.25\)

Example \(\PageIndex{2}\)

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

• \(H_{0}: \mu = 2.0\)
• \(H_{a}: \mu \neq 2.0\)

Exercise \(\PageIndex{2}\)

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 \((=, \neq, \geq, <, \leq, >)\) for the null and alternative hypotheses.

• \(H_{0}: \mu \_ 66\)
• \(H_{a}: \mu \_ 66\)
• \(H_{0}: \mu = 66\)
• \(H_{a}: \mu \neq 66\)

Example \(\PageIndex{3}\)

We want to test if college students take less than five years to graduate from college, on the average. The null and alternative hypotheses are:

• \(H_{0}: \mu \geq 5\)
• \(H_{a}: \mu < 5\)

Exercise \(\PageIndex{3}\)

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.

• \(H_{0}: \mu \_ 45\)
• \(H_{a}: \mu \_ 45\)
• \(H_{0}: \mu \geq 45\)
• \(H_{a}: \mu < 45\)

Example \(\PageIndex{4}\)

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

• \(H_{0}: p \leq 0.066\)
• \(H_{a}: p > 0.066\)

Exercise \(\PageIndex{4}\)

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

• \(H_{0}: p \_ 0.40\)
• \(H_{a}: p \_ 0.40\)
• \(H_{0}: p = 0.40\)
• \(H_{a}: p > 0.40\)

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.

In a hypothesis test , sample data is evaluated in order to arrive at a decision about some type of claim. If certain conditions about the sample are satisfied, then the claim can be evaluated for a population. In a hypothesis test, we:

• Evaluate the null hypothesis , typically denoted with \(H_{0}\). The null is not rejected unless the hypothesis test shows otherwise. The null statement must always contain some form of equality \((=, \leq \text{or} \geq)\)
• Always write the alternative hypothesis , typically denoted with \(H_{a}\) or \(H_{1}\), using less than, greater than, or not equals symbols, i.e., \((\neq, >, \text{or} <)\).
• If we reject the null hypothesis, then we can assume there is enough evidence to support the alternative hypothesis.
• Never state that a claim is proven true or false. Keep in mind the underlying fact that hypothesis testing is based on probability laws; therefore, we can talk only in terms of non-absolute certainties.

Formula Review

\(H_{0}\) and \(H_{a}\) are contradictory.

• If \(\alpha \leq p\)-value, then do not reject \(H_{0}\).
• If\(\alpha > p\)-value, then reject \(H_{0}\).

\(\alpha\) is preconceived. Its value is set before the hypothesis test starts. The \(p\)-value is calculated from the data.References

Data from the National Institute of Mental Health. Available online at http://www.nimh.nih.gov/publicat/depression.cfm .

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5 facts about religion and Americans’ views of Donald Trump

For most of the last decade, observers have been trying to understand why so many highly religious Americans have a favorable view of Donald Trump, asking how values voters can support a candidate who has been divorced twice, married three times and found liable for sexual abuse . Is Trump viewed most positively by those who might be described as “Christians in name only” – people who identify as Christians but aren’t actually religious?

The latest Pew Research Center survey sheds light on these and related questions. Here are five facts about religion and views of Trump, based on our survey of 12,693 U.S. adults conducted Feb. 13-25.

Pew Research Center conducted this analysis to explore the connection between religion and views of Donald Trump.

For this analysis, we surveyed 12,693 respondents from Feb. 13 to 25, 2024. Most of the respondents (10,642) are members of the Center’s American Trends Panel, an online survey panel recruited through national random sampling of residential addresses, which gives nearly all U.S. adults a chance of selection.

The remaining respondents (2,051) are members of three other panels, the Ipsos KnowledgePanel, the NORC AmeriSpeak panel and the SSRS opinion panel. All three are national survey panels recruited through random sampling (not “opt-in” polls). We used these additional panels to ensure that the survey would have enough Jewish and Muslim respondents to be able to report on their views.

The survey is weighted to be representative of the U.S. adult population by gender, race, ethnicity, partisan affiliation, education, religious affiliation and other categories.

For more, refer to the ATP’s methodology and the methodology for this survey .

Among religious groups, White evangelical Protestants continue to have the most positive opinion of Trump. Overall, two-thirds of White evangelical Protestants say they have a favorable view of the former president, including 30% who have a very favorable opinion of him.

Roughly half of White Catholics (51%) express positive views of Trump, as do 47% of White nonevangelical Protestants and 45% of Hispanic Protestants.

But in every other U.S. religious group large enough to be analyzed in this survey, large majorities have unfavorable opinions of Trump, including:  

• 88% of atheists
• 82% of agnostics
• 80% of Black Protestants
• 79% of Jewish Americans

These religious patterns largely reflect partisan differences . Most White evangelicals tend to vote for Republicans, as do smaller majorities of White Catholics and White nonevangelical Protestants. By contrast, most atheists, agnostics, Black Protestants and Jews tend to vote for Democrats.

Trump’s favorability rating is similar among Christians who attend church regularly and those who don’t. Some observers have pointed out that Trump’s political base consists largely of people who call themselves Christians but don’t go to church. However, our survey shows that Christians who regularly go to church express equally favorable views of Trump as those who don’t often attend religious services.

Among Christians as a whole, 47% of those who attend church at least monthly say they have a favorable view of the former president. That’s on par with the 46% of non-church-attending Christians who say the same.

Among White evangelical Protestants, 68% of regular churchgoers have a positive view of Trump – similar to the 64% among White evangelicals who don’t attend church regularly.

The only exception to this pattern is among White Protestants who do not identify as born-again or evangelical. In this group, Trump is viewed more favorably by those who don’t attend church regularly than by those who do (52% vs. 32%).

Many of the people who view Trump favorably don’t go to religious services regularly – but very few are nonreligious. Overall, 64% of respondents who have a favorable view of Trump say they attend religious services a few times a year or less often, while 35% say they go to services at least once or twice a month. (Among all respondents, 69% say they attend religious services a few times a year or less, while 30% go at least monthly.)

Religious attendance is just one way of looking at religious commitment. Another common way we measure it is to combine survey questions about attendance at religious services, how often people pray and how important religion is to them.

U.S. adults who attend religious services at least weekly, pray daily and say religion is very important in their lives are categorized as highly religious. Those who seldom or never attend services, seldom or never pray and say religion is not too important or not at all important in their lives are counted as having low religious commitment. Everyone else is counted as having medium religious commitment.

Looked at this way, 23% of U.S. adults with a favorable view of Trump are highly religious, including 11% who are highly religious White evangelical Protestants.

Another 62% of Americans with a favorable view of Trump have medium levels of religious commitment, including 13% who are White evangelicals.

Just 15% of people with a favorable view of Trump have low levels of religious commitment. By far the biggest subgroup within this category is religious “nones” – people who describe their religious identity as atheist, agnostic or “nothing in particular.” Overall, 18% of people with a positive view of Trump are religious “nones,” including 10% who are “nones” with low levels of religiousness.

Very few of the people who have a positive view of Trump are White evangelical Protestants with a low level of religiousness. Indeed, self-described White evangelical Protestants who are not religiously observant account for less than 1% of the overall U.S. population. Even if a candidate wanted to form a coalition rooted in support from nonreligious evangelicals, there just aren’t enough of them to be a national political base.

Most people who view Trump positively don’t think he is especially religious himself. But many think he stands up for people with religious beliefs like theirs. Just 8% of people who have a positive view of Trump think he is very religious, while 51% think he is somewhat religious and 38% say he is not too or not at all religious .

But 51% of those with a favorable view of Trump think he stands up for people with religious beliefs like their own, including 24% who think he does this a great deal and 27% who say he does this quite a bit.

Among White evangelical Protestants with a favorable view of Trump, just 9% view him as very religious. But roughly two-thirds think he does a great deal (32%) or quite a bit (35%) to stand up for people with religious beliefs like theirs.

Religious “nones” who are culturally Christian view Trump a bit more positively than religious “nones” who aren’t.

One way to measure for differences between “cultural” and “practicing” Christians is to compare Christians who do and don’t do go to church regularly, as we did above. Another is to look at religiously unaffiliated respondents, or “nones” – people who describe themselves, religiously, as atheist, agnostic, or “nothing in particular.” In our new survey, we asked these Americans whether they think of themselves as Christians “aside from religion … for example ethnically, culturally or because of your family’s background.”

Religious “nones” who identify as culturally Christian have a modestly more favorable opinion of Trump than “nones” who do not identify as Christian in any way. Still, large majorities in both groups express negative views of the former president.

• Donald Trump
• Religion & Politics
• Religious Identity & Affiliation

About 1 in 4 Americans have unfavorable views of both Biden and Trump

8 in 10 americans say religion is losing influence in public life, in gop contest, trump supporters stand out for dislike of compromise, most americans favor maximum age limits for federal elected officials, supreme court justices, republicans view reagan, trump as best recent presidents, most popular.

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