how do you write a null hypothesis in statistics

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How to Write a Null Hypothesis (5 Examples)

A hypothesis test uses sample data to determine whether or not some claim about a population parameter is true.

Whenever we perform a hypothesis test, we always write a null hypothesis and an alternative hypothesis, which take the following forms:

H 0 (Null Hypothesis): Population parameter =, ≤, ≥ some value

H A (Alternative Hypothesis): Population parameter <, >, ≠ some value

Note that the null hypothesis always contains the equal sign .

We interpret the hypotheses as follows:

Null hypothesis: The sample data provides no evidence to support some claim being made by an individual.

Alternative hypothesis: The sample data does provide sufficient evidence to support the claim being made by an individual.

For example, suppose it’s assumed that the average height of a certain species of plant is 20 inches tall. However, one botanist claims the true average height is greater than 20 inches.

To test this claim, she may go out and collect a random sample of plants. She can then use this sample data to perform a hypothesis test using the following two hypotheses:

H 0 : μ ≤ 20 (the true mean height of plants is equal to or even less than 20 inches)

H A : μ > 20 (the true mean height of plants is greater than 20 inches)

If the sample data gathered by the botanist shows that the mean height of this species of plants is significantly greater than 20 inches, she can reject the null hypothesis and conclude that the mean height is greater than 20 inches.

Read through the following examples to gain a better understanding of how to write a null hypothesis in different situations.

Example 1: Weight of Turtles

A biologist wants to test whether or not the true mean weight of a certain species of turtles is 300 pounds. To test this, he goes out and measures the weight of a random sample of 40 turtles.

Here is how to write the null and alternative hypotheses for this scenario:

H 0 : μ = 300 (the true mean weight is equal to 300 pounds)

H A : μ ≠ 300 (the true mean weight is not equal to 300 pounds)

Example 2: Height of Males

It’s assumed that the mean height of males in a certain city is 68 inches. However, an independent researcher believes the true mean height is greater than 68 inches. To test this, he goes out and collects the height of 50 males in the city.

H 0 : μ ≤ 68 (the true mean height is equal to or even less than 68 inches)

H A : μ > 68 (the true mean height is greater than 68 inches)

Example 3: Graduation Rates

A university states that 80% of all students graduate on time. However, an independent researcher believes that less than 80% of all students graduate on time. To test this, she collects data on the proportion of students who graduated on time last year at the university.

H 0 : p ≥ 0.80 (the true proportion of students who graduate on time is 80% or higher)

H A : μ < 0.80 (the true proportion of students who graduate on time is less than 80%)

Example 4: Burger Weights

A food researcher wants to test whether or not the true mean weight of a burger at a certain restaurant is 7 ounces. To test this, he goes out and measures the weight of a random sample of 20 burgers from this restaurant.

H 0 : μ = 7 (the true mean weight is equal to 7 ounces)

H A : μ ≠ 7 (the true mean weight is not equal to 7 ounces)

Example 5: Citizen Support

A politician claims that less than 30% of citizens in a certain town support a certain law. To test this, he goes out and surveys 200 citizens on whether or not they support the law.

H 0 : p ≥ .30 (the true proportion of citizens who support the law is greater than or equal to 30%)

H A : μ < 0.30 (the true proportion of citizens who support the law is less than 30%)

Additional Resources

Introduction to Hypothesis Testing Introduction to Confidence Intervals An Explanation of P-Values and Statistical Significance

Featured Posts

5 Tips for Interpreting P-Values Correctly in Hypothesis Testing

Hey there. My name is Zach Bobbitt. I have a Masters of Science degree in Applied Statistics and I’ve worked on machine learning algorithms for professional businesses in both healthcare and retail. I’m passionate about statistics, machine learning, and data visualization and I created Statology to be a resource for both students and teachers alike. My goal with this site is to help you learn statistics through using simple terms, plenty of real-world examples, and helpful illustrations.

2 Replies to “How to Write a Null Hypothesis (5 Examples)”

you are amazing, thank you so much

Say I am a botanist hypothesizing the average height of daisies is 20 inches, or not? Does T = (ave – 20 inches) / √ variance / (80 / 4)? … This assumes 40 real measures + 40 fake = 80 n, but that seems questionable. Please advise.

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Academic Writing

Writing Null Hypotheses in Research and Statistics

Last Updated: January 17, 2024 Fact Checked

This article was co-authored by Joseph Quinones and by wikiHow staff writer, Jennifer Mueller, JD . Joseph Quinones is a High School Physics Teacher working at South Bronx Community Charter High School. Joseph specializes in astronomy and astrophysics and is interested in science education and science outreach, currently practicing ways to make physics accessible to more students with the goal of bringing more students of color into the STEM fields. He has experience working on Astrophysics research projects at the Museum of Natural History (AMNH). Joseph recieved his Bachelor's degree in Physics from Lehman College and his Masters in Physics Education from City College of New York (CCNY). He is also a member of a network called New York City Men Teach. There are 7 references cited in this article, which can be found at the bottom of the page. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 25,992 times.

Are you working on a research project and struggling with how to write a null hypothesis? Well, you've come to the right place! Start by recognizing that the basic definition of "null" is "none" or "zero"—that's your biggest clue as to what a null hypothesis should say. Keep reading to learn everything you need to know about the null hypothesis, including how it relates to your research question and your alternative hypothesis as well as how to use it in different types of studies.

Things You Should Know

Write a research null hypothesis as a statement that the studied variables have no relationship to each other, or that there's no difference between 2 groups.

$\mu _{1}=\mu _{2}$

Adjust the format of your null hypothesis to match the statistical method you used to test it, such as using "mean" if you're comparing the mean between 2 groups.

What is a null hypothesis?

A null hypothesis states that there's no relationship between 2 variables.

Research hypothesis: States in plain language that there's no relationship between the 2 variables or there's no difference between the 2 groups being studied.
Statistical hypothesis: States the predicted outcome of statistical analysis through a mathematical equation related to the statistical method you're using.

Examples of Null Hypotheses

Null Hypothesis vs. Alternative Hypothesis

Step 1 Null hypotheses and alternative hypotheses are mutually exclusive.

For example, your alternative hypothesis could state a positive correlation between 2 variables while your null hypothesis states there's no relationship. If there's a negative correlation, then both hypotheses are false.

Step 2 Proving the null hypothesis false is a precursor to proving the alternative.

You need additional data or evidence to show that your alternative hypothesis is correct—proving the null hypothesis false is just the first step.
In smaller studies, sometimes it's enough to show that there's some relationship and your hypothesis could be correct—you can leave the additional proof as an open question for other researchers to tackle.

How do I test a null hypothesis?

Use statistical methods on collected data to test the null hypothesis.

Group means: Compare the mean of the variable in your sample with the mean of the variable in the general population. [6] X Research source
Group proportions: Compare the proportion of the variable in your sample with the proportion of the variable in the general population. [7] X Research source
Correlation: Correlation analysis looks at the relationship between 2 variables—specifically, whether they tend to happen together. [8] X Research source
Regression: Regression analysis reveals the correlation between 2 variables while also controlling for the effect of other, interrelated variables. [9] X Research source

Templates for Null Hypotheses

Research null hypothesis: There is no difference in the mean [dependent variable] between [group 1] and [group 2].

$\mu _{1}+\mu _{2}=0$

Research null hypothesis: The proportion of [dependent variable] in [group 1] and [group 2] is the same.

$p_{1}=p_{2}$

Research null hypothesis: There is no correlation between [independent variable] and [dependent variable] in the population.

$\rho =0$

Research null hypothesis: There is no relationship between [independent variable] and [dependent variable] in the population.

$\beta =0$

Expert Q&A

Expert Interview

Thanks for reading our article! If you’d like to learn more about physics, check out our in-depth interview with Joseph Quinones .

↑ https://online.stat.psu.edu/stat100/lesson/10/10.1
↑ https://online.stat.psu.edu/stat501/lesson/2/2.12
↑ https://support.minitab.com/en-us/minitab/21/help-and-how-to/statistics/basic-statistics/supporting-topics/basics/null-and-alternative-hypotheses/
↑ https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5635437/
↑ https://online.stat.psu.edu/statprogram/reviews/statistical-concepts/hypothesis-testing
↑ https://education.arcus.chop.edu/null-hypothesis-testing/
↑ https://sphweb.bumc.bu.edu/otlt/mph-modules/bs/bs704_hypothesistest-means-proportions/bs704_hypothesistest-means-proportions_print.html

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Null and Alternative Hypotheses | Definitions & Examples

Null and Alternative Hypotheses | Definitions & Examples

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

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

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

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

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

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

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

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

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

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

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

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

Examples of null hypotheses

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

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

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

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

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

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

Examples of alternative hypotheses

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

Null and alternative hypotheses are similar in some ways:

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

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

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

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

Does independent variable affect dependent variable ?

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

Test-specific

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

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

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

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

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

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

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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

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10.1 - setting the hypotheses: examples.

A significance test examines whether the null hypothesis provides a plausible explanation of the data. The null hypothesis itself does not involve the data. It is a statement about a parameter (a numerical characteristic of the population). These population values might be proportions or means or differences between means or proportions or correlations or odds ratios or any other numerical summary of the population. The alternative hypothesis is typically the research hypothesis of interest. Here are some examples.

Example 10.2: Hypotheses with One Sample of One Categorical Variable Section

About 10% of the human population is left-handed. Suppose a researcher at Penn State speculates that students in the College of Arts and Architecture are more likely to be left-handed than people found in the general population. We only have one sample since we will be comparing a population proportion based on a sample value to a known population value.

Research Question : Are artists more likely to be left-handed than people found in the general population?
Response Variable : Classification of the student as either right-handed or left-handed

State Null and Alternative Hypotheses

Null Hypothesis : Students in the College of Arts and Architecture are no more likely to be left-handed than people in the general population (population percent of left-handed students in the College of Art and Architecture = 10% or p = .10).
Alternative Hypothesis : Students in the College of Arts and Architecture are more likely to be left-handed than people in the general population (population percent of left-handed students in the College of Arts and Architecture > 10% or p > .10). This is a one-sided alternative hypothesis.

Example 10.3: Hypotheses with One Sample of One Measurement Variable Section

A generic brand of the anti-histamine Diphenhydramine markets a capsule with a 50 milligram dose. The manufacturer is worried that the machine that fills the capsules has come out of calibration and is no longer creating capsules with the appropriate dosage.

Research Question : Does the data suggest that the population mean dosage of this brand is different than 50 mg?
Response Variable : dosage of the active ingredient found by a chemical assay.
Null Hypothesis : On the average, the dosage sold under this brand is 50 mg (population mean dosage = 50 mg).
Alternative Hypothesis : On the average, the dosage sold under this brand is not 50 mg (population mean dosage ≠ 50 mg). This is a two-sided alternative hypothesis.

Example 10.4: Hypotheses with Two Samples of One Categorical Variable Section

Many people are starting to prefer vegetarian meals on a regular basis. Specifically, a researcher believes that females are more likely than males to eat vegetarian meals on a regular basis.

Research Question : Does the data suggest that females are more likely than males to eat vegetarian meals on a regular basis?
Response Variable : Classification of whether or not a person eats vegetarian meals on a regular basis
Explanatory (Grouping) Variable: Sex
Null Hypothesis : There is no sex effect regarding those who eat vegetarian meals on a regular basis (population percent of females who eat vegetarian meals on a regular basis = population percent of males who eat vegetarian meals on a regular basis or p females = p males ).
Alternative Hypothesis : Females are more likely than males to eat vegetarian meals on a regular basis (population percent of females who eat vegetarian meals on a regular basis > population percent of males who eat vegetarian meals on a regular basis or p females > p males ). This is a one-sided alternative hypothesis.

Example 10.5: Hypotheses with Two Samples of One Measurement Variable Section

Obesity is a major health problem today. Research is starting to show that people may be able to lose more weight on a low carbohydrate diet than on a low fat diet.

Research Question : Does the data suggest that, on the average, people are able to lose more weight on a low carbohydrate diet than on a low fat diet?
Response Variable : Weight loss (pounds)
Explanatory (Grouping) Variable : Type of diet
Null Hypothesis : There is no difference in the mean amount of weight loss when comparing a low carbohydrate diet with a low fat diet (population mean weight loss on a low carbohydrate diet = population mean weight loss on a low fat diet).
Alternative Hypothesis : The mean weight loss should be greater for those on a low carbohydrate diet when compared with those on a low fat diet (population mean weight loss on a low carbohydrate diet > population mean weight loss on a low fat diet). This is a one-sided alternative hypothesis.

Example 10.6: Hypotheses about the relationship between Two Categorical Variables Section

Research Question : Do the odds of having a stroke increase if you inhale second hand smoke ? A case-control study of non-smoking stroke patients and controls of the same age and occupation are asked if someone in their household smokes.
Variables : There are two different categorical variables (Stroke patient vs control and whether the subject lives in the same household as a smoker). Living with a smoker (or not) is the natural explanatory variable and having a stroke (or not) is the natural response variable in this situation.
Null Hypothesis : There is no relationship between whether or not a person has a stroke and whether or not a person lives with a smoker (odds ratio between stroke and second-hand smoke situation is = 1).
Alternative Hypothesis : There is a relationship between whether or not a person has a stroke and whether or not a person lives with a smoker (odds ratio between stroke and second-hand smoke situation is > 1). This is a one-tailed alternative.

This research question might also be addressed like example 11.4 by making the hypotheses about comparing the proportion of stroke patients that live with smokers to the proportion of controls that live with smokers.

Example 10.7: Hypotheses about the relationship between Two Measurement Variables Section

Research Question : A financial analyst believes there might be a positive association between the change in a stock's price and the amount of the stock purchased by non-management employees the previous day (stock trading by management being under "insider-trading" regulatory restrictions).
Variables : Daily price change information (the response variable) and previous day stock purchases by non-management employees (explanatory variable). These are two different measurement variables.
Null Hypothesis : The correlation between the daily stock price change (\$) and the daily stock purchases by non-management employees (\$) = 0.
Alternative Hypothesis : The correlation between the daily stock price change (\$) and the daily stock purchases by non-management employees (\$) > 0. This is a one-sided alternative hypothesis.

Example 10.8: Hypotheses about comparing the relationship between Two Measurement Variables in Two Samples Section

Calculation of a person's approximate tip for their meal

Research Question : Is there a linear relationship between the amount of the bill (\$) at a restaurant and the tip (\$) that was left. Is the strength of this association different for family restaurants than for fine dining restaurants?
Variables : There are two different measurement variables. The size of the tip would depend on the size of the bill so the amount of the bill would be the explanatory variable and the size of the tip would be the response variable.
Null Hypothesis : The correlation between the amount of the bill (\$) at a restaurant and the tip (\$) that was left is the same at family restaurants as it is at fine dining restaurants.
Alternative Hypothesis : The correlation between the amount of the bill (\$) at a restaurant and the tip (\$) that was left is the difference at family restaurants then it is at fine dining restaurants. This is a two-sided alternative hypothesis.

Null Hypothesis Examples

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In statistical analysis, the null hypothesis assumes there is no meaningful relationship between two variables. Testing the null hypothesis can tell you whether your results are due to the effect of manipulating a dependent variable or due to chance. It's often used in conjunction with an alternative hypothesis, which assumes there is, in fact, a relationship between two variables.

The null hypothesis is among the easiest hypothesis to test using statistical analysis, making it perhaps the most valuable hypothesis for the scientific method. By evaluating a null hypothesis in addition to another hypothesis, researchers can support their conclusions with a higher level of confidence. Below are examples of how you might formulate a null hypothesis to fit certain questions.

What Is the Null Hypothesis?

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

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

Examples of the Null Hypothesis

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

Other Types of Hypotheses

In addition to the null hypothesis, the alternative hypothesis is also a staple in traditional significance tests . It's essentially the opposite of the null hypothesis because it assumes the claim in question is true. For the first item in the table above, for example, an alternative hypothesis might be "Age does have an effect on mathematical ability."

Key Takeaways

In hypothesis testing, the null hypothesis assumes no relationship between two variables, providing a baseline for statistical analysis.
Rejecting the null hypothesis suggests there is evidence of a relationship between variables.
By formulating a null hypothesis, researchers can systematically test assumptions and draw more reliable conclusions from their experiments.
Difference Between Independent and Dependent Variables
Examples of Independent and Dependent Variables
What Is a Hypothesis? (Science)
What 'Fail to Reject' Means in a Hypothesis Test
Definition of a Hypothesis
Null Hypothesis Definition and Examples
Scientific Method Vocabulary Terms
Null Hypothesis and Alternative Hypothesis
Hypothesis Test for the Difference of Two Population Proportions
How to Conduct a Hypothesis Test
What Is a P-Value?
What Are the Elements of a Good Hypothesis?
What Is the Difference Between Alpha and P-Values?
Understanding Path Analysis
Hypothesis Test Example
An Example of a Hypothesis Test

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
Introductory Statistics . Authored by : Barbara Illowski, Susan Dean. Provided by : Open Stax. Located at : http://cnx.org/contents/[email protected] . License : CC BY: Attribution . License Terms : Download for free at http://cnx.org/contents/[email protected]
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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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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.

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 :

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

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

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

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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Access for free at https://openstax.org/books/introductory-statistics-2e/pages/1-introduction

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Book title: Introductory Statistics 2e
Publication date: Dec 13, 2023
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Book URL: https://openstax.org/books/introductory-statistics-2e/pages/1-introduction
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7.3: The Null Hypothesis

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The hypothesis that an apparent effect is due to chance is called the null hypothesis, written $H_0$ (“H-naught”). In the Physicians' Reactions example, the null hypothesis is that in the population of physicians, the mean time expected to be spent with obese patients is equal to the mean time expected to be spent with average-weight patients. This null hypothesis can be written as:

\[\mathrm{H}_{0}: \mu_{\mathrm{obese}}-\mu_{\mathrm{average}}=0 \]

The null hypothesis in a correlational study of the relationship between high school grades and college grades would typically be that the population correlation is 0. This can be written as

\[\mathrm{H}_{0}: \rho=0 \]

where $ρ$ is the population correlation, which we will cover in chapter 12.

Although the null hypothesis is usually that the value of a parameter is 0, there are occasions in which the null hypothesis is a value other than 0. For example, if we are working with mothers in the U.S. whose children are at risk of low birth weight, we can use 7.47 pounds, the average birthweight in the US, as our null value and test for differences against that.

For now, we will focus on testing a value of a single mean against what we expect from the population. Using birthweight as an example, our null hypothesis takes the form:

\[\mathrm{H}_{0}: \mu=7.47 \nonumber \]

The number on the right hand side is our null hypothesis value that is informed by our research question. Notice that we are testing the value for $μ$, the population parameter, NOT the sample statistic $\overline{\mathrm{X}}$. This is for two reasons: 1) once we collect data, we know what the value of $\overline{\mathrm{X}}$ is – it’s not a mystery or a question, it is observed and used for the second reason, which is 2) we are interested in understanding the population, not just our sample.

Keep in mind that the null hypothesis is typically the opposite of the researcher's hypothesis. In the Physicians' Reactions study, the researchers hypothesized that physicians would expect to spend less time with obese patients. The null hypothesis that the two types of patients are treated identically is put forward with the hope that it can be discredited and therefore rejected. If the null hypothesis were true, a difference as large or larger than the sample difference of 6.7 minutes would be very unlikely to occur. Therefore, the researchers rejected the null hypothesis of no difference and concluded that in the population, physicians intend to spend less time with obese patients.

In general, the null hypothesis is the idea that nothing is going on: there is no effect of our treatment, no relation between our variables, and no difference in our sample mean from what we expected about the population mean. This is always our baseline starting assumption, and it is what we seek to reject. If we are trying to treat depression, we want to find a difference in average symptoms between our treatment and control groups. If we are trying to predict job performance, we want to find a relation between conscientiousness and evaluation scores. However, until we have evidence against it, we must use the null hypothesis as our starting point.

Math Article

Null Hypothesis

In mathematics, Statistics deals with the study of research and surveys on the numerical data. For taking surveys, we have to define the hypothesis. Generally, there are two types of hypothesis. One is a null hypothesis, and another is an alternative hypothesis .

In probability and statistics, the null hypothesis is a comprehensive statement or default status that there is zero happening or nothing happening. For example, there is no connection among groups or no association between two measured events. It is generally assumed here that the hypothesis is true until any other proof has been brought into the light to deny the hypothesis. Let us learn more here with definition, symbol, principle, types and example, in this article.

Table of contents:

Comparison with Alternative Hypothesis

Null Hypothesis Definition

The null hypothesis is a kind of hypothesis which explains the population parameter whose purpose is to test the validity of the given experimental data. This hypothesis is either rejected or not rejected based on the viability of the given population or sample . In other words, the null hypothesis is a hypothesis in which the sample observations results from the chance. It is said to be a statement in which the surveyors wants to examine the data. It is denoted by H 0 .

Null Hypothesis Symbol

In statistics, the null hypothesis is usually denoted by letter H with subscript ‘0’ (zero), such that H 0 . It is pronounced as H-null or H-zero or H-nought. At the same time, the alternative hypothesis expresses the observations determined by the non-random cause. It is represented by H 1 or H a .

Null Hypothesis Principle

The principle followed for null hypothesis testing is, collecting the data and determining the chances of a given set of data during the study on some random sample, assuming that the null hypothesis is true. In case if the given data does not face the expected null hypothesis, then the outcome will be quite weaker, and they conclude by saying that the given set of data does not provide strong evidence against the null hypothesis because of insufficient evidence. Finally, the researchers tend to reject that.

Null Hypothesis Formula

Here, the hypothesis test formulas are given below for reference.

The formula for the null hypothesis is:

H 0 : p = p 0

The formula for the alternative hypothesis is:

H a = p >p 0 , < p 0 ≠ p 0

The formula for the test static is:

Remember that, p 0 is the null hypothesis and p – hat is the sample proportion.

Also, read:

Types of Null Hypothesis

There are different types of hypothesis. They are:

Simple Hypothesis

It completely specifies the population distribution. In this method, the sampling distribution is the function of the sample size.

Composite Hypothesis

The composite hypothesis is one that does not completely specify the population distribution.

Exact Hypothesis

Exact hypothesis defines the exact value of the parameter. For example μ= 50

Inexact Hypothesis

This type of hypothesis does not define the exact value of the parameter. But it denotes a specific range or interval. For example 45< μ <60

Null Hypothesis Rejection

Sometimes the null hypothesis is rejected too. If this hypothesis is rejected means, that research could be invalid. Many researchers will neglect this hypothesis as it is merely opposite to the alternate hypothesis. It is a better practice to create a hypothesis and test it. The goal of researchers is not to reject the hypothesis. But it is evident that a perfect statistical model is always associated with the failure to reject the null hypothesis.

How do you Find the Null Hypothesis?

The null hypothesis says there is no correlation between the measured event (the dependent variable) and the independent variable. We don’t have to believe that the null hypothesis is true to test it. On the contrast, you will possibly assume that there is a connection between a set of variables ( dependent and independent).

When is Null Hypothesis Rejected?

The null hypothesis is rejected using the P-value approach. If the P-value is less than or equal to the α, there should be a rejection of the null hypothesis in favour of the alternate hypothesis. In case, if P-value is greater than α, the null hypothesis is not rejected.

Null Hypothesis and Alternative Hypothesis

Now, let us discuss the difference between the null hypothesis and the alternative hypothesis.

Null Hypothesis Examples

Here, some of the examples of the null hypothesis are given below. Go through the below ones to understand the concept of the null hypothesis in a better way.

If a medicine reduces the risk of cardiac stroke, then the null hypothesis should be “the medicine does not reduce the chance of cardiac stroke”. This testing can be performed by the administration of a drug to a certain group of people in a controlled way. If the survey shows that there is a significant change in the people, then the hypothesis is rejected.

Few more examples are:

1). Are there is 100% chance of getting affected by dengue?

Ans: There could be chances of getting affected by dengue but not 100%.

2). Do teenagers are using mobile phones more than grown-ups to access the internet?

Ans: Age has no limit on using mobile phones to access the internet.

3). Does having apple daily will not cause fever?

Ans: Having apple daily does not assure of not having fever, but increases the immunity to fight against such diseases.

4). Do the children more good in doing mathematical calculations than grown-ups?

Ans: Age has no effect on Mathematical skills.

In many common applications, the choice of the null hypothesis is not automated, but the testing and calculations may be automated. Also, the choice of the null hypothesis is completely based on previous experiences and inconsistent advice. The choice can be more complicated and based on the variety of applications and the diversity of the objectives.

The main limitation for the choice of the null hypothesis is that the hypothesis suggested by the data is based on the reasoning which proves nothing. It means that if some hypothesis provides a summary of the data set, then there would be no value in the testing of the hypothesis on the particular set of data.

Frequently Asked Questions on Null Hypothesis

What is meant by the null hypothesis.

In Statistics, a null hypothesis is a type of hypothesis which explains the population parameter whose purpose is to test the validity of the given experimental data.

What are the benefits of hypothesis testing?

Hypothesis testing is defined as a form of inferential statistics, which allows making conclusions from the entire population based on the sample representative.

When a null hypothesis is accepted and rejected?

The null hypothesis is either accepted or rejected in terms of the given data. If P-value is less than α, then the null hypothesis is rejected in favor of the alternative hypothesis, and if the P-value is greater than α, then the null hypothesis is accepted in favor of the alternative hypothesis.

Why is the null hypothesis important?

The importance of the null hypothesis is that it provides an approximate description of the phenomena of the given data. It allows the investigators to directly test the relational statement in a research study.

How to accept or reject the null hypothesis in the chi-square test?

If the result of the chi-square test is bigger than the critical value in the table, then the data does not fit the model, which represents the rejection of the null hypothesis.

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Registered Report
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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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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.

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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 .

Foundations of intuitive power analyses in children and adults

Reducing bias, increasing transparency and calibrating confidence with preregistration

Simple nested Bayesian hypothesis testing for meta-analysis, Cox, Poisson and logistic regression models

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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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.

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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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How to Write a Null Hypothesis (5 Examples)
Whenever we perform a hypothesis test, we always write a null hypothesis and an alternative hypothesis, which take the following forms: H0 (Null Hypothesis): Population parameter =, ≤, ≥ some value. HA (Alternative Hypothesis): Population parameter <, >, ≠ some value. Note that the null hypothesis always contains the equal sign.
Null & Alternative Hypotheses
Revised on June 22, 2023. The null and alternative hypotheses are two competing claims that researchers weigh evidence for and against using a statistical test: Null hypothesis (H0): There's no effect in the population. Alternative hypothesis (Ha or H1): There's an effect in the population. The effect is usually the effect of the ...
Null Hypothesis: Definition, Rejecting & Examples
How to Write a Null Hypothesis. The null hypothesis varies by the type of statistic and hypothesis test. Remember that inferential statistics use samples to draw conclusions about populations. Consequently, when you write a null hypothesis, it must make a claim about the relevant population parameter. Further, that claim usually indicates that ...
How to Write a Null Hypothesis (with Examples and Templates)
Write a research null hypothesis as a statement that the studied variables have no relationship to each other, or that there's no difference between 2 groups. Write a statistical null hypothesis as a mathematical equation, such as. μ 1 = μ 2 {\displaystyle \mu _ {1}=\mu _ {2}} if you're comparing group means.
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.
Hypothesis Testing
Table of contents. Step 1: State your null and alternate hypothesis. Step 2: Collect data. Step 3: Perform a statistical test. Step 4: Decide whether to reject or fail to reject your null hypothesis. Step 5: Present your findings. Other interesting articles. Frequently asked questions about hypothesis testing.
Null Hypothesis Definition and Examples, How to State
Step 1: Figure out the hypothesis from the problem. The hypothesis is usually hidden in a word problem, and is sometimes a statement of what you expect to happen in the experiment. The hypothesis in the above question is "I expect the average recovery period to be greater than 8.2 weeks.". Step 2: Convert the hypothesis to math.
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 ...
Null hypothesis
Basic definitions. The null hypothesis and the alternative hypothesis are types of conjectures used in statistical tests to make statistical inferences, which are formal methods of reaching conclusions and separating scientific claims from statistical noise.. The statement being tested in a test of statistical significance is called the null hypothesis. . The test of significance is designed ...
Examples of null and alternative hypotheses
It is the opposite of your research hypothesis. The alternative hypothesis--that is, the research hypothesis--is the idea, phenomenon, observation that you want to prove. If you suspect that girls take longer to get ready for school than boys, then: Alternative: girls time > boys time. Null: girls time <= boys time.
8.1.1: Null and Alternative Hypotheses
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.
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\)). Null Hypothesis. The statement that there is not a difference in the population (s), denoted as \ (H_0\)
8.3: The Null Hypothesis
8.3: The Null Hypothesis. The hypothesis that an apparent effect is due to chance is called the null hypothesis, written ("H-naught"). In the Physicians' Reactions example, the null hypothesis is that in the population of physicians, the mean time expected to be spent with obese patients is equal to the mean time expected to be spent with ...
10.1
10.1 - Setting the Hypotheses: Examples. A significance test examines whether the null hypothesis provides a plausible explanation of the data. The null hypothesis itself does not involve the data. It is a statement about a parameter (a numerical characteristic of the population). These population values might be proportions or means or ...
How to Formulate a Null Hypothesis (With Examples)
The null hypothesis states there is no relationship between the measured phenomenon (the dependent variable) and the independent variable, which is the variable an experimenter typically controls or changes.You do not need to believe that the null hypothesis is true to test it. On the contrary, you will likely suspect there is a relationship between a set of variables.
16.3: The Process of Null Hypothesis Testing
16.3.5 Step 5: Determine the probability of the data under the null hypothesis. This is the step where NHST starts to violate our intuition - rather than determining the likelihood that the null hypothesis is true given the data, we instead determine the likelihood of the data under the null hypothesis - because we started out by assuming that the null hypothesis is true!
Null and Alternative Hypotheses
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.
What is a Null Hypothesis?
Overview of null hypothesis, examples of null and alternate hypotheses, and how to write a null hypothesis statement.
Null & Alternative Hypotheses
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. Examples: Null Hypothesis: H 0: There is no difference in the salary ...
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.
7.3: The Null Hypothesis
The null hypothesis in a correlational study of the relationship between high school grades and college grades would typically be that the population correlation is 0. This can be written as. H0: ρ = 0 (7.3.2) (7.3.2) H 0: ρ = 0. where ρ ρ is the population correlation, which we will cover in chapter 12. Although the null hypothesis is ...
Null & Alternative Hypothesis
The general procedure for testing the null hypothesis is as follows: Suppose you perform a statistical test of the null hypothesis with α = .05 and obtain a p-value of p = .04, thereby rejecting the null hypothesis. This does not mean there is a 4% probability of the null hypothesis being true, i.e. P(H0) =.04.
Null Hypothesis
In other words, the null hypothesis is a hypothesis in which the sample observations results from the chance. It is said to be a statement in which the surveyors wants to examine the data. It is denoted by H 0. Null Hypothesis Symbol. In statistics, the null hypothesis is usually denoted by letter H with subscript '0' (zero), such that H 0 ...
Comparing researchers' degree of dichotomous thinking using ...
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 ...

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Example 10.2: Hypotheses with One Sample of One Categorical Variable Section

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9.1 Null and Alternative Hypotheses

Example 9.1

Example 9.2

Example 9.3

Example 9.4

Collaborative Exercise

Margin Size

7.3: The Null Hypothesis

Null Hypothesis

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Null Hypothesis Formula

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