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5.2 - writing hypotheses.

The first step in conducting a hypothesis test is to write the hypothesis statements that are going to be tested. For each test you will have a null hypothesis ($$H_0$$) and an alternative hypothesis ($$H_a$$).

When writing hypotheses there are three things that we need to know: (1) the parameter that we are testing (2) the direction of the test (non-directional, right-tailed or left-tailed), and (3) the value of the hypothesized parameter.

• At this point we can write hypotheses for a single mean ($$\mu$$), paired means($$\mu_d$$), a single proportion ($$p$$), the difference between two independent means ($$\mu_1-\mu_2$$), the difference between two proportions ($$p_1-p_2$$), a simple linear regression slope ($$\beta$$), and a correlation ($$\rho$$).
• The research question will give us the information necessary to determine if the test is two-tailed (e.g., "different from," "not equal to"), right-tailed (e.g., "greater than," "more than"), or left-tailed (e.g., "less than," "fewer than").
• The research question will also give us the hypothesized parameter value. This is the number that goes in the hypothesis statements (i.e., $$\mu_0$$ and $$p_0$$). For the difference between two groups, regression, and correlation, this value is typically 0.

Hypotheses are always written in terms of population parameters (e.g., $$p$$ and $$\mu$$).  The tables below display all of the possible hypotheses for the parameters that we have learned thus far. Note that the null hypothesis always includes the equality (i.e., =).

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## 10.2: 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.

• The null hypothesis ($$H_{0}$$) is a statement about the population that either is believed to be true or is used to put forth an argument unless it can be shown to be incorrect beyond a reasonable doubt.
• The alternative hypothesis ($$H_{a}$$) is 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.

$$H_{0}$$ 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.

## Chapter Review

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 .

## Module 9: Hypothesis Testing With One Sample

Null and alternative hypotheses, learning outcomes.

• Describe hypothesis testing in general and in practice

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 about the population that either is believed to be true or is used to put forth an argument unless it can be shown to be incorrect beyond a reasonable doubt.

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 .

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 adecision. There are two options for a  decision . They are “reject H 0 ” if the sample information favors the alternative hypothesis or “do not reject H 0 ” or “decline to reject H 0 ” if the sample information is insufficient to reject the null hypothesis.

Mathematical Symbols Used in  H 0 and H a :

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

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

H a : More than 30% of the registered voters in Santa Clara County voted in the primary election. p > 30

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

H 0 : The drug reduces cholesterol by 25%. p = 0.25

H a : The drug does not reduce cholesterol by 25%. p ≠ 0.25

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

H 0 : μ = 2.0

H a : μ ≠ 2.0

We want to test whether the mean height of eighth graders is 66 inches. State the null and alternative hypotheses. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses. H 0 : μ __ 66 H a : μ __ 66

• H 0 : μ = 66
• H a : μ ≠ 66

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

H 0 : μ ≥ 5

H a : μ < 5

We want to test if it takes fewer than 45 minutes to teach a lesson plan. State the null and alternative hypotheses. Fill in the correct symbol ( =, ≠, ≥, <, ≤, >) for the null and alternative hypotheses. H 0 : μ __ 45 H a : μ __ 45

• H 0 : μ ≥ 45
• H a : μ < 45

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

H 0 : p ≤ 0.066

H a : p > 0.066

On a state driver’s test, about 40% pass the test on the first try. We want to test if more than 40% pass on the first try. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses. H 0 : p __ 0.40 H a : p __ 0.40

• H 0 : p = 0.40
• H a : p > 0.40

## Concept Review

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 (=, ≤ or ≥) Always write the alternative hypothesis , typically denoted with H a or H 1 , using less than, greater than, or not equals symbols, i.e., (≠, >, 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.

• OpenStax, Statistics, Null and Alternative Hypotheses. Provided by : OpenStax. Located at : http://cnx.org/contents/[email protected]:58/Introductory_Statistics . License : CC BY: Attribution

Hypothesis Testing with One Sample

## Null and Alternative Hypotheses

OpenStaxCollege

[latexpage]

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 about the population that either is believed to be true or is used to put forth an argument unless it can be shown to be incorrect beyond a reasonable doubt.

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 .

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

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

Mathematical Symbols Used in H 0 and H a :

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

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

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

H 0 : The drug reduces cholesterol by 25%. p = 0.25

H a : The drug does not reduce cholesterol by 25%. p ≠ 0.25

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

H 0 : μ = 2.0

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

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

H 0 : μ ≥ 5

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

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

H 0 : p ≤ 0.066

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

• H 0 : p = 0.40
• H a : p > 0.40

<!– ??? –>

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.

## Chapter Review

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:

## Formula Review

H 0 and H a are contradictory.

If α ≤ p -value, then do not reject H 0 .

If α > p -value, then reject H 0 .

α is preconceived. Its value is set before the hypothesis test starts. The p -value is calculated from the data.

You are testing that the mean speed of your cable Internet connection is more than three Megabits per second. What is the random variable? Describe in words.

The random variable is the mean Internet speed in Megabits per second.

You are testing that the mean speed of your cable Internet connection is more than three Megabits per second. State the null and alternative hypotheses.

The American family has an average of two children. What is the random variable? Describe in words.

The random variable is the mean number of children an American family has.

The mean entry level salary of an employee at a company is 💲58,000. You believe it is higher for IT professionals in the company. State the null and alternative hypotheses.

A sociologist claims the probability that a person picked at random in Times Square in New York City is visiting the area is 0.83. You want to test to see if the proportion is actually less. What is the random variable? Describe in words.

The random variable is the proportion of people picked at random in Times Square visiting the city.

A sociologist claims the probability that a person picked at random in Times Square in New York City is visiting the area is 0.83. You want to test to see if the claim is correct. State the null and alternative hypotheses.

In a population of fish, approximately 42% are female. A test is conducted to see if, in fact, the proportion is less. State the null and alternative hypotheses.

Suppose that a recent article stated that the mean time spent in jail by a first–time convicted burglar is 2.5 years. A study was then done to see if the mean time has increased in the new century. A random sample of 26 first-time convicted burglars in a recent year was picked. The mean length of time in jail from the survey was 3 years with a standard deviation of 1.8 years. Suppose that it is somehow known that the population standard deviation is 1.5. If you were conducting a hypothesis test to determine if the mean length of jail time has increased, what would the null and alternative hypotheses be? The distribution of the population is normal.

A random survey of 75 death row inmates revealed that the mean length of time on death row is 17.4 years with a standard deviation of 6.3 years. If you were conducting a hypothesis test to determine if the population mean time on death row could likely be 15 years, what would the null and alternative hypotheses be?

• H 0 : __________
• H a : __________
• H 0 : μ = 15
• H a : μ ≠ 15

The National Institute of Mental Health published an article stating that in any one-year period, approximately 9.5 percent of American adults suffer from depression or a depressive illness. Suppose that in a survey of 100 people in a certain town, seven of them suffered from depression or a depressive illness. If you were conducting a hypothesis test to determine if the true proportion of people in that town suffering from depression or a depressive illness is lower than the percent in the general adult American population, what would the null and alternative hypotheses be?

Some of the following statements refer to the null hypothesis, some to the alternate hypothesis.

State the null hypothesis, H 0 , and the alternative hypothesis. H a , in terms of the appropriate parameter ( μ or p ).

• The mean number of years Americans work before retiring is 34.
• At most 60% of Americans vote in presidential elections.
• The mean starting salary for San Jose State University graduates is at least 💲100,000 per year.
• Twenty-nine percent of high school seniors get drunk each month.
• Fewer than 5% of adults ride the bus to work in Los Angeles.
• The mean number of cars a person owns in her lifetime is not more than ten.
• About half of Americans prefer to live away from cities, given the choice.
• Europeans have a mean paid vacation each year of six weeks.
• The chance of developing breast cancer is under 11% for women.
• Private universities’ mean tuition cost is more than 💲20,000 per year.
• H 0 : μ = 34; H a : μ ≠ 34
• H 0 : p ≤ 0.60; H a : p > 0.60
• H 0 : μ ≥ 100,000; H a : μ < 100,000
• H 0 : p = 0.29; H a : p ≠ 0.29
• H 0 : p = 0.05; H a : p < 0.05
• H 0 : μ ≤ 10; H a : μ > 10
• H 0 : p = 0.50; H a : p ≠ 0.50
• H 0 : μ = 6; H a : μ ≠ 6
• H 0 : p ≥ 0.11; H a : p < 0.11
• H 0 : μ ≤ 20,000; H a : μ > 20,000

Over the past few decades, public health officials have examined the link between weight concerns and teen girls’ smoking. Researchers surveyed a group of 273 randomly selected teen girls living in Massachusetts (between 12 and 15 years old). After four years the girls were surveyed again. Sixty-three said they smoked to stay thin. Is there good evidence that more than thirty percent of the teen girls smoke to stay thin? The alternative hypothesis is:

• p < 0.30
• p > 0.30

A statistics instructor believes that fewer than 20% of Evergreen Valley College (EVC) students attended the opening night midnight showing of the latest Harry Potter movie. She surveys 84 of her students and finds that 11 attended the midnight showing. An appropriate alternative hypothesis is:

• p > 0.20
• p < 0.20

Previously, an organization reported that teenagers spent 4.5 hours per week, on average, on the phone. The organization thinks that, currently, the mean is higher. Fifteen randomly chosen teenagers were asked how many hours per week they spend on the phone. The sample mean was 4.75 hours with a sample standard deviation of 2.0. Conduct a hypothesis test. The null and alternative hypotheses are:

• H o : $$\overline{x}$$ = 4.5, H a : $$\overline{x}$$ > 4.5
• H o : μ ≥ 4.5, H a : μ < 4.5
• H o : μ = 4.75, H a : μ > 4.75
• H o : μ = 4.5, H a : μ > 4.5

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

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## AP®︎/College Statistics

Course: ap®︎/college statistics   >   unit 10.

• Idea behind hypothesis testing
• Examples of null and alternative hypotheses

## Writing null and alternative hypotheses

• P-values and significance tests
• Comparing P-values to different significance levels
• Estimating a P-value from a simulation
• Estimating P-values from simulations
• Using P-values to make conclusions

• (Choice A)   H 0 : p = 0.1 H a : p ≠ 0.1 ‍   A H 0 : p = 0.1 H a : p ≠ 0.1 ‍
• (Choice B)   H 0 : p ≠ 0.1 H a : p = 0.1 ‍   B H 0 : p ≠ 0.1 H a : p = 0.1 ‍
• (Choice C)   H 0 : p = 0.1 H a : p > 0.1 ‍   C H 0 : p = 0.1 H a : p > 0.1 ‍
• (Choice D)   H 0 : p = 0.1 H a : p < 0.1 ‍   D H 0 : p = 0.1 H a : p < 0.1 ‍

Neag School of Education

## Educational Research Basics by Del Siegle

Null and alternative hypotheses.

Converting research questions to hypothesis is a simple task. Take the questions and make it a positive statement that says a relationship exists (correlation studies) or a difference exists between the groups (experiment study) and you have the alternative hypothesis. Write the statement such that a relationship does not exist or a difference does not exist and you have the null hypothesis. You can reverse the process if you have a hypothesis and wish to write a research question.

When you are comparing two groups, the groups are the independent variable. When you are testing whether something affects something else, the cause is the independent variable. The independent variable is the one you manipulate.

Teachers given higher pay will have more positive attitudes toward children than teachers given lower pay. The first step is to ask yourself “Are there two or more groups being compared?” The answer is “Yes.” What are the groups? Teachers who are given higher pay and teachers who are given lower pay. The independent variable is teacher pay. The dependent variable (the outcome) is attitude towards school.

You could also approach is another way. “Is something causing something else?” The answer is “Yes.”  What is causing what? Teacher pay is causing attitude towards school. Therefore, teacher pay is the independent variable (cause) and attitude towards school is the dependent variable (outcome).

By tradition, we try to disprove (reject) the null hypothesis. We can never prove a null hypothesis, because it is impossible to prove something does not exist. We can disprove something does not exist by finding an example of it. Therefore, in research we try to disprove the null hypothesis. When we do find that a relationship (or difference) exists then we reject the null and accept the alternative. If we do not find that a relationship (or difference) exists, we fail to reject the null hypothesis (and go with it). We never say we accept the null hypothesis because it is never possible to prove something does not exist. That is why we say that we failed to reject the null hypothesis, rather than we accepted it.

Del Siegle, Ph.D. Neag School of Education – University of Connecticut [email protected] www.delsiegle.com

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• Knowledge Base
• Methodology
• How to Write a Strong Hypothesis | Guide & Examples

## How to Write a Strong Hypothesis | Guide & Examples

Published on 6 May 2022 by Shona McCombes .

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.

What is a hypothesis, developing a hypothesis (with example), hypothesis examples, 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 variables . An independent variable is something the researcher changes or controls. A dependent variable is something the researcher observes and measures.

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 .

## Prevent plagiarism, run a free check.

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

At this stage, you might construct a conceptual framework to identify which variables you will study and what you think the relationships are between them. Sometimes, you’ll have to operationalise 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.

## Step 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

## Step 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.

## Step 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 .

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.

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).

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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Once you have developed a clear and focused research question or set of research questions, you’ll be ready to conduct further research, a literature review, on the topic to help you make an educated guess about the answer to your question(s). This educated guess is called a hypothesis.

In research, there are two types of hypotheses: null and alternative. They work as a complementary pair, each stating that the other is wrong.

• Null Hypothesis (H 0 ) – This can be thought of as the implied hypothesis. “Null” meaning “nothing.”  This hypothesis states that there is no difference between groups or no relationship between variables. The null hypothesis is a presumption of status quo or no change.
• Alternative Hypothesis (H a ) – This is also known as the claim. This hypothesis should state what you expect the data to show, based on your research on the topic. This is your answer to your research question.

Null Hypothesis:   H 0 : There is no difference in the salary of factory workers based on gender. Alternative Hypothesis :  H a : Male factory workers have a higher salary than female factory workers.

Null Hypothesis :  H 0 : There is no relationship between height and shoe size. Alternative Hypothesis :  H a : There is a positive relationship between height and shoe size.

Null Hypothesis :  H 0 : Experience on the job has no impact on the quality of a brick mason’s work. Alternative Hypothesis :  H a : The quality of a brick mason’s work is influenced by on-the-job experience.

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## Research Hypothesis In Psychology: Types, & Examples

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A research hypothesis, in its plural form “hypotheses,” is a specific, testable prediction about the anticipated results of a study, established at its outset. It is a key component of the scientific method .

Hypotheses connect theory to data and guide the research process towards expanding scientific understanding

## Some key points about hypotheses:

• A hypothesis expresses an expected pattern or relationship. It connects the variables under investigation.
• It is stated in clear, precise terms before any data collection or analysis occurs. This makes the hypothesis testable.
• A hypothesis must be falsifiable. It should be possible, even if unlikely in practice, to collect data that disconfirms rather than supports the hypothesis.
• Hypotheses guide research. Scientists design studies to explicitly evaluate hypotheses about how nature works.
• For a hypothesis to be valid, it must be testable against empirical evidence. The evidence can then confirm or disprove the testable predictions.
• Hypotheses are informed by background knowledge and observation, but go beyond what is already known to propose an explanation of how or why something occurs.
Predictions typically arise from a thorough knowledge of the research literature, curiosity about real-world problems or implications, and integrating this to advance theory. They build on existing literature while providing new insight.

## Types of Research Hypotheses

Alternative hypothesis.

The research hypothesis is often called the alternative or experimental hypothesis in experimental research.

It typically suggests a potential relationship between two key variables: the independent variable, which the researcher manipulates, and the dependent variable, which is measured based on those changes.

The alternative hypothesis states a relationship exists between the two variables being studied (one variable affects the other).

A hypothesis is a testable statement or prediction about the relationship between two or more variables. It is a key component of the scientific method. Some key points about hypotheses:

• Important hypotheses lead to predictions that can be tested empirically. The evidence can then confirm or disprove the testable predictions.

In summary, a hypothesis is a precise, testable statement of what researchers expect to happen in a study and why. Hypotheses connect theory to data and guide the research process towards expanding scientific understanding.

An experimental hypothesis predicts what change(s) will occur in the dependent variable when the independent variable is manipulated.

It states that the results are not due to chance and are significant in supporting the theory being investigated.

The alternative hypothesis can be directional, indicating a specific direction of the effect, or non-directional, suggesting a difference without specifying its nature. It’s what researchers aim to support or demonstrate through their study.

## Null Hypothesis

The null hypothesis states no relationship exists between the two variables being studied (one variable does not affect the other). There will be no changes in the dependent variable due to manipulating the independent variable.

It states results are due to chance and are not significant in supporting the idea being investigated.

The null hypothesis, positing no effect or relationship, is a foundational contrast to the research hypothesis in scientific inquiry. It establishes a baseline for statistical testing, promoting objectivity by initiating research from a neutral stance.

Many statistical methods are tailored to test the null hypothesis, determining the likelihood of observed results if no true effect exists.

This dual-hypothesis approach provides clarity, ensuring that research intentions are explicit, and fosters consistency across scientific studies, enhancing the standardization and interpretability of research outcomes.

## Nondirectional Hypothesis

A non-directional hypothesis, also known as a two-tailed hypothesis, predicts that there is a difference or relationship between two variables but does not specify the direction of this relationship.

It merely indicates that a change or effect will occur without predicting which group will have higher or lower values.

For example, “There is a difference in performance between Group A and Group B” is a non-directional hypothesis.

## Directional Hypothesis

A directional (one-tailed) hypothesis predicts the nature of the effect of the independent variable on the dependent variable. It predicts in which direction the change will take place. (i.e., greater, smaller, less, more)

It specifies whether one variable is greater, lesser, or different from another, rather than just indicating that there’s a difference without specifying its nature.

For example, “Exercise increases weight loss” is a directional hypothesis.

## Falsifiability

The Falsification Principle, proposed by Karl Popper , is a way of demarcating science from non-science. It suggests that for a theory or hypothesis to be considered scientific, it must be testable and irrefutable.

Falsifiability emphasizes that scientific claims shouldn’t just be confirmable but should also have the potential to be proven wrong.

It means that there should exist some potential evidence or experiment that could prove the proposition false.

However many confirming instances exist for a theory, it only takes one counter observation to falsify it. For example, the hypothesis that “all swans are white,” can be falsified by observing a black swan.

For Popper, science should attempt to disprove a theory rather than attempt to continually provide evidence to support a research hypothesis.

## Can a Hypothesis be Proven?

Hypotheses make probabilistic predictions. They state the expected outcome if a particular relationship exists. However, a study result supporting a hypothesis does not definitively prove it is true.

All studies have limitations. There may be unknown confounding factors or issues that limit the certainty of conclusions. Additional studies may yield different results.

In science, hypotheses can realistically only be supported with some degree of confidence, not proven. The process of science is to incrementally accumulate evidence for and against hypothesized relationships in an ongoing pursuit of better models and explanations that best fit the empirical data. But hypotheses remain open to revision and rejection if that is where the evidence leads.
• Disproving a hypothesis is definitive. Solid disconfirmatory evidence will falsify a hypothesis and require altering or discarding it based on the evidence.
• However, confirming evidence is always open to revision. Other explanations may account for the same results, and additional or contradictory evidence may emerge over time.

We can never 100% prove the alternative hypothesis. Instead, we see if we can disprove, or reject the null hypothesis.

If we reject the null hypothesis, this doesn’t mean that our alternative hypothesis is correct but does support the alternative/experimental hypothesis.

Upon analysis of the results, an alternative hypothesis can be rejected or supported, but it can never be proven to be correct. We must avoid any reference to results proving a theory as this implies 100% certainty, and there is always a chance that evidence may exist which could refute a theory.

## How to Write a Hypothesis

• Identify variables . The researcher manipulates the independent variable and the dependent variable is the measured outcome.
• Operationalized the variables being investigated . Operationalization of a hypothesis refers to the process of making the variables physically measurable or testable, e.g. if you are about to study aggression, you might count the number of punches given by participants.
• Decide on a direction for your prediction . If there is evidence in the literature to support a specific effect of the independent variable on the dependent variable, write a directional (one-tailed) hypothesis. If there are limited or ambiguous findings in the literature regarding the effect of the independent variable on the dependent variable, write a non-directional (two-tailed) hypothesis.
• Make it Testable : Ensure your hypothesis can be tested through experimentation or observation. It should be possible to prove it false (principle of falsifiability).
• Clear & concise language . A strong hypothesis is concise (typically one to two sentences long), and formulated using clear and straightforward language, ensuring it’s easily understood and testable.

Consider a hypothesis many teachers might subscribe to: students work better on Monday morning than on Friday afternoon (IV=Day, DV= Standard of work).

Now, if we decide to study this by giving the same group of students a lesson on a Monday morning and a Friday afternoon and then measuring their immediate recall of the material covered in each session, we would end up with the following:

• The alternative hypothesis states that students will recall significantly more information on a Monday morning than on a Friday afternoon.
• The null hypothesis states that there will be no significant difference in the amount recalled on a Monday morning compared to a Friday afternoon. Any difference will be due to chance or confounding factors.

## More Examples

• Memory : Participants exposed to classical music during study sessions will recall more items from a list than those who studied in silence.
• Social Psychology : Individuals who frequently engage in social media use will report higher levels of perceived social isolation compared to those who use it infrequently.
• Developmental Psychology : Children who engage in regular imaginative play have better problem-solving skills than those who don’t.
• Clinical Psychology : Cognitive-behavioral therapy will be more effective in reducing symptoms of anxiety over a 6-month period compared to traditional talk therapy.
• Cognitive Psychology : Individuals who multitask between various electronic devices will have shorter attention spans on focused tasks than those who single-task.
• Health Psychology : Patients who practice mindfulness meditation will experience lower levels of chronic pain compared to those who don’t meditate.
• Organizational Psychology : Employees in open-plan offices will report higher levels of stress than those in private offices.
• Behavioral Psychology : Rats rewarded with food after pressing a lever will press it more frequently than rats who receive no reward.

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What Is a Focus Group?

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What Is Face Validity In Research? Importance & How To Measure

Criterion Validity: Definition & Examples

## Null Hypothesis and Alternative Hypothesis

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

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

## The Null Hypothesis

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

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

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

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

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

## The Alternative Hypothesis

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

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

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

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

• Null hypothesis: “ x is equal to y .” Alternative hypothesis “ x is not equal to y .”
• Null hypothesis: “ x is at least y .” Alternative hypothesis “ x is less than y .”
• Null hypothesis: “ x is at most y .” Alternative hypothesis “ x is greater than y .”
• Null Hypothesis Examples
• An Example of a Hypothesis Test
• Hypothesis Test for the Difference of Two Population Proportions
• What Is a P-Value?
• How to Conduct a Hypothesis Test
• Hypothesis Test Example
• What Level of Alpha Determines Statistical Significance?
• Chi-Square Goodness of Fit Test
• Maslow's Hierarchy of Needs Explained
• Popular Math Terms and Definitions
• The Difference Between Type I and Type II Errors in Hypothesis Testing
• How to Do Hypothesis Tests With the Z.TEST Function in Excel
• The Runs Test for Random Sequences
• Type I and Type II Errors in Statistics
• What 'Fail to Reject' Means in a Hypothesis Test
• What Is the Difference Between Alpha and P-Values?

## 9.1 Null and Alternative Hypotheses

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

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

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

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

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

Mathematical Symbols Used in H 0 and H a :

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

## Example 9.1

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

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

## Example 9.2

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

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

• H 0 : μ __ 66
• H a : μ __ 66

## Example 9.3

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

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

• H 0 : μ __ 45
• H a : μ __ 45

## Example 9.4

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

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

• H 0 : p __ 0.40
• H a : p __ 0.40

## Collaborative Exercise

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

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## Alternative Hypothesis

Alternative hypothesis defines there is a statistically important relationship between two variables. Whereas null hypothesis states there is no statistical relationship between the two variables. In statistics, we usually come across various kinds of hypotheses. A statistical hypothesis is supposed to be a working statement which is assumed to be logical with given data. It should be noticed that a hypothesis is neither considered true nor false.

The alternative hypothesis is a statement used in statistical inference experiment. It is contradictory to the null hypothesis and denoted by H a or H 1 . We can also say that it is simply an alternative to the null. In hypothesis testing, an alternative theory is a statement which a researcher is testing. This statement is true from the researcher’s point of view and ultimately proves to reject the null to replace it with an alternative assumption. In this hypothesis, the difference between two or more variables is predicted by the researchers, such that the pattern of data observed in the test is not due to chance.

To check the water quality of a river for one year, the researchers are doing the observation. As per the null hypothesis, there is no change in water quality in the first half of the year as compared to the second half. But in the alternative hypothesis, the quality of water is poor in the second half when observed.

## Difference Between Null and Alternative Hypothesis

Basically, there are three types of the alternative hypothesis, they are;

Left-Tailed : Here, it is expected that the sample proportion (π) is less than a specified value which is denoted by π 0 , such that;

H 1 : π < π 0

Right-Tailed: It represents that the sample proportion (π) is greater than some value, denoted by π 0 .

H 1 : π > π 0

Two-Tailed: According to this hypothesis, the sample proportion (denoted by π) is not equal to a specific value which is represented by π 0 .

H 1 : π ≠ π 0

Note: The null hypothesis for all the three alternative hypotheses, would be H 1 : π = π 0 .

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## 9.2: 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.

• The null hypothesis ($$H_{0}$$) is a statement about the population that either is believed to be true or is used to put forth an argument unless it can be shown to be incorrect beyond a reasonable doubt.
• The alternative hypothesis ($$H_{a}$$) is 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.

$$H_{0}$$ 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.

## Chapter Review

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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• Published: 27 May 2024

## Comparing researchers’ degree of dichotomous thinking using frequentist versus Bayesian null hypothesis testing

• Jasmine Muradchanian   ORCID: orcid.org/0000-0002-2914-9197 1 ,
• Rink Hoekstra 1 ,
• Henk Kiers 1 ,
• Dustin Fife 2 &
• Don van Ravenzwaaij 1

Scientific Reports volume  14 , Article number:  12120 ( 2024 ) Cite this article

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• Human behaviour
• Neuroscience

A large amount of scientific literature in social and behavioural sciences bases their conclusions on one or more hypothesis tests. As such, it is important to obtain more knowledge about how researchers in social and behavioural sciences interpret quantities that result from hypothesis test metrics, such as p -values and Bayes factors. In the present study, we explored the relationship between obtained statistical evidence and the degree of belief or confidence that there is a positive effect in the population of interest. In particular, we were interested in the existence of a so-called cliff effect: A qualitative drop in the degree of belief that there is a positive effect around certain threshold values of statistical evidence (e.g., at p  = 0.05). We compared this relationship for p -values to the relationship for corresponding degrees of evidence quantified through Bayes factors, and we examined whether this relationship was affected by two different modes of presentation (in one mode the functional form of the relationship across values was implicit to the participant, whereas in the other mode it was explicit). We found evidence for a higher proportion of cliff effects in p -value conditions than in BF conditions (N = 139), but we did not get a clear indication whether presentation mode had an effect on the proportion of cliff effects.

## Protocol registration

The stage 1 protocol for this Registered Report was accepted in principle on 2 June 2023. The protocol, as accepted by the journal, can be found at: https://doi.org/10.17605/OSF.IO/5CW6P .

## Introduction

In applied science, researchers typically conduct statistical tests to learn whether an effect of interest differs from zero. Such tests typically tend to quantify evidence by means of p -values (but see e.g., Lakens 1 who warns against such an interpretation of p -values). A Bayesian alternative to the p -value is the Bayes factor (BF), which is a tool used for quantifying statistical evidence in hypothesis testing 2 , 3 . P -values and BFs are related to one another 4 , with BFs being used much less frequently. Having two contrasting hypotheses (i.e., a null hypothesis, H 0 , and an alternative hypothesis, H 1 ), a p -value is the probability of getting a result as extreme or more extreme than the actual observed sample result, given that H 0 were true (and given that the assumptions hold). A BF on the other hand, quantifies the probability of the data given H 1 relative to the probability of the data given H 0 (called BF 10 3 ).

There is ample evidence that researchers often find it difficult to interpret quantities such as p -values 5 , 6 , 7 . Although there has been growing awareness of the dangers of misinterpreting p -values, these dangers seem to remain prevalent. One of the key reasons for these misinterpretations is that these concepts are not simple or intuitive, and the correct interpretation of them would require more cognitive effort. Because of this high cognitive demand academics have been using shortcut interpretations, which are simply wrong 6 . An example of such a misinterpretation is that the p -value would represent the probability of the null hypothesis being true 6 . Research is typically conducted in order to reduce uncertainty around the existence of an effect in the population of interest. To do this, we use measures such as p -values and Bayes factors as a tool. Hence, it might be interesting (especially given the mistakes that are made by researchers when interpreting quantities such as p -values) to study how these measures affect people’s beliefs regarding the existence of an effect in the population of interest, so one can study how outcomes like p -values and Bayes factors translate to subjective beliefs about the existence of an effect in practice.

One of the first studies that focused on how researchers interpret statistical quantities was conducted by Rosenthal and Gaito 8 , in which they specifically studied how researchers interpret p -values of varying magnitude. Nineteen researchers and graduate students at their psychology faculty were requested to indicate their degree of belief or confidence in 14 p -values, varying from 0.001 to 0.90, on a 6-point scale ranging from “5 extreme confidence or belief” to “0 complete absence of confidence or belief” 8 , pp. 33–34 . These individuals were shown p -values for sample sizes of 10 and 100. The authors wanted to measure the degree of belief or confidence in research findings as a function of associated p -values, but stated as such it is not really clear what is meant here. We assume that the authors actually wanted to assess degree of belief or confidence in the existence of an effect, given the p -value. Their findings suggested that subjects’ degree of belief or confidence appeared to be a decreasing exponential function of the p- value. Additionally, for any p -value, self-rated confidence was greater for the larger sample size (i.e., n  = 100). Furthermore, the authors argued in favor of the existence of a cliff effect around p  = 0.05, which refers to an abrupt drop in the degree of belief or confidence in a p -value just beyond the 0.05 level 8 , 9 . This finding has been confirmed in several subsequent studies 10 , 11 , 12 . The studies described so far have been focusing on the average, and have not taken individual differences into account.

The cliff effect suggests p -values invite dichotomous thinking, which according to some authors seems to be a common type of reasoning when interpreting p -values in the context of Null Hypothesis Significance Testing (NHST 13 ). The outcome of the significance test seems to be usually interpreted dichotomously such as suggested by studies focusing on the cliff effect 8 , 9 , 10 , 11 , 12 , 13 , where one makes a binary choice between rejecting or not rejecting a null hypothesis 14 . This practice has taken some academics away from the main task of finding out the size of the effect of interest and the level of precision with which it has been measured 5 . However, Poitevineau and Lecoutre 15 argued that the cliff effect around p  = 0.05 is probably overstated. According to them, previous studies paid insufficient attention to individual differences. To demonstrate this, they explored the individual data and found qualitative heterogeneity in the respondents’ answers. The authors identified three categories of functions based on 12 p -values: (1) a decreasing exponential curve, (2) a decreasing linear curve, and (3) an all-or-none curve representing a very high degree of confidence when p  ≤ 0.05 and quasi-zero confidence otherwise. Out of 18 participants, they found that the responses of 10 participants followed a decreasing exponential curve, 4 participants followed a decreasing linear curve, and 4 participants followed an all-or-none curve. The authors concluded that the cliff effect may be an artifact of averaging, resulting from the fact that a few participants have an all-or-none interpretation of statistical significance 15 .

Although NHST has been used frequently, it has been argued that it should be replaced by effect sizes, confidence intervals (CIs), and meta-analyses. Doing so may allegedly invite a shift from dichotomous thinking to estimation and meta-analytic thinking 14 . Lai et al. 13 studied whether using CIs rather than p -values would reduce the cliff effect, and thereby dichotomous thinking. Similar to the classification by Poitevineau and Lecoutre 15 , the responses were divided into three classes: decreasing exponential, decreasing linear, or all-or-none. In addition, Lai et al. 13 found patterns in the responses of some of the participants that corresponded with what they called a “moderate cliff model”, which refers to using statistical significance as both a decision-making criterion and a measure of evidence 13 .

In contrast to Poitevineau and Lecoutre 15 , Lai et al. 13 concluded that the cliff effect is probably not just a byproduct resulting from the all-or-none class, because the cliff models were accountable for around 21% of the responses in NHST interpretation and for around 33% of the responses in CI interpretation. Furthermore, a notable finding was that the cliff effect prevalence in CI interpretations was more than 50% higher than that of NHST 13 . Something similar was found in a study by Hoekstra, Johnson, and Kiers 16 . They also predicted that the cliff effect would be stronger for results presented in the NHST format compared to the CI format, and like Lai et al. 13 , they actually found more evidence of a cliff effect in the CI format compared to the NHST format 16 .

The studies discussed so far seem to provide evidence for the existence of a cliff effect around p  = 0.05. Table 1 shows an overview of evidence related to the cliff effect. Interestingly, in a recent study, Helske et al. 17 examined how various visualizations can aim in reducing the cliff effect when interpreting inferential statistics among researchers. They found that compared to textual representation of the CI with p -values and classic CI visualization, including more complex visual information to classic CI representation seemed to decrease the cliff effect (i.e., dichotomous interpretations 17 ).

Although Bayesian methods have become more popular within different scientific fields 18 , 19 , we know of no studies that have examined whether self-reported degree of belief of the existence of an effect when interpreting BFs by researchers results in a similar cliff effect to those obtained for p -values and CIs. Another matter that seems to be conspicuously absent in previous examinations of the cliff effect is a comparison between the presentation methods that are used to investigate the cliff effect. In some cliff effect studies the p -values were presented to the participants on separate pages 15 and in other cliff effect studies the p -values were presented on the same page 13 . It is possible that the cliff effect manifests itself in (some) researchers without explicit awareness. It is possible that for those researchers presenting p -values/Bayes factors in isolation would lead to a cliff effect, whereas presenting all p -values/Bayes factors at once would lead to a cognitive override. Perhaps when participants see their cliff effect, they might think that they should not think dichotomously, and might change their results to be more in line with how they believe they should think, thereby removing their cliff effect. To our knowledge, no direct comparison of p -values/Bayes factors in isolation and all p -values/Bayes factors at once has yet been conducted. Therefore, to see whether the method matters, both types of presentation modes will be included in the present study.

All of this gives rise to the following three research questions: (1) What is the relation between obtained statistical evidence and the degree of belief or confidence that there is a positive effect in the population of interest across participants? (2) What is the difference in this relationship when the statistical evidence is quantified through p -values versus Bayes factors? (3) What is the difference in this relationship when the statistical evidence is presented in isolation versus all at once?

In the present study, we will investigate the relationship between method (i.e., p -values and Bayes factors) and the degree of belief or confidence that there is a positive effect in the population of interest, with special attention for the cliff effect. We choose this specific wording (“positive effect in the population of interest”) as we believe that this way of phrasing is more specific than those used in previous cliff effect studies. We will examine the relationship between different levels of strength of evidence using p -values or corresponding Bayes factors and measure participants' degree of belief or confidence in the following two scenarios: (1) the scenario in which values will be presented in isolation (such that the functional form of the relationship across values is implicit to the participant) and (2) the scenario in which all values will be presented simultaneously (such that the functional form of the relationship across values is explicit to the participant).

In what follows, we will first describe the set-up of the present study. In the results section, we will explore the relationship between obtained statistical evidence and the degree of belief or confidence, and in turn, we will compare this relationship for p -values to the corresponding relationship for BFs. All of this will be done in scenarios in which researchers are either made aware or not made aware of the functional form of the relationship. In the discussion, we will discuss implications for applied researchers using p -values and/or BFs in order to quantify statistical evidence.

## Ethics information

Our study protocol has been approved by the ethics committee of the University of Groningen and our study complies with all relevant ethical regulations of the University of Groningen. Informed consent will be obtained from all participants. As an incentive for participating, we will raffle 10 Amazon vouchers with a worth of 25USD among participants that successfully completed our study.

## Sampling plan

Our target population will consist of researchers in the social and behavioural sciences who are at least somewhat familiar with interpreting Bayes factors. We will obtain our prospective sample by collecting the e-mail addresses of (approximately) 2000 corresponding authors from 20 different journals in social and behavioural sciences with the highest impact factor. Specifically, we will collect the e-mail addresses of 100 researchers who published an article in the corresponding journal in 2021. We will start with the first issue and continue until we have 100 e-mail addresses per journal. We will contact the authors by e-mail. In the e-mail we will mention that we are looking for researchers who are familiar with interpreting Bayes factors. If they are familiar with interpreting Bayes factors, then we will ask them to participate in our study. If they are not familiar with interpreting Bayes factors, then we will ask them to ignore our e-mail.

If the currently unknown response rate is too low to answer our research questions, we will collect additional e-mail addresses of corresponding authors from articles published in 2022 in the same 20 journals. Based on a projected response rate of 10%, we expect a final completion rate of 200 participants. This should be enough to obtain a BF higher than 10 in favor of an effect if the proportions differ by 0.2 (see section “ Planned analyses ” for details).

## Materials and procedure

The relationship between the different magnitudes of p -values/BFs and the degree of belief or confidence will be examined in a scenario in which values will be presented in isolation and in a scenario in which the values will be presented simultaneously. This all will result in four different conditions: (1) p -value questions in the isolation scenario (isolated p -value), (2) BF questions in the isolation scenario (isolated BF), (3) p -value questions in the simultaneous scenario (all at once p -value), and (4) BF questions in the simultaneous scenario (all at once BF). To reduce boredom, and to try to avoid making underlying goals of the study too apparent, each participant will receive randomly one out of four scenarios (i.e., all at once p -value, all at once BF, isolated p -value, or isolated BF), so the study has a between-person design.

The participants will receive an e-mail with an anonymous Qualtrics survey link. The first page of the survey will consist of the informed consent. We will ask all participants to indicate their level of familiarity with both Bayes factors and p -values on a 3-point scale with “completely unfamiliar/somewhat familiar/very familiar” and we will include everyone who is at least somewhat familiar on both. To have a better picture of our sample population, we will include the following demographic variables in the survey: gender, main continent, career stage, and broad research area. Then we will randomly assign respondents to one of four conditions (see below for a detailed description). After completing the content-part of the survey, all respondents will receive a question about providing their e-mail address if they are interested in (1) being included in the random draw of the Amazon vouchers; or (2) receiving information on our study outcomes.

In the isolated p -value condition, the following fabricated experimental scenario will be presented:

“Suppose you conduct an experiment comparing two independent groups, with n = 250 in each group. The null hypothesis states that the population means of the two groups do not differ. The alternative hypothesis states that the population mean in group 1 is larger than the population mean in group 2. Suppose a two-sample t test was conducted and a one-sided p value calculated.”

Then a set of possible findings of the fabricated experiment will be presented at different pages. We varied the strength of evidence for the existence of a positive effect with the following ten p -values in a random order: 0.001, 0.002, 0.004, 0.008, 0.016, 0.032, 0.065, 0.131, 0.267, and 0.543. A screenshot of a part of the isolated p -value questions is presented in S1 in the Supplementary Information.

In the all at once BF condition, a fabricated experimental scenario will be presented identical to that in the isolated p -value condition, except the last part is replaced by:

“Suppose a Bayesian two-sample t test was conducted and a one-sided Bayes factor (BF) calculated, with the alternative hypothesis in the numerator and the null hypothesis in the denominator, denoted BF 10 .”

A set of possible findings of the fabricated experiment will be presented at the same page. These findings vary in terms of the strength of evidence for the existence of a positive effect, quantified with the following ten BF 10 values in the following order: 22.650, 12.008, 6.410, 3.449, 1.873, 1.027, 0.569, 0.317, 0.175, and 0.091. These BF values correspond one-on-one to the p -values presented in the isolated p -value condition (the R code for the findings of the fabricated experiment can be found on https://osf.io/sq3fp ). A screenshot of a part of the all at once BF questions can be found in S2 in the Supplementary Information.

In both conditions, the respondents will be asked to rate their degree of belief or confidence that there is a positive effect in the population of interest based on these findings on a scale ranging from 0 (completely convinced that there is no effect), through 50 (somewhat convinced that there is a positive effect), to 100 (completely convinced that there is a positive effect).

The other two conditions (i.e., isolated BF condition and the all at once p -value condition) will be the same as the previously described conditions. The only difference between these two conditions and the previously described conditions is that in the isolated BF condition, the findings of the fabricated experiment for the BF questions will be presented at different pages in a random order, and in the all at once p -value condition, the findings for the p -value questions will be presented at the same page in a non-random order.

To keep things as simple as possible for the participants, all fictitious scenarios will include a two-sample t test with either a one-tailed p -value or a BF. The total sample size will be large ( n  = 250 in each group) in order to have sufficiently large power to detect even small effects.

## Planned analyses

Poitevineau and Lecoutre 15 have suggested the following three models for the relationships between the different levels of statistical evidence and researchers’ subjective belief that a non-zero effect exists: all-or-none ( y  =  a for p  < 0.05, y  =  b for p  ≥ 0.05), linear ( y  =  a  +  bp ), and exponential ( y  = exp( a  +  bp )). In addition, Lai et al. 13 have suggested the moderate cliff model (a more gradual version of all-or-none), which they did not define more specifically. In the study by Lai et al. 13 (Fig.  4 ), the panel that represents the moderate cliff seems to be a combination of the exponential and the all-or-none function. In the present study, we will classify responses as moderate cliff if we observe a steep drop in the degree of belief or confidence around a certain p -value/BF, while for the remaining p -values/BFs the decline in confidence is more gradual. So, for example, a combination of the decreasing linear and the all-or-none function will also be classified as moderate cliff in the present study. Plots of the four models with examples of reasonable choices for the parameters are presented in Fig.  1 (the R code for Fig.  1 can be found on https://osf.io/j6d8c ).

Plots are shown for fictitious outcomes for the four models (all-or-none, linear, exponential, and moderate cliff). The x-axis represents the different p -values. In the two BF conditions, the x-axis represents the different BF values. The y-axis represents the proportion of degree of belief or confidence that there is a positive effect in the population of interest. Note that these are prototype responses; different variations on these response patterns are possible.

We will manually classify data for each participant for each scenario as one of the relationship models. We will do so by blinding the coders as to the conditions associated with the data. Specifically, author JM will organize the data from each of the four conditions and remove the p -value or BF labels. Subsequently, authors DvR and RH will classify the data independently from one another. In order to improve objectivity regarding the classification, authors DvR and RH will classify the data according to specific instructions that are constructed before collecting the data (see Appendix 1 ). After coding, we will compute Cohen’s kappa for these data. For each set of scores per condition per subject for which there was no agreement on classification, authors DvR and RH will try to reach consensus in a discussion of no longer than 5 min. If after this discussion no agreement is reached, then author DF will classify these data. If author DF will choose the same class as either DvR or RH, then the data will be classified accordingly. However, if author DF will choose another class, then the data will be classified in a so-called rest category. This rest category will also include data that extremely deviate from the four relationship models, and we will assess these data by running exploratory analyses. Before classifying the real data, we will conduct a small pilot study in order to provide authors DvR and RH with the possibility to practice classifying the data. In the Qualtrics survey, the respondents cannot continue with the next question without answering the current question. However, it might be possible that some of the respondents quit filling out the survey. The responses of the participants who did not answer all questions will be removed from the dataset. This means that we will use complete case analysis in order to deal with missing data, because we do not expect to find specific patterns in the missing values.

Our approach to answer Research Question 1 (RQ1; “What is the relation between obtained statistical evidence and the degree of belief or confidence that there is a positive effect in the population of interest across participants?”) will be descriptive in nature. We will explore the results visually, by assessing the four models (i.e., all-or-none, linear, exponential, and moderate cliff) in each of the four conditions (i.e., isolated p -value, all at once p -value, isolated BF, and all at once BF), followed by zooming in on the classification ‘cliff effect’. This means that we will compare the frequency of the four classification models with one another within each of the four conditions.

In order to answer Research Question 2 (RQ2; “What is the difference in this relationship when the statistical evidence is quantified through p -values versus Bayes factors?”), we will first combine categories as follows: the p -value condition will encompass the data from both the isolated and the all at once p -value conditions, and the BF condition will encompass the data from both the isolated and the all at once BF conditions. Furthermore, the cliff condition will encompass the all-or-none and the moderate cliff models, and the non-cliff condition will encompass the linear and the exponential models. This classification ensures that we distinguish between curves that reflect a sudden change in the relationship between the level of statistical evidence and the degree of confidence that a positive effect exists in the population of interest, and those that represent a gradual relationship between the level of statistical evidence and the degree of confidence. We will then compare the proportions of cases with a cliff in the p -value conditions to those in the BF conditions, and we will add inferential information for this comparison by means of a Bayesian chi square test on the 2 × 2 table ( p -value/BF x cliff/non-cliff), as will be specified below.

Finally, in order to answer Research Question 3 (RQ3; “What is the difference in this relationship when the statistical evidence is presented in isolation versus all at once?”), we will first combine categories again, as follows: the isolation condition will encompass the data from both the isolated p -value and the isolated BF conditions, and the all at once condition will encompass the data from both the all at once p -value and the all at once BF conditions. The cliff/non-cliff distinction is made analogous to the one employed for RQ2. We will then compare the proportions of cases with a cliff in the isolated conditions to those in the all at once conditions, and we will add inferential information for this comparison by means of a Bayesian chi square test on the 2 × 2 table (all at once/isolated x cliff/non-cliff), as will be specified below.

For both chi square tests, the null hypothesis states that there is no difference in the proportion of cliff classifications between the two conditions, and the alternative hypothesis states that there is a difference in the proportion of cliff classifications between the two conditions. Under the null hypothesis, we specify a single beta(1,1) prior for the proportion of cliff classifications and under the alternative hypothesis we specify two independent beta(1,1) priors for the proportion of cliff classifications 20 , 21 . A beta(1,1) prior is a flat or uniform prior from 0 to 1. The Bayes factor that will result from both chi square tests gives the relative evidence for the alternative hypothesis over the null hypothesis (BF 10 ) provided by the data. Both tests will be carried out in RStudio 22 (the R code for calculating the Bayes factors can be found on https://osf.io/5xbzt ). Additionally, the posterior of the difference in proportions will be provided (the R code for the posterior of the difference in proportions can be found on https://osf.io/3zhju ).

If, after having computed results on the obtained sample, we observe that our BFs are not higher than 10 or smaller than 0.1, we will expand our sample in the way explained at the end of section “Sampling Plan”. To see whether this approach will likely lead to useful results, we have conducted a Bayesian power simulation study for the case of population proportions of 0.2 and 0.4 (e.g., 20% cliff effect in the p -value group, and 40% cliff effect in the BF group) in order to determine how large the Bayesian power would be for reaching the BF threshold for a sample size of n  = 200. Our results show that for values 0.2 and 0.4 in both populations respectively, our estimated sample size of 200 participants (a 10% response rate) would lead to reaching a BF threshold 96% of the time, suggesting very high power under this alternative hypothesis. We have also conducted a Bayesian power simulation study for the case of population proportions of 0.3 (i.e., 30% cliff effect in the p -value group, and 30% cliff effect in the BF group) in order to determine how long sampling takes for a zero effect. The results show that for values of 0.3 in both populations, our estimated sample size of 200 participants would lead to reaching a BF threshold 7% of the time. Under the more optimistic scenario of a 20% response rate, a sample size of 400 participants would lead to reaching a BF threshold 70% of the time (the R code for the power can be found on https://osf.io/vzdce ). It is well known that it is harder to find strong evidence for the absence of an effect than for the presence of an effect 23 . In light of this, we deem a 70% chance of reaching a BF threshold under the null hypothesis given a 20% response rate acceptable. If, after sampling the first 2000 participants and factoring in the response rate, we have not reached either BF threshold, we will continue sampling participants in increments of 200 (10 per journal) until we reach a BF threshold or until we have an effective sample size of 400, or until we reach a total of 4000 participants.

In sum, RQ1 is exploratory in nature, so we will descriptively explore the patterns in our data. For RQ2, we will determine what proportion of applied researchers make a binary distinction regarding the existence of a positive effect in the population of interest, and we will test whether this binary distinction is different when research results are expressed in the p -value versus the BF condition. Finally, for RQ3, we will determine whether this binary distinction is different in the isolated versus all at once condition (see Table 2 for a summary of the study design).

## Sampling process

We deviated from our preregistered sampling plan in the following ways: we collected the e-mail address of all corresponding authors who published in the 20 journals in social and behavioural sciences in 2021 and 2022 at the same time . In total, we contacted 3152 academics, and 89 of them completed our survey (i.e., 2.8% of the contacted academics). We computed the BFs based on the responses of these 89 academics, and it turned out that the BF for RQ2 was equal to BF 10  = 16.13 and the BF for RQ3 was equal to BF 10  = 0.39, so the latter was neither higher than 10 nor smaller than 0.1.

In order to reach at least 4000 potential participants (see “ Planned analyses ” section), we decided to collect additional e-mail addresses of corresponding authors from articles published in 2019 and 2020 in the same 20 journals. In total, we thus reached another 2247 academics (total N = 5399), and 50 of them completed our survey (i.e., 2.2% of the contacted academics, effective N = 139).

In light of the large number of academics we had contacted at this point, we decided to do an ‘interim power analysis’ to calculate the upper and lower bounds of the BF for RQ3 to see if it made sense to continue collecting data up to N = 200. The already collected data of 21 cliffs out of 63 in the isolated conditions and 13 out of 65 in the all-at-once conditions yields a Bayes factor of 0.8 (see “ Results ” section below). We analytically verified that by increasing the number of participants to a total of 200, the strongest possible pro-null evidence we can get given the data we already had would be BF 10  = 0.14, or BF 01  = 6.99 (for 21 cliffs out of 100 in both conditions). In light of this, our judgment was that it was not the best use of human resources to continue collecting data, so we proceeded with a final sample of N = 139.

To summarize our sampling procedure, we contacted 5399 academics in total. Via Qualtrics, 220 participants responded. After removing the responses of the participants who did not complete the content part of our survey (i.e., the questions about the p -values or BFs), 181 cases remained. After removing the cases who were completely unfamiliar with p -values, 177 cases remained. After removing the cases who were completely unfamiliar with BFs, 139 cases remained. Note that there were also many people who responded via e-mail informing us that they were not familiar with interpreting BFs. Since the Qualtrics survey was anonymous, it was impossible for us to know the overlap between people who contacted us via e-mail and via Qualtrics that they were unfamiliar with interpreting BFs.

We contacted a total number of 5399 participants. The total number of participants who filled out the survey completely was N = 139, so 2.6% of the total sample (note that this is a result of both response rate and our requirement that researchers needed to self-report familiarity with interpreting BFs). Our entire Qualtrics survey can be found on https://osf.io/6gkcj . Five “difficult to classify” pilot plots were created such that authors RH and DvR could practice before classifying the real data. These plots can be found on https://osf.io/ndaw6/ (see folder “Pilot plots”). Authors RH and DvR had a qualitative discussion about these plots; however, no adjustments were made to the classification protocol. We manually classified data for each participant for each scenario as one of the relationship models (i.e., all-or-none, moderate cliff, linear, and exponential). Author JM organized the data from each of the four conditions and removed the p -value or BF labels. Authors RH and DvR classified the data according to the protocol provided in Appendix 1 , and the plot for each participant (including the condition each participant was in and the model in which each participant was classified) can be found in Appendix 2 . After coding, Cohen’s kappa was determined for these data, which was equal to κ = 0.47. Authors RH and DvR independently reached the same conclusion for 113 out of 139 data sets (i.e., 81.3%). For the remaining 26 data sets, RH and DvR were able to reach consensus within 5 min per data set, as laid out in the protocol. In Fig.  2 , plots are provided which include the prototype lines as well as the actual responses plotted along with them. This way, all responses can be seen at once along with how they match up with the prototype response for each category. To have a better picture of our sample population, we included the following demographic variables in the survey: gender, main continent, career stage, and broad research area. The results are presented in Table 3 . Based on these results it appeared that most of the respondents who filled out our survey were male (71.2%), living in Europe (51.1%), had a faculty position (94.1%), and were working in the field of psychology (56.1%). The total responses (i.e., including the responses of the respondents who quit filling out our survey) were very similar to the responses of the respondents who did complete our survey.

Plots including the prototype lines and the actual responses.

To answer RQ1 (“What is the relation between obtained statistical evidence and the degree of belief or confidence that there is a positive effect in the population of interest across participants?”), we compared the frequency of the four classification models (i.e., all-or-none, moderate cliff, linear, and exponential) with one another within each of the four conditions (i.e., all at once and isolated p -values, and all at once and isolated BFs). The results are presented in Table 4 . In order to enhance the interpretability of the results in Table 4 , we have plotted them in Fig.  3 .

Plotted frequency of classification models within each condition.

We observe that within the all at once p -value condition, the cliff models accounted for a proportion of (0 + 11)/33 = 0.33 of the responses. The non-cliff models accounted for a proportion of (1 + 21)/33 = 0.67 of the responses. Looking at the isolated p -value condition, we can see that the cliff models accounted for a proportion of (1 + 15)/35 = 0.46 of the responses. The non-cliff models accounted for a proportion of (0 + 19)/35 = 0.54 of the responses. In the all at once BF condition, we observe that the cliff models accounted for a proportion of (2 + 0)/32 = 0.06 of the responses. The non-cliff models accounted for a proportion of (0 + 30)/32 = 0.94 of the responses. Finally, we observe that within the isolated BF condition, the cliff models accounted for a proportion of (2 + 3)/28 = 0.18 of the responses. The non-cliff models accounted for a proportion of (0 + 23)/28 = 0.82 of the responses.

Thus, we observed a higher proportion of cliff models in p -value conditions than in BF conditions (27/68 = 0.40 vs 7/60 = 0.12), and we observed a higher proportion of cliff models in isolated conditions than in all-at-once conditions (21/63 = 0.33 vs 13/65 = 0.20). Next, we conducted statistical inference to dive deeper into these observations.

To answer RQ2 (“What is the difference in this relationship when the statistical evidence is quantified through p -values versus Bayes factors?”), we compared the sample proportions mentioned above (27/68 = 0.40 and 7/60 = 0.12, respectively, with a difference between these proportions equal to 0.40–0.12 = 0.28), and we tested whether the proportion of cliff classifications in the p -value conditions differed from that in the BF conditions in the population by means of a Bayesian chi square test. For the chi square test, the null hypothesis was that there is no difference in the proportion of cliff classifications between the two conditions, and the alternative hypothesis was that there is a difference in the proportion of cliff classifications between the two conditions.

The BF that resulted from the chi square test was equal to BF 10  = 140.01 and gives the relative evidence for the alternative hypothesis over the null hypothesis provided by the data. This means that the data are 140.01 times more likely under the alternative hypothesis than under the null hypothesis: we found strong support for the alternative hypothesis that there is a difference in the proportion of cliff classifications between the p -value and BF condition. Inspection of Table 4 or Fig.  3 shows that the proportion of cliff classifications is higher in the p -value conditions.

Additionally, the posterior distribution of the difference in proportions is provided in Fig.  4 , and the 95% credible interval was found to be [0.13, 0.41]. This means that there is a 95% probability that the population parameter for the difference of proportions of cliff classifications between p -value conditions and BF conditions lies within this interval, given the evidence provided by the observed data.

The posterior density of difference of proportions of cliff models in p -value conditions versus BF conditions.

To answer RQ3 (“What is the difference in this relationship when the statistical evidence is presented in isolation versus all at once?”), we compared the sample proportions mentioned above (21/63 = 0.33 vs 13/65 = 0.20, respectively with a difference between these proportions equal to 0.33–0.20 = 0.13), and we tested whether the proportion of cliff classifications in the all or none conditions differed from that in the isolated conditions in the population by means of a Bayesian chi square test analogous to the test above.

The BF that resulted from the chi square test was equal to BF 10  = 0.81, and gives the relative evidence for the alternative hypothesis over the null hypothesis provided by the data. This means that the data are 0.81 times more likely under the alternative hypothesis than under the null hypothesis: evidence on whether there is a difference in the proportion of cliff classifications between the isolation and all at once conditions is ambiguous.

Additionally, the posterior distribution of the difference in proportions is provided in Fig.  5 . The 95% credible interval is [− 0.28, 0.02].

The posterior density of difference of proportions of cliff models in all at once conditions versus isolated conditions.

There were 11 respondents who provided responses that extremely deviated from the four relationship models, so they were included in the rest category, and were left out of the analyses. Eight of these were in the isolated BF condition, one was in the isolated p -value condition, one was in the all at once BF condition, and one was in the all at once p -value condition. For five of these, their outcomes resulted in a roughly decreasing trend with significant large bumps. For four of these, there were one or more considerable increases in the plotted outcomes. For two of these, the line was flat. All these graphs are available in Appendix 2 .

In the present study, we explored the relationship between obtained statistical evidence and the degree of belief or confidence that there is a positive effect in the population of interest. We were in particular interested in the existence of a cliff effect. We compared this relationship for p -values to the relationship for corresponding degrees of evidence quantified through Bayes factors, and we examined whether this relationship was affected by two different modes of presentation. In the isolated presentation mode a possible clear functional form of the relationship across values was not visible to the participants, whereas in the all-at-once presentation mode, such a functional form could easily be seen by the participants.

The observed proportions of cliff models was substantially higher for the p -values than for the BFs, and the credible interval as well as the high BF test value indicate that a (substantial) difference will also hold more generally at the population level. Based on our literature review (summarized in Table 1 ), we did not know of studies that have compared the prevalence of cliff effect when interpreting p -values to that when interpreting BFs, so we think that this part is new in the literature. However, our findings are consistent with previous literature regarding the presence of a cliff effect when using p -values. Although we observed a higher proportion of cliff models for isolated presentations than for all-at-once presentation, we did not get a clear indication from the present results whether or not, at the population level, these proportion differences will also hold. We believe that this comparison between the presentation methods that have been used to investigate the cliff effect is also new. In previous research, the p -values were presented on separate pages in some studies 15 , while in other studies the p -values were presented on the same page 13 .

We deviated from our preregistered sampling plan by collecting the e-mail addresses of all corresponding authors who published in the 20 journals in social and behavioural sciences in 2021 and 2022 simultaneously, rather than sequentially. We do not believe that this approach created any bias in our study results. Furthermore, we decided that it would not make sense to collect additional data (after approaching 5399 academics who published in 2019, 2020, 2021, and 2022 in the 20 journals) in order to reach an effective sample size of 200. Based on our interim power analysis, the strongest possible pro-null evidence we could get if we continued collecting data up to an effective sample size of 200 given the data we already had would be BF 10  = 0.14 or BF 01  = 6.99. Therefore, we decided that it would be unethical to continue collecting additional data.

There were several limitations in this study. Firstly, the response rate was very low. This was probably the case because many academics who we contacted mentioned that they were not familiar with interpreting Bayes factors. It is important to note that our findings apply only to researchers who are at least somewhat familiar with interpreting Bayes factors, and our sample does probably not represent the average researcher in the social and behavioural sciences. Indeed, it is well possible that people who are less familiar with Bayes factors (and possibly with statistics in general) would give responses that were even stronger in line with cliff models, because we expect that researchers who exhibit a cliff effect will generally have less statistical expertise or understanding: there is nothing special about certain p -value or Bayes factor thresholds that merits a qualitative drop in the perceived strength of evidence. Furthermore, a salient finding was that the proportion of graduate students was very small. In our sample, the proportion of graduate students showing a cliff effect is 25% and the proportion of more senior researchers showing a cliff effect is 23%. Although we see no clear difference in our sample, we cannot rule out that our findings might be different if the proportion of graduate students in our sample would be higher.

There were several limitations related to the survey. Some of the participants mentioned via e-mail that in the scenarios insufficient information was provided. For example, we did not provide effect sizes and any information about the research topic. We had decided to leave out this information to make sure that the participants could only focus on the p -values and the Bayes factors. Furthermore, the questions in our survey referred to posterior probabilities. A respondent noted that without being able to evaluate the prior plausibility of the rival hypotheses, the questions were difficult to answer. Although this observation is correct, we do think that many respondents think they can do this nevertheless.

The respondents could indicate their degree of belief or confidence that there is a positive effect in the population of interest based on the fictitious findings on a scale ranging from 0 (completely convinced that there is no effect), through 50 (somewhat convinced that there is a positive effect), to 100 (completely convinced that there is a positive effect). A respondent mentioned that it might be unclear where the midpoint is between somewhat convinced that there is no effect and somewhat convinced that there is a positive effect, so biasing the scale towards yes response. Another respondent mentioned that there was no possibility to indicate no confidence in either the null or the alternative hypothesis. Although this is true, we do not think that many participants experienced this as problematic.

In our exploratory analyses we observed that eight out of eleven unclassifiable responses were in the isolated BF condition. In our survey, the all at once and isolated presentation conditions did not only differ in the way the pieces of statistical evidence were presented, but they also differed in the order. In all at once, the different pieces were presented in sequential order, while in the isolated condition, they were presented in a random order. Perhaps this might be an explanation for why the isolated BF condition contained most of the unclassifiable responses. Perhaps academics are more familiar with single p -values and can more easily place them along a line of “possible values” even if they are presented out of order.

This study indicates that a substantial proportion of researchers who are at least somewhat familiar with interpreting BFs experience a sharp drop in confidence when an effect exists around certain p -values and to a much smaller extent around certain Bayes factor values. But how do people act on these beliefs? In a recent study by Muradchanian et al. 24 , it was shown that editors, reviewers, and authors alike are much less likely to accept for publication, endorse, and submit papers with non-significant results than with significant results, suggesting these believes about the existence of an effect translate into considering certain findings more publication-worthy.

Allowing for these caveats, our findings showed that cliff models were more prevalent when interpreting p -values than when interpreting BFs, based on a sample of academics who were at least somewhat familiar with interpreting BFs. However, the high prevalence of the non-cliff models (i.e., linear and exponential) implied that p -values do not necessarily entail dichotomous thinking for everyone. Nevertheless, it is important to note that the cliff models were still accountable for 37.5% of responses in p -values, whereas in BFs, the cliff models were only accountable for 12.3% of the responses.

We note that dichotomous thinking has a place in interpreting scientific evidence, for instance in the context of decision criteria (if the evidence is more compelling than some a priori agreed level, then we bring this new medicine to the market), or in the context of sampling plans (we stop collecting data once the evidence or level of certainty hits some a priori agreed level). However, we claim that it is not rational for someone’s subjective belief that some effect is non-zero to make a big jump around for example a p -value of 0.05 or a BF of 10, but not at any other point along the range of potential values.

Based on our findings, one might think replacing p -values with BFs might be sufficient to overcome dichotomous thinking. We think that this is probably too simplistic. We believe that rejecting or not rejecting a null hypothesis is probably so deep-seated in the academic culture that dichotomous thinking might become more and more prevalent in the interpretation of BFs in time. In addition to using tools such as p -values or BFs, we agree with Lai et al. 13 that several ways to overcome dichotomous thinking in p -values, BFs, etc. are to focus on teaching (future) academics to formulate research questions requiring quantitative answers such as, for example, evaluating the extent to which therapy A is superior to therapy B rather than only evaluating that therapy A is superior to therapy B, and adopting effect size estimation in addition to statistical hypotheses in both thinking and communication.

In light of the results regarding dichotomous thinking among researchers, future research can focus on, for example, the development of comprehensive teaching methods aimed at cultivating the skills necessary for formulating research questions that require quantitative answers. Pedagogical methods and curricula can be investigated that encourage adopting effect size estimation in addition to statistical hypotheses in both thinking and communication.

## Data availability

The raw data are available within the OSF repository: https://osf.io/ndaw6/ .

## Code availability

For the generation of the p -values and BFs, the R file “2022-11-04 psbfs.R” can be used; for Fig.  1 , the R file “2021-06-03 ProtoCliffPlots.R” can be used; for the posterior for the difference between the two proportions in RQ2 and RQ3, the R file “2022-02-17 R script posterior for difference between two proportions.R” can be used; for the Bayesian power simulation, the R file “2022-11-04 Bayes Power Sim Cliff.R” can be used; for calculating the Bayes factors in RQ2 and RQ3 the R file “2022-10-21 BFs RQ2 and RQ3.R” can be used; for the calculation of Cohen’s kappa, the R file “2023-07-23 Cohens kappa.R” can be used; for data preparation, the R file “2023-07-23 data preparation.R” can be used; for Fig.  2 , the R file “2024-03-11 data preparation including Fig.  2 .R” can be used; for the interim power analysis, the R file “2024-03-16 Interim power analysis.R” can be used; for Fig.  3 , the R file “2024-03-16 Plot for Table 4 R” can be used. The R codes were written in R version 2022.2.0.443, and are uploaded as part of the supplementary material. These R codes are made available within the OSF repository: https://osf.io/ndaw6/ .

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## Acknowledgements

We would like to thank Maximilian Linde for writing R code which we could use to collect the e-mail addresses of our potential participants. We would also like to thank Julia Bottesini and an anonymous reviewer for helping us improve the quality of our manuscript.

## Author information

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Jasmine Muradchanian, Rink Hoekstra, Henk Kiers & Don van Ravenzwaaij

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## Contributions

J.M., R.H., H.K., D.F., and D.v.R. meet the following authorship conditions: substantial contributions to the conception or design of the work; or the acquisition, analysis, or interpretation of data; or the creation of new software used in the work; or have drafted the work or substantively revised it; and approved the submitted version (and any substantially modified version that involves the author's contribution to the study); and agreed both to be personally accountable for the author's own contributions and to ensure that questions related to the accuracy or integrity of any part of the work, even ones in which the author was not personally involved, are appropriately investigated, resolved, and the resolution documented in the literature. J.M. participated in data/statistical analysis, participated in the design of the study, drafted the manuscript and critically revised the manuscript; R.H. participated in data/statistical analysis, participated in the design of the study, and critically revised the manuscript; H.K. participated in the design of the study, and critically revised the manuscript; D.F. participated in the design of the study, and critically revised the manuscript; D.v.R. participated in data/statistical analysis, participated in the design of the study, and critically revised the manuscript.

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Muradchanian, J., Hoekstra, R., Kiers, H. et al. Comparing researchers’ degree of dichotomous thinking using frequentist versus Bayesian null hypothesis testing. Sci Rep 14 , 12120 (2024). https://doi.org/10.1038/s41598-024-62043-w

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#### IMAGES

1. Research Hypothesis Generator

2. Null Hypothesis and Alternative Hypothesis

3. Null and Alternative Hypothesis: Symbolic Notation

4. Hypothesis Testing

5. Difference between Null and Alternative Hypothesis

6. Alternative hypothesis

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3. How to frame the Hypothesis statement in your Research

4. Null Hypothesis vs Alternate Hypothesis

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1. Null & 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 ( H0) answers "No, there's no effect in the population.". The alternative hypothesis ( Ha) answers "Yes, there is an effect in the ...

2. 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.

3. 9.1: Null and Alternative Hypotheses

The actual test begins by considering two hypotheses.They are called the null hypothesis and the alternative hypothesis.These hypotheses contain opposing viewpoints. $$H_0$$: The null hypothesis: It is a statement of no difference between the variables—they are not related. This can often be considered the status quo and as a result if you cannot accept the null it requires some action.

4. Null and Alternative Hypotheses

The null and alternative hypotheses are two competing claims that researchers weigh evidence for and against using a statistical test: Null hypothesis (H0): There's no effect in the population. Alternative hypothesis (HA): There's an effect in the population. The effect is usually the effect of the independent variable on the dependent ...

5. What is an Alternative Hypothesis in Statistics?

Null hypothesis: µ ≥ 70 inches. Alternative hypothesis: µ < 70 inches. A two-tailed hypothesis involves making an "equal to" or "not equal to" statement. For example, suppose we assume the mean height of a male in the U.S. is equal to 70 inches. The null and alternative hypotheses in this case would be: Null hypothesis: µ = 70 inches.

6. 5.2

5.2 - Writing Hypotheses. The first step in conducting a hypothesis test is to write the hypothesis statements that are going to be tested. For each test you will have a null hypothesis ( H 0) and an alternative hypothesis ( H a ). When writing hypotheses there are three things that we need to know: (1) the parameter that we are testing (2) the ...

7. Examples of null and alternative hypotheses

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8. 10.2: Null and Alternative Hypotheses

The alternative hypothesis ( Ha H a) is a claim about the population that is contradictory to H0 H 0 and what we conclude when we reject H0 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 ...

9. Null and Alternative Hypotheses

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10. Null and Alternative Hypotheses

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Take the questions and make it a positive statement that says a relationship exists (correlation studies) or a difference exists between the groups (experiment study) and you have the alternative hypothesis. Write the statement such that a relationship does not exist or a difference does not exist and you have the null hypothesis.

14. 9.1 Null and Alternative Hypotheses

The actual test begins by considering two hypotheses.They are called the null hypothesis and the alternative hypothesis.These hypotheses contain opposing viewpoints. H 0: The null hypothesis: It is a statement of no difference between the variables-they are not related. This can often be considered the status quo and as a result if you cannot accept the null it requires some action.

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21. Alternative Hypothesis-Definition, Types and Examples

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23. 9.2: Null and Alternative Hypotheses

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