null and alternative hypothesis statements

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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.

Candela Citations

• 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]
• Simple hypothesis testing | Probability and Statistics | Khan Academy. Authored by : Khan Academy. Located at : https://youtu.be/5D1gV37bKXY . License : All Rights Reserved . License Terms : Standard YouTube License

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

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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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Microbe Notes

Null hypothesis and alternative hypothesis with 9 differences

Table of Contents

Interesting Science Videos

Null hypothesis definition

The null hypothesis is a general statement that states that there is no relationship between two phenomenons under consideration or that there is no association between two groups.

• A hypothesis, in general, is an assumption that is yet to be proved with sufficient pieces of evidence. A null hypothesis thus is the hypothesis a researcher is trying to disprove.
• A null hypothesis is a hypothesis capable of being objectively verified, tested, and even rejected.
• If a study is to compare method A with method B about their relationship, and if the study is preceded on the assumption that both methods are equally good, then this assumption is termed as the null hypothesis.
• The null hypothesis should always be a specific hypothesis, i.e., it should not state about or approximately a certain value.

Null hypothesis symbol

• The symbol for the null hypothesis is H 0, and it is read as H-null, H-zero, or H-naught.
• The null hypothesis is usually associated with just ‘equals to’ sign as a null hypothesis can either be accepted or rejected.

Null hypothesis purpose

• The main purpose of a null hypothesis is to verify/ disprove the proposed statistical assumptions.
• Some scientific null hypothesis help to advance a theory.
• The null hypothesis is also used to verify the consistent results of multiple experiments. For e.g., the null hypothesis stating that there is no relation between some medication and age of the patients supports the general effectiveness conclusion, and allows recommendations.

Null hypothesis principle

• The principle of the null hypothesis is collecting the data and determining the chances of the collected data in the study of a random sample, proving that the null hypothesis is true.
• In situations or studies where the collected data doesn’t complete the expectation of the null hypothesis, it is concluded that the data doesn’t provide sufficient or reliable pieces of evidence to support the null hypothesis and thus, it is rejected.
• The data collected is tested through some statistical tool which is designed to measure the extent of departure of the date from the null hypothesis.
• The procedure decides whether the observed departure obtained from the statistical tool is larger than a defined value so that the probability of occurrence of a high departure value is very small under the null hypothesis.
• However, some data might not contradict the null hypothesis which explains that only a weak conclusion can be made and that the data doesn’t provide strong pieces of evidence against the null hypothesis and the null hypothesis might or might not be true.
• Under some other conditions, if the data collected is sufficient and is capable of providing enough evidence, the null hypothesis can be considered valid, indicating no relationship between the phenomena.

When to reject null hypothesis?

• When the p-value of the data is less than the significant level of the test, the null hypothesis is rejected, indicating the test results are significant.
• However, if the p-value is higher than the significant value, the null hypothesis is not rejected, and the results are considered not significant.
• The level of significance is an important concept while hypothesis testing as it determines the percentage risk of rejecting the null hypothesis when H 0 might happen to be true.
• In other words, if we take the level of significance at 5%, it means that the researcher is willing to take as much as a 5 percent risk of rejecting the null hypothesis when it (H 0 ) happens to be true.
• The null hypothesis cannot be accepted because the lack of evidence only means that the relationship is not proven. It doesn’t prove that something doesn’t exist, but it just means that there are not enough shreds of evidence and the study might have missed it.

Null hypothesis examples

The following are some examples of null hypothesis:

• If the hypothesis is that “the consumption of a particular medicine reduces the chances of heart arrest”, the null hypothesis will be “the consumption of the medicine doesn’t reduce the chances of heart arrest.”
• If the hypothesis is that, “If random test scores are collected from men and women, does the score of one group differ from the other?” a possible null hypothesis will be that the mean test score of men is the same as that of the women.

H 0 : µ 1 = µ 2

H 0 = null hypothesis µ 1 = mean score of men µ 2 = mean score of women

Alternative hypothesis definition

An alternative hypothesis is a statement that describes that there is a relationship between two selected variables in a study.

• An alternative hypothesis is usually used to state that a new theory is preferable to the old one (null hypothesis).
• This hypothesis can be simply termed as an alternative to the null hypothesis.
• The alternative hypothesis is the hypothesis that is to be proved that indicates that the results of a study are significant and that the sample observation is not results just from chance but from some non-random cause.
• If a study is to compare method A with method B about their relationship and we assume that the method A is superior or the method B is inferior, then such a statement is termed as an alternative hypothesis.
• Alternative hypotheses should be clearly stated, considering the nature of the research problem.

Alternative hypothesis symbol

• The symbol of the alternative hypothesis is either H 1 or H a while using less than, greater than or not equal signs.

Alternative hypothesis purpose

• An alternative hypothesis provides the researchers with some specific restatements and clarifications of the research problem.
• An alternative hypothesis provides a direction to the study, which then can be utilized by the researcher to obtain the desired results.
• Since the alternative hypothesis is selected before conducting the study, it allows the test to prove that the study is supported by evidence, separating it from the researchers’ desires and values.
• An alternative hypothesis provides a chance of discovering new theories that can disprove an existing one that might not be supported by evidence.
• The alternative hypothesis is important as they prove that a relationship exists between two variables selected and that the results of the study conducted are relevant and significant.

Alternative hypothesis principle

• The principle behind the alternative hypothesis is similar to that of the null hypothesis.
• The alternative hypothesis is based on the concept that when sufficient evidence is collected from the data of random sample, it provides a basis for proving the assumption made by the researcher regarding the study.
• Like in the null hypothesis, the data collected from a random sample is passed through a statistical tool that measures the extent of departure of the data from the null hypothesis.
• If the departure is small under the selected level of significance, the alternative hypothesis is accepted, and the null hypothesis is rejected.
• If the data collected don’t have chances of being in the study of the random sample and are instead decided by the relationship within the sample of the study, an alternative hypothesis stands true.

Alternative hypothesis examples

The following are some examples of alternative hypothesis:

1. If a researcher is assuming that the bearing capacity of a bridge is more than 10 tons, then the hypothesis under this study will be:

Null hypothesis H 0 : µ= 10 tons Alternative hypothesis H a : µ>10 tons

2. Under another study that is trying to test whether there is a significant difference between the effectiveness of medicine against heart arrest, the alternative hypothesis will be that there is a relationship between the medicine and chances of heart arrest.

Null hypothesis vs Alternative hypothesis

• R. Kothari (1990) Research Methodology. Vishwa Prakasan. India.
• https://www.statisticssolutions.com/null-hypothesis-and-alternative-hypothesis/
• https://byjus.com/maths/null-hypothesis/
• https://en.wikipedia.org/wiki/Null_hypothesis
• https://keydifferences.com/difference-between-null-and-alternative-hypothesis.html
• 5% – https://en.wikipedia.org/wiki/Null_hypothesis
• 3% – https://keydifferences.com/difference-between-null-and-alternative-hypothesis.html
• 2% – https://byjus.com/maths/null-hypothesis/
• 1% – https://www.wisdomjobs.com/e-university/research-methodology-tutorial-355/procedure-for-hypothesis-testing-11525.html
• 1% – https://www.thoughtco.com/definition-of-null-hypothesis-and-examples-605436
• 1% – https://www.quora.com/What-are-the-different-types-of-hypothesis-and-what-are-some-examples-of-them
• 1% – https://www.dummies.com/education/math/statistics/what-a-p-value-tells-you-about-statistical-data/
• 1% – https://www.coursehero.com/file/p7jfbal5/These-are-hypotheses-capable-of-being-objectively-verified-and-tested-Thus-we/
• 1% – https://support.minitab.com/en-us/minitab/18/help-and-how-to/modeling-statistics/anova/how-to/one-way-anova/interpret-the-results/all-statistics-and-graphs/methods/
• 1% – https://stats.stackexchange.com/questions/105319/test-whether-there-is-a-significant-difference-between-two-groups
• 1% – https://statisticsbyjim.com/hypothesis-testing/failing-reject-null-hypothesis/
• 1% – https://quizlet.com/45299306/statistics-flash-cards/
• <1% – https://www.thoughtco.com/significance-level-in-hypothesis-testing-1147177
• <1% – https://www.thoughtco.com/null-hypothesis-vs-alternative-hypothesis-3126413
• <1% – https://www.sagepub.com/sites/default/files/upm-binaries/40007_Chapter8.pdf
• <1% – https://www.differencebetween.com/difference-between-hypothesis-and-vs-assumption/
• <1% – https://www.coursehero.com/file/18076181/introduction-to-hypothesis/
• <1% – https://statisticsbyjim.com/glossary/significance-level/
• <1% – https://quizlet.com/164755799/research-methods-midterm-2-flash-cards/
• <1% – https://online.stat.psu.edu/statprogram/reviews/statistical-concepts/hypothesis-testing/p-value-approach

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Difference between Null and Alternate Hypothesis

Hypothesis is a statement or an assumption that may be true or false. There are six types of hypotheses mainly the Simple hypothesis, Complex hypothesis, Directional hypothesis, Associative hypothesis, and Null hypothesis. Usually, the hypothesis is the start point of any scientific investigation, It gives the right direction to the process of investigation. It avoids the blind search and gives direction to the search. It acts as a compass in the process.

Null hypothesis suggests that there is no relationship between the two variables. Null hypothesis is also exactly the opposite of the alternative hypothesis. Null hypothesis is generally what researchers or scientists try to disprove and if the null hypothesis gets accepted then we have to make changes in our opinion i.e. we have to make changes in our original opinion or statement in order to match null hypothesis. Null hypothesis is represented as H0. If my alternative hypothesis is that 55% of boys in my town are taller than girls then my alternative hypothesis will be that 55% of boys in my town are not taller than girls. 

Alternative hypothesis is a method for reaching a conclusion and making inferences and judgements about certain facts or a statement. This is done on the basis of the data which is available. Usually, the statement which we check regarding the null hypothesis is commonly known as the alternative hypothesis. Most of the times alternative hypothesis is exactly the opposite of the null hypothesis. This is what generally researchers or scientists try to approve. Alternative hypothesis is represented as Ha or H1. If my null hypothesis is that 55% of boys in my town are not taller than girls then my alternative hypothesis will be that 55% of boys in my town are taller than girls.

Following is the difference between the null hypothesis and alternate hypothesis:

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Difference Between Null and Alternative Hypothesis

Null hypothesis implies a statement that expects no difference or effect. On the contrary, an alternative hypothesis is one that expects some difference or effect. Null hypothesis This article excerpt shed light on the fundamental differences between null and alternative hypothesis.

Content: Null Hypothesis Vs Alternative Hypothesis

Comparison chart, definition of null hypothesis.

A null hypothesis is a statistical hypothesis in which there is no significant difference exist between the set of variables. It is the original or default statement, with no effect, often represented by H 0 (H-zero). It is always the hypothesis that is tested. It denotes the certain value of population parameter such as µ, s, p. A null hypothesis can be rejected, but it cannot be accepted just on the basis of a single test.

Definition of Alternative Hypothesis

A statistical hypothesis used in hypothesis testing, which states that there is a significant difference between the set of variables. It is often referred to as the hypothesis other than the null hypothesis, often denoted by H 1 (H-one). It is what the researcher seeks to prove in an indirect way, by using the test. It refers to a certain value of sample statistic, e.g., x¯, s, p

The acceptance of alternative hypothesis depends on the rejection of the null hypothesis i.e. until and unless null hypothesis is rejected, an alternative hypothesis cannot be accepted.

Key Differences Between Null and Alternative Hypothesis

The important points of differences between null and alternative hypothesis are explained as under:

• A null hypothesis is a statement, in which there is no relationship between two variables. An alternative hypothesis is a statement; that is simply the inverse of the null hypothesis, i.e. there is some statistical significance between two measured phenomenon.
• A null hypothesis is what, the researcher tries to disprove whereas an alternative hypothesis is what the researcher wants to prove.
• A null hypothesis represents, no observed effect whereas an alternative hypothesis reflects, some observed effect.
• If the null hypothesis is accepted, no changes will be made in the opinions or actions. Conversely, if the alternative hypothesis is accepted, it will result in the changes in the opinions or actions.
• As null hypothesis refers to population parameter, the testing is indirect and implicit. On the other hand, the alternative hypothesis indicates sample statistic, wherein, the testing is direct and explicit.
• A null hypothesis is labelled as H 0 (H-zero) while an alternative hypothesis is represented by H 1 (H-one).
• The mathematical formulation of a null hypothesis is an equal sign but for an alternative hypothesis is not equal to sign.
• In null hypothesis, the observations are the outcome of chance whereas, in the case of the alternative hypothesis, the observations are an outcome of real effect.

There are two outcomes of a statistical test, i.e. first, a null hypothesis is rejected and alternative hypothesis is accepted, second, null hypothesis is accepted, on the basis of the evidence. In simple terms, a null hypothesis is just opposite of alternative hypothesis.

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Zipporah Thuo says

February 22, 2018 at 6:06 pm

The comparisons between the two hypothesis i.e Null hypothesis and the Alternative hypothesis are the best.Thank you.

Getu Gamo says

March 4, 2019 at 3:42 am

Thank you so much for the detail explanation on two hypotheses. Now I understood both very well, including their differences.

Jyoti Bhardwaj says

May 28, 2019 at 6:26 am

Thanks, Surbhi! Appreciate the clarity and precision of this content.

January 9, 2020 at 6:16 am

John Jenstad says

July 20, 2020 at 2:52 am

Thanks very much, Surbhi, for your clear explanation!!

Navita says

July 2, 2021 at 11:48 am

Thanks for the Comparison chart! it clears much of my doubt.

GURU UPPALA says

July 21, 2022 at 8:36 pm

Thanks for the Comparison chart!

Enock kipkoech says

September 22, 2022 at 1:57 pm

What are the examples of null hypothesis and substantive hypothesis

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Null and Alternative Hypothesis: Research Guidelines

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When undertaking a qualitative or quantitative research project, researchers must first formulate a research question, from which they develop their theories. By definition, an assumption is a prediction that a researcher makes about an actual research question and can either be affirmative or negative. In this case, writing a research question has three main components: variables (independent and dependent), a population sample, and the relation between these variables. To find null and alternative hypotheses, scholars identify a specific research question, determine the variables involved, and state H0 as no effect or difference and H1 or Ha as a significant effect or difference. When the prediction contradicts the research question, it is referred to as a null assumption. In short, an initial theory is a statement that implies there is no relationship between independent and dependent variables. Hence, researchers need to learn how to write a good null and alternative hypothesis to present quality studies.

General Aspects

Students with qualitative or quantitative research assignments must learn how to formulate and write good research questions and proposition statements. In essence, hypothesis testing is a statistical method used to determine if there is enough evidence to reject an initial theory and support an alternative assumption based on sample data. By definition, a research proposition is an assumption or prediction that a scholar makes before undertaking an experimental investigation. Basically, academic standards require such a prediction to be a precise and testable statement, meaning that researchers must prove or disapprove of it in the course of their assignment and provide alternatives if possible. In this case, the main components of a typical research assumption are variables (independent and dependent), a population sample, and the relation between these variables. To formulate a null hypothesis (H0) in quantitative research, researchers state there is no effect or difference between variables (e.g., µ1 = µ2), and, for an alternative hypothesis (H1 or Ha), they posit there is a significant effect or difference (e.g., µ1 ≠ µ2). Therefore, a research proposition is a prediction that scholars write about the relationship between two or more variables. In turn, a standard research inquiry is a particular process that seeks to answer a specific research question and, in the process, test a particular theory by confirming or disapproving it.

Types of Hypotheses

There are several types of hypotheses, including null, alternative, directional, and non-directional assumptions. Basically, a directional hypothesis is a prediction of how an independent variable affects a dependent variable. In contrast, a non-directional hypothesis predicts that an independent variable influences a dependent variable but does not specify how. Regardless of the type, all propositions are about predicting the relationship between independent and dependent variables. To write H0 (null assumption) and H1 or Ha (alternative prediction), researchers clearly state H0 as a central assumption of no effect or no difference (e.g., µ1 = µ2) and H1 or Ha as a secondary assumption of a significant effect or difference (e.g., µ1 ≠ µ2).

What Is a Null Hypothesis (H0) and Its Purpose

According to its definition, a null hypothesis is a foundational statement in statistical testing that posits there is no significant effect, relationship, or difference between groups or variables within a given study. In simple words, a null hypothesis, usually symbolized as “H0,” is a statement that contradicts an actual research theory (Watt & Collins, 2019). The main purpose of writing a null hypothesis is to provide a basis for comparison, allowing researchers to determine whether there is sufficient evidence to reject this assumption in favor of an alternative theory, which suggests a real effect or relationship. As such, it is a negative statement, indicating that there is no relationship or connection between independent and dependent variables (Harrison et al., 2020). By starting with a null proposition, researchers can also employ various statistical tests to evaluate an entire data, ensuring the objectivity of findings and minimizing their bias. The process helps to ensure the validity of scientific research, minimizing the likelihood of drawing incorrect conclusions from the data collected. Moreover, by testing an initial theory, researchers can determine whether the inquiry results are due to the chance or the effect of manipulating a dependent variable (McNulty, 2022). In most instances, a null assumption corresponds with an alternative theory, a positive statement that covers a relationship that exists between independent and dependent variables. Finally, it is highly recommended that researchers should write an alternative assumption first before a null proposition.

What Is an Alternative Hypothesis (H1 or Ha) and Its Purpose

According to its definition and opposite to a null assumption, an alternative hypothesis in research is another statement in statistical testing that suggests there is a significant effect, relationship, or difference between groups or variables in a given study. Basically, this statement contrasts with what a null theory posits, which asserts that no such effect or relationship exists (Baker, 2021). The main purpose of writing an alternative hypothesis is to guide researchers in testing and validating new theories or effects and determine whether the observed data can provide evidence against a null proposition. The process involves comparing observed results to what would be expected under a null assumption. When statistical tests provide enough evidence to reject an initial postulation, an alternative theory becomes true, indicating that the observed effect or relationship is likely real and not due to random variation (Jawlik, 2016). By framing their research around an alternative hypothesis, scientists can focus their investigations on discovering meaningful effects and relationships, thereby advancing knowledge and understanding in their study fields. Hence, writing good null and alternative hypotheses is important because they provide a structured framework for statistical testing, allowing researchers to objectively evaluate evidence and draw conclusions about the presence of significant effects or relationships in an entire data.

Null vs. Alternative Hypothesis Formats

Steps on how to write a good null and alternative hypothesis.

• Identify a Specific Research Question: Start with clearly defining a particular problem or phenomenon you want to study.
• Determine Key Variables: Identify independent and dependent variables involved in your study.
• State a Specific Null Hypothesis (H0): Formulate a concrete statement that suggests no effect, no difference, or no relationship between your variables. This is usually a statement of equality (e.g., µ1 = µ2).
• State a Clear Alternative Hypothesis (H1 or Ha): Formulate another statement that suggests a significant effect, difference, or relationship between your variables. This is usually a statement of inequality (e.g., µ1 ≠ µ2, µ1 > µ2, or µ1 < µ2).
• Means: H0: µ1 = µ2 vs. H1: µ1 ≠ µ2
• Proportions: H0: p1 = p2 vs. H1: p1 ≠ p2
• One-tailed test: If you are testing for a specific direction of effect (e.g., H1: µ1 > µ2).
• Two-tailed test: If you are testing for any difference, regardless of direction (e.g., H1: µ1 ≠ µ2).
• Consult Literature: Review existing research to see how similar or alternative theories have been formulated. This can provide guidance and ensure your expectations are aligned with standard practices in your field.
• Write in Simple Terms: Ensure both null and alternative theories are stated clearly and concisely, making them easy to understand.
• Review and Refine: Double-check your propositions for clarity and correctness. Make sure they are mutually exclusive and collectively exhaustive, covering all possible outcomes.
• Seek Feedback: Discuss your approaches with peers or advisors to ensure they are logical, relevant, and testable. Adjust as necessary based on their input.

Note: A null hypothesis is a specific statement assuming no effect or difference, while other hypotheses refer to general statements that include writing null and alternative hypotheses and proposing possible outcomes to be tested.

Written Examples of Research Questions With H0 and H1 Hypotheses

Before developing any study proposition, a researcher must formulate a specific research question. In this case, a research hypothesis is a broad, testable statement about the expected relationship between variables, while a statistical hypothesis specifically refers to writing null and alternative hypotheses used in statistical testing to validate or refute an initial study assumption (O’Donnell et al., 2023). Then, the next step is to transform this study question into a negative statement that claims the lack of a relationship between independent and dependent variables. Alternatively, researchers can change the question into a positive statement that includes a relationship that exists between the variables. In turn, this latter statement becomes an alternative hypothesis and is symbolized as H1 or Ha. Hence, some of the examples of research questions and hull and alternative hypotheses are as follows:

Research Question (RQ) 1: Do physical exercises help individuals to age gracefully?

• A Null Hypothesis (H0): Physical exercises are not a guarantee for graceful old age.
• An Alternative Hypothesis (H1): Engaging in physical exercises enables individuals to remain healthy and active into old age.

RQ 2: What are the implications of therapeutic interventions in the fight against substance abuse?

• H0: Therapeutic interventions are of no help in the fight against substance abuse.
• H1: Exposing individuals with substance abuse disorders to therapeutic interventions helps to control and even stop their addictions.

RQ 3: How do sexual orientation and gender identity affect the experiences of late adolescents in foster care?

• H0: Sexual orientation and gender identity have no effects on the experiences of late adolescents in foster care.
• H1: The reality of stereotypes in society makes sexual orientation and gender identity factors complicate the experiences of late adolescents in foster care.

RQ 4: Does income inequality contribute to crime in high-density urban areas?

• H0: There is no correlation between income inequality and incidences of crime in high-density urban areas.
• H1: The high crime rates in high-density urban areas are due to the incidence of income inequality in those areas.

RQ 5: Does placement in foster care impact individuals’ mental health?

• H0: There is no correlation between being in foster care and having mental health problems.
• H1: Individuals placed in foster care experience anxiety and depression at one point in their life.

RQ 6: Do assistive devices and technologies lessen the mobility challenges of older adults with a stroke?

• H0: Assistive devices and technologies do not provide any assistance to the mobility of older adults diagnosed with a stroke.
• H1: Assistive devices and technologies enhance the mobility of older adults diagnosed with a stroke.

RQ 7: Does race identity undermine classroom participation?

• H0: There is no correlation between racial identity and the ability to participate in classroom learning.
• H1: Students from racial minorities are not as active as white students in classroom participation.

RQ 8: Do high school grades determine future success?

• H0: There is no correlation between how one performs in high school and their success level in life.
• H1: Attaining high grades in high school positions one for greater success in the future personal and professional lives.

RQ 9: Does critical thinking predict academic achievement?

• H0: There is no correlation between critical thinking and academic achievement.
• H1: Being a critical thinker is a pathway to academic success.

RQ 10: What benefits does group therapy provide to victims of domestic violence?

• H0: Group therapy does not help victims of domestic violence because individuals prefer to hide rather than expose their shame.
• H1: Group therapy provides domestic violence victims with a platform to share their hurt and connect with others with similar experiences.

Symbols and Signs in Writing

Common Mistakes

• Ambiguity in Theories: Writing vague or unclear null and alternative assumptions.
• Directional vs. Non-Directional Confusion: Confusing one-tailed (directional) and two-tailed (non-directional) claims.
• Using Sample Statistics: Stating initial and alternative propositions in terms of sample statistics instead of population parameters.
• Overlapping Assumptions: Creating null and alternative statements that are not mutually exclusive.
• Testing Multiple Variables: Including multiple variables or conditions in a single theory.
• Misinterpreting a Null Proposition: Assuming an initial statement is what you want to prove.
• Incorrect Symbols and Signs: Using incorrect or inconsistent symbols and signs for writing null and alternative propositions.
• Ignoring Context: Writing initial and alternative theories that are not relevant to an assigned research question or context.
• Not Testable Hypotheses: Formulating null and alternative statements that are not testable with the available data or methods.
• Confusing Null and Alternative Hypotheses: Swapping the roles of null and alternative assumptions.

The formulation of research questions in qualitative and quantitative assignments helps students to develop a specific theory for their experiments. In this case, learning how to write a good null and alternative hypothesis helps students and researchers to make their research relevant. Basically, the difference between a null and alternative hypothesis is that the former contradicts an entire research question, while the latter affirms it. In short, an initial proposition is a negative statement relative to a particular research question, and an alternative theory is a positive assumption. Moreover, it is important to note that developing a null hypothesis at the beginning of the assignment is for prediction purposes. As such, the research work must answer a specific research question and confirm or disapprove of an initial proposition. Hence, some of the tips that students and researchers need to know when developing any theory include:

• Formulate a research question that specifies the relationship between an independent variable and a dependent variable.
• Develop an alternative assumption that says a relationship exists between the variables.
• Develop a null proposition that says a relationship does not exist between the variables.
• Conduct an experiment to answer a research question under analysis, which allows the confirmation of a disapproval of a null theory or considering alternative options.

Baker, L. (2021). Hypothesis testing: How to choose the correct test (Getting started with statistics) . Chi-Squared Innovations.

Harrison, A. J., McErlain-Naylor, S. A., Bradshaw, E. J., Dai, B., Nunome, H., Hughes, G. T. G., Kong, P. W., Vanwanseele, B., Vilas-Boas, J. P., & Fong, D. T. (2020). Recommendations for statistical analysis involving null hypothesis significance testing. Sports Biomechanics , 19 (5), 561–568. https://doi.org/10.1080/14763141.2020.1782555

Jawlik, A. (2016). Statistics from A to Z: Confusing concepts clarified . John Wiley & Sons, Inc.

McNulty, R. (2022). A logical analysis of null hypothesis significance testing using popular terminology. BMC Medical Research Methodology , 22 (1), 1–9. https://doi.org/10.1186/s12874-022-01696-5

O’Donnell, C. T., Fielding-Singh, V., & Vanneman, M. W. (2023). The art of the null hypothesis — Considerations for study design and scientific reporting. Journal of Cardiothoracic and Vascular Anesthesia , 37 (6), 867–869. https://doi.org/10.1053/j.jvca.2023.02.026

Watt, R., & Collins, E. (2019). Null hypothesis testing . SAGE Publications Ltd.

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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  

• 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.

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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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Null vs. Alternative Hypothesis

04.28.2023 • 5 min read

Sarah Thomas

Subject Matter Expert

Learn about a null versus alternative hypothesis and what they show with examples for each. Also go over the main differences and similarities between them.

In This Article

What Is a Null Hypothesis?

What is an alternative hypothesis, outcomes of a hypothesis test.

Main Differences Between Null & Alternative Hypothesis

Similarities Between Null & Alternative Hypothesis

Hypothesis Testing & Errors

In statistics, you’ll draw insights or “inferences” about population parameters using data from a sample. This process is called inferential statistics.

To make statistical inferences, you need to determine if you have enough evidence to support a certain hypothesis about the population. This is where null and alternative hypotheses come into play!

In this article, we’ll explain the differences between these two types of hypotheses, and we’ll explain the role they play in hypothesis testing.

Imagine you want to know what percent of Americans are vegetarians. You find a Gallup poll claiming ‌5% of the population was vegetarian in 2018, but your intuition tells you vegetarianism is on the rise and that ‌far more than 5% of Americans are vegetarian today.

To investigate further, you collect your own sample data by surveying 1,000 randomly selected Americans. You’ll use this random sample to determine whether it’s likely ‌the true population proportion of vegetarians is, in fact, 5% (as the Gallup data suggests) or whether it could be the case that the percentage of vegetarians is now higher.

Notice ‌that your investigation involves two rival hypotheses about the population. One hypothesis is that the proportion of vegetarians is 5%. The other hypothesis is that the proportion of vegetarians is greater than 5%. In statistics, we would call the first hypothesis the null hypothesis, and the second hypothesis the alternative hypothesis. The null hypothesis ( H 0 H_0 H 0 ​ ) represents the status quo or what is assumed to be true about the population at the start of your investigation.

Null Hypothesis

In hypothesis testing, the null hypothesis ( H 0 H_0 H 0 ​ ) is the default hypothesis.

It's what the status quo assumes to be true about the population.

The alternative hypothesis ( H a H_a H a ​ or H 1 H_1 H 1 ​ ) is the hypothesis that stands contrary to the null hypothesis. The alternative hypothesis ‌represents the research hypothesis—what you as the statistician are trying to prove with your data .

In medical studies, where scientists are trying to demonstrate whether a treatment has a significant effect on patient outcomes, the alternative hypothesis represents the hypothesis that the treatment does have an effect, while the null hypothesis represents the assumption that the treatment has no effect.

Alternative Hypothesis

The alternative hypothesis ( H a H_a H a ​ or H 1 H_1 H 1 ​ ) is the hypothesis being proposed in opposition to the null hypothesis.

Examples of Null and Alternative Hypotheses

In a hypothesis test, the null and alternative hypotheses must be mutually exclusive statements, meaning both hypotheses cannot be true at the same time. For example, if the null hypothesis includes an equal sign, the alternative hypothesis must state that the values being mentioned are “not equal” in some way.

Your hypotheses will also depend on the formulation of your test—are you running a one-sample T-test, a two-sample T-test, F-test for ANOVA , or a Chi-squared test? It also matters whether you are conducting a directional one-tailed test or a nondirectional two-tailed test.

Example 1: Two-Tailed T-test

Null Hypothesis: The population mean is equal to some number, x. 𝝁 = x

Alternative Hypothesis: The population mean is not equal to x. 𝝁 ≠ x

Example 2: One-tailed T-test (Right-Tailed)

Null Hypothesis: The population mean is less than or equal to some number, x. 𝝁 ≤ x Alternative Hypothesis: The population mean is greater than x. 𝝁 > x

Example 3: One-tailed T-test (Left-Tailed)

Null Hypothesis: The population mean is greater than or equal to some number, x. 𝝁 ≥ x

Alternative Hypothesis: The population mean is less than x. 𝝁 < x

By the end of a hypothesis test, you will have reached one of two conclusions.

You will run into either 2 outcomes:

Fail to reject the null hypothesis on the grounds that there's insufficient evidence to move away from the null hypothesis

Reject the null hypothesis in favor of the alternative.

If you’re ‌confused about the outcomes of a hypothesis test, a good analogy is a jury trial. In a jury trial, the defendant is innocent until proven guilty. To reach a verdict of guilt, the jury must find strong evidence (beyond a reasonable doubt) that the defendant committed the crime.

This is analogous to a statistician who must assume the null hypothesis is true unless they can uncover strong evidence ( a p-value less than or equal to the significance level) in support of the alternative hypothesis.

Notice also, that a jury never concludes a defendant is innocent—only that the defendant is guilty or not guilty. This is similar to how we never conclude that the null hypothesis is true. In a hypothesis test, we never conclude ‌that the null hypothesis is true. We can only “reject” the null hypothesis or “fail to reject” it.

In this video, let’s look at the jury example again, the reasoning behind hypothesis testing, and how to form a test. It starts by stating your null and alternative hypotheses.

Main Differences Between Null and Alternative Hypothesis

Here is a summary of the key differences between the null and the alternative hypothesis test.

The null hypothesis represents the status quo; the alternative hypothesis represents an alternative statement about the population.

The null and the alternative are mutually exclusive statements, meaning both statements cannot be true at the same time.

In a medical study, the null hypothesis represents the assumption that a treatment has no statistically significant effect on the outcome being studied. The alternative hypothesis represents the belief that the treatment does have an effect.

The null hypothesis is denoted by H_0 ; the alternative hypothesis is denoted by H_a H_1

You “fail to reject” the null hypothesis when the p-value is larger than the significance level. You “reject” the null hypothesis in favor of the alternative hypothesis when the p-value is less than or equal to your test’s significance level.

Similarities Between Null and Alternative Hypothesis

The similarities between the null and alternative hypotheses are as follows.

Both the null and the alternative are statements about the same underlying data.

Both statements provide a possible answer to a statistician’s research question.

The same hypothesis test will provide evidence for or against the null and alternative hypotheses.

Hypothesis Testing and Errors

Always remember that statistical inference provides you with inferences based on probability rather than hard truths. Anytime you conduct a hypothesis test, there is a chance that you’ll reach the wrong conclusion about your data.

In statistics, we categorize these wrong conclusions into two types of errors:

Type I Errors

Type II Errors

Type I Error (ɑ)

A Type I error occurs when you reject the null hypothesis when, in fact, the null hypothesis is true. This is sometimes called a false positive and is analogous to a jury that falsely convicts an innocent defendant. The probability of making this type of error is represented by alpha, ɑ.

Type II Error (ꞵ)

A Type II error occurs when you fail to reject the null hypothesis when, in fact, the null hypothesis is false. This is sometimes called a false negative and is analogous to a jury that reaches a verdict of “not guilty,” when, in fact, the defendant has committed the crime. The probability of making this type of error is represented by beta, ꞵ.

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