null hypothesis statistics symbol

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
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Hypothesis Testing with One Sample

Null and Alternative Hypotheses

OpenStaxCollege

[latexpage]

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

H 0 : The null hypothesis: It is a statement about the population that either is believed to be true or is used to put forth an argument unless it can be shown to be incorrect beyond a reasonable doubt.

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

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

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

Mathematical Symbols Used in H 0 and H a :

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

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

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

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

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

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

H 0 : μ = 2.0

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

• H 0 : μ = 66
• H a : μ ≠ 66

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

H 0 : μ ≥ 5

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

• H 0 : μ ≥ 45
• H a : μ < 45

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

H 0 : p ≤ 0.066

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

• H 0 : p = 0.40
• H a : p > 0.40

<!– ??? –>

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

Chapter Review

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

Formula Review

H 0 and H a are contradictory.

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

If α > p -value, then reject H 0 .

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

• p < 0.30
• p > 0.30

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

• p > 0.20
• p < 0.20

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

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

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

Null and Alternative Hypotheses Copyright © 2013 by OpenStaxCollege is licensed under a Creative Commons Attribution 4.0 International License , except where otherwise noted.

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• Knowledge Base
• Null and Alternative Hypotheses | Definitions & Examples

Null and Alternative Hypotheses | Definitions & Examples

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

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

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

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

Table of contents

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

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

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

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

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

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

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

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

Examples of null hypotheses

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

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

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

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

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

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

Examples of alternative hypotheses

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

Null and alternative hypotheses are similar in some ways:

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

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

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

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

Does independent variable affect dependent variable ?

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

Test-specific

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

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

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

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

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

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

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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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Video transcript

• 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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7.3: The Research Hypothesis and the Null Hypothesis

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• Page ID 18038

• Michelle Oja
• Taft College

Hypotheses are predictions of expected findings.

The Research Hypothesis

A research hypothesis is a mathematical way of stating a research question.  A research hypothesis names the groups (we'll start with a sample and a population), what was measured, and which we think will have a higher mean.  The last one gives the research hypothesis a direction.  In other words, a research hypothesis should include:

• The name of the groups being compared.  This is sometimes considered the IV.
• What was measured.  This is the DV.
• Which group are we predicting will have the higher mean.  

There are two types of research hypotheses related to sample means and population means:  Directional Research Hypotheses and Non-Directional Research Hypotheses

Directional Research Hypothesis

If we expect our obtained sample mean to be above or below the other group's mean (the population mean, for example), we have a directional hypothesis. There are two options:

• Symbol:       \( \displaystyle \bar{X} > \mu \)
• (The mean of the sample is greater than than the mean of the population.)
• Symbol:     \( \displaystyle \bar{X} < \mu \)
• (The mean of the sample is less than than mean of the population.)

Example \(\PageIndex{1}\)

A study by Blackwell, Trzesniewski, and Dweck (2007) measured growth mindset and how long the junior high student participants spent on their math homework.  What’s a directional hypothesis for how scoring higher on growth mindset (compared to the population of junior high students) would be related to how long students spent on their homework?  Write this out in words and symbols.

Answer in Words:            Students who scored high on growth mindset would spend more time on their homework than the population of junior high students.

Answer in Symbols:         \( \displaystyle \bar{X} > \mu \) 

Non-Directional Research Hypothesis

A non-directional hypothesis states that the means will be different, but does not specify which will be higher.  In reality, there is rarely a situation in which we actually don't want one group to be higher than the other, so we will focus on directional research hypotheses.  There is only one option for a non-directional research hypothesis: "The sample mean differs from the population mean."  These types of research hypotheses don’t give a direction, the hypothesis doesn’t say which will be higher or lower.

A non-directional research hypothesis in symbols should look like this:    \( \displaystyle \bar{X} \neq \mu \) (The mean of the sample is not equal to the mean of the population).

Exercise \(\PageIndex{1}\)

What’s a non-directional hypothesis for how scoring higher on growth mindset higher on growth mindset (compared to the population of junior high students) would be related to how long students spent on their homework (Blackwell, Trzesniewski, & Dweck, 2007)?  Write this out in words and symbols.

Answer in Words:            Students who scored high on growth mindset would spend a different amount of time on their homework than the population of junior high students.

Answer in Symbols:        \( \displaystyle \bar{X} \neq \mu \) 

See how a non-directional research hypothesis doesn't really make sense?  The big issue is not if the two groups differ, but if one group seems to improve what was measured (if having a growth mindset leads to more time spent on math homework).  This textbook will only use directional research hypotheses because researchers almost always have a predicted direction (meaning that we almost always know which group we think will score higher).

The Null Hypothesis

The hypothesis that an apparent effect is due to chance is called the null hypothesis, written \(H_0\) (“H-naught”). We usually test this through comparing an experimental group to a comparison (control) group.  This null hypothesis can be written as:

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

For most of this textbook, the null hypothesis is that the means of the two groups are similar.  Much later, the null hypothesis will be that there is no relationship between the two groups.  Either way, remember that a null hypothesis is always saying that nothing is different.  

This is where descriptive statistics diverge from inferential statistics.  We know what the value of \(\overline{\mathrm{X}}\) is – it’s not a mystery or a question, it is what we observed from the sample.  What we are using inferential statistics to do is infer whether this sample's descriptive statistics probably represents the population's descriptive statistics.  This is the null hypothesis, that the two groups are similar.  

Keep in mind that the null hypothesis is typically the opposite of the research hypothesis. A research hypothesis for the ESP example is that those in my sample who say that they have ESP would get more correct answers than the population would get correct, while the null hypothesis is that the average number correct for the two groups will be similar. 

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.

In sum, the null hypothesis is always : There is no difference between the groups’ means OR There is no relationship between the variables .

In the next chapter, the null hypothesis is that there’s no difference between the sample mean   and population mean.  In other words:

• There is no mean difference between the sample and population.
• The mean of the sample is the same as the mean of a specific population.
• \(\mathrm{H}_{0}: \bar{X} = \mu \nonumber \)
• We expect our sample’s mean to be same as the population mean.

Exercise \(\PageIndex{2}\)

A study by Blackwell, Trzesniewski, and Dweck (2007) measured growth mindset and how long the junior high student participants spent on their math homework.  What’s the null hypothesis for scoring higher on growth mindset (compared to the population of junior high students) and how long students spent on their homework?  Write this out in words and symbols.

Answer in Words:            Students who scored high on growth mindset would spend a similar amount of time on their homework as the population of junior high students.

Answer in Symbols:    \( \bar{X} = \mu \)

Contributors and Attributions

Foster et al.  (University of Missouri-St. Louis, Rice University, & University of Houston, Downtown Campus)

Dr. MO ( Taft College )

• Null hypothesis

by Marco Taboga , PhD

In a test of hypothesis , a sample of data is used to decide whether to reject or not to reject a hypothesis about the probability distribution from which the sample was extracted.

The hypothesis is called the null hypothesis, or simply "the null".

Table of contents

The null is like the defendant in a criminal trial

How is the null hypothesis tested, example 1 - proportion of defective items, measurement, test statistic, critical region, interpretation, example 2 - reliability of a production plant, rejection and failure to reject, not rejecting and accepting are not the same thing, failure to reject can be due to lack of power, rejections are easier to interpret, but be careful, takeaways - how to (and not to) formulate a null hypothesis, more examples, more details, best practices in science, keep reading the glossary.

Formulating null hypotheses and subjecting them to statistical testing is one of the workhorses of the scientific method.

Scientists in all fields make conjectures about the phenomena they study, translate them into null hypotheses and gather data to test them.

This process resembles a trial:

the defendant (the null hypothesis) is accused of being guilty (wrong);

evidence (data) is gathered in order to prove the defendant guilty (reject the null);

if there is evidence beyond any reasonable doubt, the defendant is found guilty (the null is rejected);

otherwise, the defendant is found not guilty (the null is not rejected).

Keep this analogy in mind because it helps to better understand statistical tests, their limitations, use and misuse, and frequent misinterpretation.

Before collecting the data:

we decide how to summarize the relevant characteristics of the sample data in a single number, the so-called test statistic ;

we derive the probability distribution of the test statistic under the hypothesis that the null is true (the data is regarded as random; therefore, the test statistic is a random variable);

we decide what probability of incorrectly rejecting the null we are willing to tolerate (the level of significance , or size of the test ); the level of significance is typically a small number, such as 5% or 1%.

we choose one or more intervals of values (collectively called rejection region) such that the probability that the test statistic falls within these intervals is equal to the desired level of significance; the rejection region is often a tail of the distribution of the test statistic (one-tailed test) or the union of the left and right tails (two-tailed test).

Then, the data is collected and used to compute the value of the test statistic.

A decision is taken as follows:

if the test statistic falls within the rejection region, then the null hypothesis is rejected;

otherwise, it is not rejected.

We now make two examples of practical problems that lead to formulate and test a null hypothesis.

A new method is proposed to produce light bulbs.

The proponents claim that it produces less defective bulbs than the method currently in use.

To check the claim, we can set up a statistical test as follows.

We keep the light bulbs on for 10 consecutive days, and then we record whether they are still working at the end of the test period.

The probability that a light bulb produced with the new method is still working at the end of the test period is the same as that of a light bulb produced with the old method.

100 light bulbs are tested:

50 of them are produced with the new method (group A)

the remaining 50 are produced with the old method (group B).

The final data comprises 100 observations of:

an indicator variable which is equal to 1 if the light bulb is still working at the end of the test period and 0 otherwise;

a categorical variable that records the group (A or B) to which each light bulb belongs.

We use the data to compute the proportions of working light bulbs in groups A and B.

The proportions are estimates of the probabilities of not being defective, which are equal for the two groups under the null hypothesis.

We then compute a z-statistic (see here for details) by:

taking the difference between the proportion in group A and the proportion in group B;

standardizing the difference:

we subtract the expected value (which is zero under the null hypothesis);

we divide by the standard deviation (it can be derived analytically).

The distribution of the z-statistic can be approximated by a standard normal distribution .

We decide that the level of confidence must be 5%. In other words, we are going to tolerate a 5% probability of incorrectly rejecting the null hypothesis.

The critical region is the right 5%-tail of the normal distribution, that is, the set of all values greater than 1.645 (see the glossary entry on critical values if you are wondering how this value was obtained).

If the test statistic is greater than 1.645, then the null hypothesis is rejected; otherwise, it is not rejected.

A rejection is interpreted as significant evidence that the new production method produces less defective items; failure to reject is interpreted as insufficient evidence that the new method is better.

A production plant incurs high costs when production needs to be halted because some machinery fails.

The plant manager has decided that he is not willing to tolerate more than one halt per year on average.

If the expected number of halts per year is greater than 1, he will make new investments in order to improve the reliability of the plant.

A statistical test is set up as follows.

The reliability of the plant is measured by the number of halts.

The number of halts in a year is assumed to have a Poisson distribution with expected value equal to 1 (using the Poisson distribution is common in reliability testing).

The manager cannot wait more than one year before taking a decision.

There will be a single datum at his disposal: the number of halts observed during one year.

The number of halts is used as a test statistic. By assumption, it has a Poisson distribution under the null hypothesis.

The manager decides that the probability of incorrectly rejecting the null can be at most 10%.

A Poisson random variable with expected value equal to 1 takes values:

larger than 1 with probability 26.42%;

larger than 2 with probability 8.03%.

Therefore, it is decided that the critical region will be the set of all values greater than or equal to 3.

If the test statistic is strictly greater than or equal to 3, then the null is rejected; otherwise, it is not rejected.

A rejection is interpreted as significant evidence that the production plant is not reliable enough (the average number of halts per year is significantly larger than tolerated).

Failure to reject is interpreted as insufficient evidence that the plant is unreliable.

This section discusses the main problems that arise in the interpretation of the outcome of a statistical test (reject / not reject).

When the test statistic does not fall within the critical region, then we do not reject the null hypothesis.

Does this mean that we accept the null? Not really.

In general, failure to reject does not constitute, per se, strong evidence that the null hypothesis is true .

Remember the analogy between hypothesis testing and a criminal trial. In a trial, when the defendant is declared not guilty, this does not mean that the defendant is innocent. It only means that there was not enough evidence (not beyond any reasonable doubt) against the defendant.

In turn, lack of evidence can be due:

either to the fact that the defendant is innocent ;

or to the fact that the prosecution has not been able to provide enough evidence against the defendant, even if the latter is guilty .

This is the very reason why courts do not declare defendants innocent, but they use the locution "not guilty".

In a similar fashion, statisticians do not say that the null hypothesis has been accepted, but they say that it has not been rejected.

To better understand why failure to reject does not in general constitute strong evidence that the null hypothesis is true, we need to use the concept of statistical power .

The power of a test is the probability (calculated ex-ante, i.e., before observing the data) that the null will be rejected when another hypothesis (called the alternative hypothesis ) is true.

Let's consider the first of the two examples above (the production of light bulbs).

In that example, the null hypothesis is: the probability that a light bulb is defective does not decrease after introducing a new production method.

Let's make the alternative hypothesis that the probability of being defective is 1% smaller after changing the production process (assume that a 1% decrease is considered a meaningful improvement by engineers).

How much is the ex-ante probability of rejecting the null if the alternative hypothesis is true?

If this probability (the power of the test) is small, then it is very likely that we will not reject the null even if it is wrong.

If we use the analogy with criminal trials, low power means that most likely the prosecution will not be able to provide sufficient evidence, even if the defendant is guilty.

Thus, in the case of lack of power, failure to reject is almost meaningless (it was anyway highly likely).

This is why, before performing a test, it is good statistical practice to compute its power against a relevant alternative .

If the power is found to be too small, there are usually remedies. In particular, statistical power can usually be increased by increasing the sample size (see, e.g., the lecture on hypothesis tests about the mean ).

As we have explained above, interpreting a failure to reject the null hypothesis is not always straightforward. Instead, interpreting a rejection is somewhat easier.

When we reject the null, we know that the data has provided a lot of evidence against the null. In other words, it is unlikely (how unlikely depends on the size of the test) that the null is true given the data we have observed.

There is an important caveat though. The null hypothesis is often made up of several assumptions, including:

the main assumption (the one we are testing);

other assumptions (e.g., technical assumptions) that we need to make in order to set up the hypothesis test.

For instance, in Example 2 above (reliability of a production plant), the main assumption is that the expected number of production halts per year is equal to 1. But there is also a technical assumption: the number of production halts has a Poisson distribution.

It must be kept in mind that a rejection is always a joint rejection of the main assumption and all the other assumptions .

Therefore, we should always ask ourselves whether the null has been rejected because the main assumption is wrong or because the other assumptions are violated.

In the case of Example 2 above, is a rejection of the null due to the fact that the expected number of halts is greater than 1 or is it due to the fact that the distribution of the number of halts is very different from a Poisson distribution?

When we suspect that a rejection is due to the inappropriateness of some technical assumption (e.g., assuming a Poisson distribution in the example), we say that the rejection could be due to misspecification of the model .

The right thing to do when these kind of suspicions arise is to conduct so-called robustness checks , that is, to change the technical assumptions and carry out the test again.

In our example, we could re-run the test by assuming a different probability distribution for the number of halts (e.g., a negative binomial or a compound Poisson - do not worry if you have never heard about these distributions).

If we keep obtaining a rejection of the null even after changing the technical assumptions several times, the we say that our rejection is robust to several different specifications of the model .

What are the main practical implications of everything we have said thus far? How does the theory above help us to set up and test a null hypothesis?

What we said can be summarized in the following guiding principles:

A test of hypothesis is like a criminal trial and you are the prosecutor . You want to find evidence that the defendant (the null hypothesis) is guilty. Your job is not to prove that the defendant is innocent. If you find yourself hoping that the defendant is found not guilty (i.e., the null is not rejected) then something is wrong with the way you set up the test. Remember: you are the prosecutor.

Compute the power of your test against one or more relevant alternative hypotheses. Do not run a test if you know ex-ante that it is unlikely to reject the null when the alternative hypothesis is true.

Beware of technical assumptions that you add to the main assumption you want to test. Make robustness checks in order to verify that the outcome of the test is not biased by model misspecification.

More examples of null hypotheses and how to test them can be found in the following lectures.

The lecture on Hypothesis testing provides a more detailed mathematical treatment of null hypotheses and how they are tested.

This lecture on the null hypothesis was featured in Stanford University's Best practices in science .

Previous entry: Normal equations

Next entry: Parameter

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Please cite as:

Taboga, Marco (2021). "Null hypothesis", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/glossary/null-hypothesis.

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5.5 Introduction to Hypothesis Tests

One job of a statistician is to make statistical inferences about populations based on samples taken from the population. Confidence intervals are one way to estimate a population parameter.

Another way to make a statistical inference is to make a decision about a parameter. For instance, a car dealer advertises that its new small truck gets 35 miles per gallon, on average. A tutoring service claims that its method of tutoring helps 90% of its students get an A or a B. A company says that women managers in their company earn an average of $60,000 per year.  A statistician may want to make a decision about or evaluate these claims. A hypothesis test can be used to do this.

A hypothesis test involves collecting data from a sample and evaluating the data. Then, the statistician makes a decision as to whether or not there is sufficient evidence, based upon analyses of the data, to reject the null hypothesis.

In this section you will conduct hypothesis tests on single means when the population standard deviation is known.

Hypothesis testing consists of two contradictory hypotheses or statements, a decision based on the data, and a conclusion. To perform a hypothesis test, a statistician will perform some variation of these steps:

• Define hypotheses.
• Collect and/OR use the sample data to determine the correct distribution to use.
• Calculate Test Statistic.
• Make a decision
• Write a conclusion.

Defining your hypotheses

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

The null hypothesis ( H 0 ): It is often a statement of the accepted historical value or norm. This is your starting point that you must assume from the beginning in order to show an effect exists.

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

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

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

Mathematical symbols used in H 0 and H a :

Note: 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.

We want to test whether the mean GPA of students in American colleges is different from 2.0 (out of 4.0). The null hypothesis is: H 0 : μ = 2.0. What is the alternative hypothesis?

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

Using the Sample to Test the Null Hypothesis

Once you have defined your hypotheses the next step in the process, is to collect sample data.  In a classroom context most of the time the data or summary statistics will be given to you.

Then you will have to determine the correct distribution to perform the hypothesis test, given the assumptions you are able to make about the situation.  Right now we are demonstrating these ideas in a test for a mean when the population standard deviation is known using the Z distribution.  We will see other scenarios in the future.

Calculating a Test Statistic

Next, you will start evaluating the data.  This begins with calculating your test statistic , which is a measure of how far what you observed is from what you are assuming to he true.  In this context, your test statistic, z ο , quantifies the number of standard deviations between the sample mean x and the population mean µ.  Calculating the test statistic is analogous to standardizing observations with Z-scores as discussed previously:

where µ o   is the value assumed to be true in the null hypothesis.

Making a Decision

Once you have your test statistic there are two methods to use it to make your decision:

• Critical value method—This is one way you can make a decision, but will not be discussed in detail at this time.
• P-Value method—This is the preferred method we will focus on.

P-Value Method

To find a p-value we use the test statistic to calculate the actual probability of getting the test result. Formally, the p -value is the probability that, if the null hypothesis is true, the results from another randomly selected sample will be as extreme or more extreme as the results obtained from the given sample.

A large p -value calculated from the data indicates that we should not reject the null hypothesis. The smaller the p -value, the more unlikely the outcome, and the stronger the evidence is against the null hypothesis. We would reject the null hypothesis if the evidence is strongly against it.

Draw a graph that shows the p -value. The hypothesis test is easier to perform if you use a graph because you see the problem more clearly.

Suppose a baker claims that his bread height is more than 15 cm, on average. Several of his customers do not believe him. To persuade his customers that he is right, the baker decides to do a hypothesis test. He bakes 10 loaves of bread. The mean height of the sample loaves is 17 cm. The baker knows from baking hundreds of loaves of bread that the standard deviation for the height is 0.5 cm. and the distribution of heights is normal.

The null hypothesis could be H 0 : μ ≤ 15

The alternate hypothesis is H a : μ > 15

The words “is more than” translates as a “>” so “ μ > 15″ goes into the alternate hypothesis. The null hypothesis must contradict the alternate hypothesis.

Suppose the null hypothesis is true (the mean height of the loaves is no more than 15 cm). Then is the mean height (17 cm) calculated from the sample unexpectedly large? The hypothesis test works by asking the question how unlikely the sample mean would be if the null hypothesis were true. The graph shows how far out the sample mean is on the normal curve. The p -value is the probability that, if we were to take other samples, any other sample mean would fall at least as far out as 17 cm.

The p -value, then, is the probability that a sample mean is the same or greater than 17 cm. when the population mean is, in fact, 15 cm. We can calculate this probability using the normal distribution for means.

A p -value of approximately zero tells us that it is highly unlikely that a loaf of bread rises no more than 15 cm, on average. That is, almost 0% of all loaves of bread would be at least as high as 17 cm. purely by CHANCE had the population mean height really been 15 cm. Because the outcome of 17 cm. is so unlikely (meaning it is happening NOT by chance alone), we conclude that the evidence is strongly against the null hypothesis (the mean height is at most 15 cm.). There is sufficient evidence that the true mean height for the population of the baker’s loaves of bread is greater than 15 cm.

A normal distribution has a standard deviation of 1. We want to verify a claim that the mean is greater than 12. A sample of 36 is taken with a sample mean of 12.5.

Find The P-value:

Decision and Conclusion

A systematic way to make a decision of whether to reject or not reject the null hypothesis is to compare the p -value and a preset or preconceived α (also called a significance level ). A preset α is the probability of a Type I error (rejecting the null hypothesis when the null hypothesis is true). It may or may not be given to you at the beginning of the problem.  If there is no given preconceived α , then use α = 0.05.

When you make a decision to reject or not reject H 0 , do as follows:

• If α > p -value, reject H 0 . The results of the sample data are statistically significant . You can say there is sufficient evidence to conclude that H 0 is an incorrect belief and that the alternative hypothesis, H a , may be correct.
• If α ≤ p -value, fail to reject H 0 . The results of the sample data are not significant. There is not sufficient evidence to conclude that the alternative hypothesis, H a , may be correct.

After you make your decision, write a thoughtful conclusion in the context of the scenario incorporating the hypotheses.

NOTE: When you “do not reject H 0 “, it does not mean that you should believe that H 0 is true. It simply means that the sample data have failed to provide sufficient evidence to cast serious doubt about the truthfulness of H o .

When using the p -value to evaluate a hypothesis test, it is sometimes useful to use the following memory device

If the p -value is low, the null must go.

If the p -value is high, the null must fly.

This memory aid relates a p -value less than the established alpha (the p is low) as rejecting the null hypothesis and, likewise, relates a p -value higher than the established alpha (the p is high) as not rejecting the null hypothesis.

Fill in the blanks:

• Reject the null hypothesis when _____.
• The results of the sample data _____.
• Do not reject the null when hypothesis when _____.

It’s a Boy Genetics Labs claim their procedures improve the chances of a boy being born. The results for a test of a single population proportion are as follows:

H 0 : p = 0.50, H a : p > 0.50

p -value = 0.025

Interpret the results and state a conclusion in simple, non-technical terms.

MORE RESOURCES

Figure References

Figure 5.11: Alora Griffiths. 2019. Public domain. https://unsplash.com/photos/7aRQZtLsvqw

Figure 5.13: Kindred Grey. 2020. CC BY-SA 4.0.

A decision making procedure for determining whether sample evidence supports a hypothesis

The claim that is assumed to be true and is tested in a hypothesis test

A working hypothesis that is contradictory to the null hypothesis

A measure of how far what you observed is from the hypothesized (or claimed) value

The probability that an event will occur, assuming the null hypothesis is true

Probability that a true null hypothesis will be rejected, also known as Type I error and denoted by α

Finding sufficient evidence that the effect we see is not just due to variability, often from rejecting the null hypothesis

Significant Statistics Copyright © 2024 by John Morgan Russell, OpenStaxCollege, OpenIntro is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License , except where otherwise noted.

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Statistics Made Easy

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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5.2 - writing hypotheses.

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

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

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

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

9.1 Null and Alternative Hypotheses

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

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

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

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

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

Mathematical Symbols Used in H 0 and H a :

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

Example 9.1

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

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

Example 9.2

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

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

• H 0 : μ __ 66
• H a : μ __ 66

Example 9.3

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

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

• H 0 : μ __ 45
• H a : μ __ 45

Example 9.4

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

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

• H 0 : p __ 0.40
• H a : p __ 0.40

Collaborative Exercise

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

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7.1: Null and Alternative Hypotheses

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

• The null hypothesis (\(H_{0}\)) is a statement about the population that either is believed to be true or is used to put forth an argument unless it can be shown to be incorrect beyond a reasonable doubt.
• The alternative hypothesis (\(H_{a}\)) is a claim about the population that is contradictory to \(H_{0}\) and what we conclude when we reject \(H_{0}\).

Since the null and alternative hypotheses are contradictory, you must examine evidence to decide if you have enough evidence to reject the null hypothesis or not. The evidence is in the form of sample data. After you have determined which hypothesis the sample supports, you make a decision. There are two options for a decision. They are "reject \(H_{0}\)" if the sample information favors the alternative hypothesis or "do not reject \(H_{0}\)" or "decline to reject \(H_{0}\)" if the sample information is insufficient to reject the null hypothesis.

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

Example \(\PageIndex{1}\)

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

Exercise \(\PageIndex{1}\)

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

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

Example \(\PageIndex{2}\)

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

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

Exercise \(\PageIndex{2}\)

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

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

Example \(\PageIndex{3}\)

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

• \(H_{0}: \mu \geq 66\)
• \(H_{a}: \mu < 66\)

Exercise \(\PageIndex{3}\)

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

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

Example \(\PageIndex{4}\)

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

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

Exercise \(\PageIndex{4}\)

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

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

COLLABORATIVE EXERCISE

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

Chapter Review

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

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

Formula Review

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

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

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

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

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Null Hypothesis

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Null Hypothesis , often denoted as H 0, is a foundational concept in statistical hypothesis testing. It represents an assumption that no significant difference, effect, or relationship exists between variables within a population. It serves as a baseline assumption, positing no observed change or effect occurring. The null is t he truth or falsity of an idea in analysis.

In this article, we will discuss the null hypothesis in detail, along with some solved examples and questions on the null hypothesis.

Table of Content

What is Null Hypothesis?

Null hypothesis symbol, formula of null hypothesis, types of null hypothesis, null hypothesis examples, principle of null hypothesis, how do you find null hypothesis, null hypothesis in statistics, null hypothesis and alternative hypothesis, null hypothesis and alternative hypothesis examples, null hypothesis – practice problems.

Null Hypothesis in statistical analysis suggests the absence of statistical significance within a specific set of observed data. Hypothesis testing, using sample data, evaluates the validity of this hypothesis. Commonly denoted as H 0 or simply “null,” it plays an important role in quantitative analysis, examining theories related to markets, investment strategies, or economies to determine their validity.

Null Hypothesis Meaning

Null Hypothesis represents a default position, often suggesting no effect or difference, against which researchers compare their experimental results. The Null Hypothesis, often denoted as H 0 asserts a default assumption in statistical analysis. It posits no significant difference or effect, serving as a baseline for comparison in hypothesis testing.

The null Hypothesis is represented as H 0 , the Null Hypothesis symbolizes the absence of a measurable effect or difference in the variables under examination.

Certainly, a simple example would be asserting that the mean score of a group is equal to a specified value like stating that the average IQ of a population is 100.

The Null Hypothesis is typically formulated as a statement of equality or absence of a specific parameter in the population being studied. It provides a clear and testable prediction for comparison with the alternative hypothesis. The formulation of the Null Hypothesis typically follows a concise structure, stating the equality or absence of a specific parameter in the population.

Mean Comparison (Two-sample t-test)

H 0 : μ 1 = μ 2

This asserts that there is no significant difference between the means of two populations or groups.

Proportion Comparison

H 0 : p 1 − p 2 = 0

This suggests no significant difference in proportions between two populations or conditions.

Equality in Variance (F-test in ANOVA)

H 0 : σ 1 = σ 2

This states that there’s no significant difference in variances between groups or populations.

Independence (Chi-square Test of Independence):

H 0 : Variables are independent

This asserts that there’s no association or relationship between categorical variables.

Null Hypotheses vary including simple and composite forms, each tailored to the complexity of the research question. Understanding these types is pivotal for effective hypothesis testing.

Equality Null Hypothesis (Simple Null Hypothesis)

The Equality Null Hypothesis, also known as the Simple Null Hypothesis, is a fundamental concept in statistical hypothesis testing that assumes no difference, effect or relationship between groups, conditions or populations being compared.

Non-Inferiority Null Hypothesis

In some studies, the focus might be on demonstrating that a new treatment or method is not significantly worse than the standard or existing one.

Superiority Null Hypothesis

The concept of a superiority null hypothesis comes into play when a study aims to demonstrate that a new treatment, method, or intervention is significantly better than an existing or standard one.

Independence Null Hypothesis

In certain statistical tests, such as chi-square tests for independence, the null hypothesis assumes no association or independence between categorical variables.

Homogeneity Null Hypothesis

In tests like ANOVA (Analysis of Variance), the null hypothesis suggests that there’s no difference in population means across different groups.

• Medicine: Null Hypothesis: “No significant difference exists in blood pressure levels between patients given the experimental drug versus those given a placebo.”
• Education: Null Hypothesis: “There’s no significant variation in test scores between students using a new teaching method and those using traditional teaching.”
• Economics: Null Hypothesis: “There’s no significant change in consumer spending pre- and post-implementation of a new taxation policy.”
• Environmental Science: Null Hypothesis: “There’s no substantial difference in pollution levels before and after a water treatment plant’s establishment.”

The principle of the null hypothesis is a fundamental concept in statistical hypothesis testing. It involves making an assumption about the population parameter or the absence of an effect or relationship between variables.

In essence, the null hypothesis (H 0 ) proposes that there is no significant difference, effect, or relationship between variables. It serves as a starting point or a default assumption that there is no real change, no effect or no difference between groups or conditions.

The null hypothesis is usually formulated to be tested against an alternative hypothesis (H 1 or H [Tex]\alpha [/Tex] ) which suggests that there is an effect, difference or relationship present in the population.

Null Hypothesis Rejection

Rejecting the Null Hypothesis occurs when statistical evidence suggests a significant departure from the assumed baseline. It implies that there is enough evidence to support the alternative hypothesis, indicating a meaningful effect or difference. Null Hypothesis rejection occurs when statistical evidence suggests a deviation from the assumed baseline, prompting a reconsideration of the initial hypothesis.

Identifying the Null Hypothesis involves defining the status quotient, asserting no effect and formulating a statement suitable for statistical analysis.

When is Null Hypothesis Rejected?

The Null Hypothesis is rejected when statistical tests indicate a significant departure from the expected outcome, leading to the consideration of alternative hypotheses. It occurs when statistical evidence suggests a deviation from the assumed baseline, prompting a reconsideration of the initial hypothesis.

In statistical hypothesis testing, researchers begin by stating the null hypothesis, often based on theoretical considerations or previous research. The null hypothesis is then tested against an alternative hypothesis (Ha), which represents the researcher’s claim or the hypothesis they seek to support.

The process of hypothesis testing involves collecting sample data and using statistical methods to assess the likelihood of observing the data if the null hypothesis were true. This assessment is typically done by calculating a test statistic, which measures the difference between the observed data and what would be expected under the null hypothesis.

In the realm of hypothesis testing, the null hypothesis (H 0 ) and alternative hypothesis (H₁ or Ha) play critical roles. The null hypothesis generally assumes no difference, effect, or relationship between variables, suggesting that any observed change or effect is due to random chance. Its counterpart, the alternative hypothesis, asserts the presence of a significant difference, effect, or relationship between variables, challenging the null hypothesis. These hypotheses are formulated based on the research question and guide statistical analyses.

Difference Between Null Hypothesis and Alternative Hypothesis

The null hypothesis (H 0 ) serves as the baseline assumption in statistical testing, suggesting no significant effect, relationship, or difference within the data. It often proposes that any observed change or correlation is merely due to chance or random variation. Conversely, the alternative hypothesis (H 1 or Ha) contradicts the null hypothesis, positing the existence of a genuine effect, relationship or difference in the data. It represents the researcher’s intended focus, seeking to provide evidence against the null hypothesis and support for a specific outcome or theory. These hypotheses form the crux of hypothesis testing, guiding the assessment of data to draw conclusions about the population being studied.

Let’s envision a scenario where a researcher aims to examine the impact of a new medication on reducing blood pressure among patients. In this context:

Null Hypothesis (H 0 ): “The new medication does not produce a significant effect in reducing blood pressure levels among patients.”

Alternative Hypothesis (H 1 or Ha): “The new medication yields a significant effect in reducing blood pressure levels among patients.”

The null hypothesis implies that any observed alterations in blood pressure subsequent to the medication’s administration are a result of random fluctuations rather than a consequence of the medication itself. Conversely, the alternative hypothesis contends that the medication does indeed generate a meaningful alteration in blood pressure levels, distinct from what might naturally occur or by random chance.

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Example 1: A researcher claims that the average time students spend on homework is 2 hours per night.

Null Hypothesis (H 0 ): The average time students spend on homework is equal to 2 hours per night. Data: A random sample of 30 students has an average homework time of 1.8 hours with a standard deviation of 0.5 hours. Test Statistic and Decision: Using a t-test, if the calculated t-statistic falls within the acceptance region, we fail to reject the null hypothesis. If it falls in the rejection region, we reject the null hypothesis. Conclusion: Based on the statistical analysis, we fail to reject the null hypothesis, suggesting that there is not enough evidence to dispute the claim of the average homework time being 2 hours per night.

Example 2: A company asserts that the error rate in its production process is less than 1%.

Null Hypothesis (H 0 ): The error rate in the production process is 1% or higher. Data: A sample of 500 products shows an error rate of 0.8%. Test Statistic and Decision: Using a z-test, if the calculated z-statistic falls within the acceptance region, we fail to reject the null hypothesis. If it falls in the rejection region, we reject the null hypothesis. Conclusion: The statistical analysis supports rejecting the null hypothesis, indicating that there is enough evidence to dispute the company’s claim of an error rate of 1% or higher.

Q1. A researcher claims that the average time spent by students on homework is less than 2 hours per day. Formulate the null hypothesis for this claim?

Q2. A manufacturing company states that their new machine produces widgets with a defect rate of less than 5%. Write the null hypothesis to test this claim?

Q3. An educational institute believes that their online course completion rate is at least 60%. Develop the null hypothesis to validate this assertion?

Q4. A restaurant claims that the waiting time for customers during peak hours is not more than 15 minutes. Formulate the null hypothesis for this claim?

Q5. A study suggests that the mean weight loss after following a specific diet plan for a month is more than 8 pounds. Construct the null hypothesis to evaluate this statement?

Summary – Null Hypothesis and Alternative Hypothesis

The null hypothesis (H 0 ) and alternative hypothesis (H a ) are fundamental concepts in statistical hypothesis testing. The null hypothesis represents the default assumption, stating that there is no significant effect, difference, or relationship between variables. It serves as the baseline against which the alternative hypothesis is tested. In contrast, the alternative hypothesis represents the researcher’s hypothesis or the claim to be tested, suggesting that there is a significant effect, difference, or relationship between variables. The relationship between the null and alternative hypotheses is such that they are complementary, and statistical tests are conducted to determine whether the evidence from the data is strong enough to reject the null hypothesis in favor of the alternative hypothesis. This decision is based on the strength of the evidence and the chosen level of significance. Ultimately, the choice between the null and alternative hypotheses depends on the specific research question and the direction of the effect being investigated.

FAQs on Null Hypothesis

What does null hypothesis stands for.

The null hypothesis, denoted as H 0 ​, is a fundamental concept in statistics used for hypothesis testing. It represents the statement that there is no effect or no difference, and it is the hypothesis that the researcher typically aims to provide evidence against.

How to Form a Null Hypothesis?

A null hypothesis is formed based on the assumption that there is no significant difference or effect between the groups being compared or no association between variables being tested. It often involves stating that there is no relationship, no change, or no effect in the population being studied.

When Do we reject the Null Hypothesis?

In statistical hypothesis testing, if the p-value (the probability of obtaining the observed results) is lower than the chosen significance level (commonly 0.05), we reject the null hypothesis. This suggests that the data provides enough evidence to refute the assumption made in the null hypothesis.

What is a Null Hypothesis in Research?

In research, the null hypothesis represents the default assumption or position that there is no significant difference or effect. Researchers often try to test this hypothesis by collecting data and performing statistical analyses to see if the observed results contradict the assumption.

What Are Alternative and Null Hypotheses?

The null hypothesis (H0) is the default assumption that there is no significant difference or effect. The alternative hypothesis (H1 or Ha) is the opposite, suggesting there is a significant difference, effect or relationship.

What Does it Mean to Reject the Null Hypothesis?

Rejecting the null hypothesis implies that there is enough evidence in the data to support the alternative hypothesis. In simpler terms, it suggests that there might be a significant difference, effect or relationship between the groups or variables being studied.

How to Find Null Hypothesis?

Formulating a null hypothesis often involves considering the research question and assuming that no difference or effect exists. It should be a statement that can be tested through data collection and statistical analysis, typically stating no relationship or no change between variables or groups.

How is Null Hypothesis denoted?

The null hypothesis is commonly symbolized as H 0 in statistical notation.

What is the Purpose of the Null hypothesis in Statistical Analysis?

The null hypothesis serves as a starting point for hypothesis testing, enabling researchers to assess if there’s enough evidence to reject it in favor of an alternative hypothesis.

What happens if we Reject the Null hypothesis?

Rejecting the null hypothesis implies that there is sufficient evidence to support an alternative hypothesis, suggesting a significant effect or relationship between variables.

What are Test for Null Hypothesis?

Various statistical tests, such as t-tests or chi-square tests, are employed to evaluate the validity of the Null Hypothesis in different scenarios.

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• Null Hypothesis

Before understanding the Null Hypothesis in detail, let us first understand the definition of the Hypothesis.

In Statistics, a hypothesis can be defined as a formal statement, which gives an explanation about the relationship between any two or more variables of the specified population. 

Hypothesis helps the researcher to translate any given problem to a clear explanation for the outcome of the study.

 Hypothesis clearly explains and predicts the expected outcome and it indicates the types of experimental design and directs the study of any research process.

What Is The Null Hypothesis?

We can define a null hypothesis as a general statement or a default position that says there is no relationship between two measured phenomena or there is no association among groups. 

Why is Null Hypothesis Important?

Testing (which involves accepting, approving, rejecting, or disproving) the null hypothesis and thus concluding that there are or we can say that there are no grounds for believing that there is any relationship between two phenomena is basically a central task in the modern practice of science; in the field of statistics.

To be more specific, hypothesis testing gives precise criteria for rejecting or accepting a null hypothesis within a level known as the confidence level.

Null Hypothesis Symbol-

A Null Hypothesis is denoted by the symbol H 0  in statistics. It is usually pronounced as “h-nought” or “H-null”. The Subscript in H is the digit 0.

Null Hypothesis Principle and When is A Null Hypothesis Rejected?

The principle followed for null hypothesis testing is basically collecting the data and determining the chances of a given set of data during the study on any given random sample, assuming that the null hypothesis is true.

Suppose, if the given data does not face the expected null hypothesis, then the outcome we will get will be quite weaker and they conclude that by saying that the given set of data does not provide strong evidence against the null hypothesis which is because of insufficient evidence.

Finally, this leads to null hypothesis rejection.

Null Hypothesis Formula or How Do You Find The Null Hypothesis :

So far, understanding the concept of null hypothesis, let’s now discuss the null hypothesis formula:

Here, the null hypothesis formula is given below.

H 0 :  p = p 0

The formula for the alternative hypothesis can be written as: H a = p >p 0 , and < p 0 ≠ p 0

The formula for the test static is denoted by:

\[Z=\frac{P-P_{0}}{\sqrt{\frac{P-P_{0}}{n}}}\]

Remember that, p 0 here is the null hypothesis.

Types of Null Hypothesis.

There are 4 different types of Null hypothesis. Each of them is explained below with examples.

Simple Hypothesis: A simple Hypothesis is the one in which the relationship between two variables is predicted. One is an independent variable and the other is a dependent variable. A simple Hypothesis completely specifies the population distribution. An example of a simple Hypothesis is “Consuming sugar drinks daily leads to being overweight.”

Composite hypothesis. A Composite hypothesis describes the relationship between two or more independent variables and two or more dependent variables. In this hypothesis, the population distribution is not specified. An example of the Composite hypothesis is stated below. “ Individuals who consume sugar drinks daily and have a family history of health issues are more likely to become overweight and develop diabetes.”

Exact hypothesis. In this hypothesis, the exact value of the parameter or variable is defined. All the assumptions made during the derivation of the hypothesis are met in the exact hypothesis. An example of an exact hypothesis is, “ Students in a division score an average 17 out of 25 in exams. Hence, μ=17.”

Inexact hypothesis. Unlike the exact hypothesis, the exact value of the parameter or variable is not defined in the Inexact type of Hypothesis. Instead, a specific range or interval of the parameter is stated. For example, Students in class score an average between 12 to 15 out of 20 in exams. Hence, 12< μ< 15.

Null Hypothesis Problem Problem : A researcher thinks that if the knee surgery patients go to physical therapy twice a week (instead of going 3 times), their recovery period will be longer. Given average recovery time knee surgery patients are given 8.2 weeks. 

Step 1: First, we need to figure out the hypothesis from the problem . The hypothesis is usually hidden in a word problem that you need to figure out. The hypothesis that has been given in the above question is “I expect the average recovery period to be greater than 8.2 weeks.”

Step 2: You need to convert the hypothesis to math . Remember that the average can be sometimes written as μ.

H 1 : μ > 8.2(average)

Step 3: Now state what will happen if the hypothesis doesn’t come true. If the recovery time is not greater than the given average that is 8.2 weeks, there are only two possibilities, that the recovery time is equal to 8.2 weeks or it is less than 8.2 weeks.

H 0 : μ ≤ 8.2

H 0 (The null hypothesis): μ (the average) ≤ (is less than or equal to) 8.2

H 0 🡪 μ 1 = μ 2 where

H 0 is the null hypothesis,

μ 1 is the mean of population 1, and

μ 2 is the mean of population 2.

A stronger null hypothesis denotes that if two samples are drawn from the same given population, such that the variances and shapes of the given distributions are also equal.

Examples of situations where hypothesis tests are used in the real world.

Is there a 100% chance of getting affected by dengue?

Answer : There could be chances of getting affected by dengue but the chances of getting affected by dengue are not 100%.

Do teenagers these days use mobile phones more than adults to access the internet?

Answer : Age has no limit and is not a factor that we can use mobile phones to access the internet.

Does having the fruit apple daily will not cause fever?

Answer : Having an apple daily does not assure of not having fever, but on the other hand it does increase the immunity to fight against such diseases.

Are children better at performing mathematical calculations than adults?

Answer : Again in this example too, age has no effect on Mathematical skills.

Hypothesis tests are often conducted in botany to determine whether some new treatment, fertilizer or chemical can cause increased growth, stamina or immunity in plants and animals. 

Suppose, a botanist believes that a certain fertilizer will cause plants to grow more during a 1 month period than what they actually grow. Currently, they grow to 20 inches. To test the effectiveness of the fertilizer, the botanist applies the fertilizer to each of the plants in her lab for a month. Then performs a hypothesis test using the following hypothesis: H 0 : μ = 20 inches as the effect is nil on the mean plant growth by the fertilizer.

H A : μ > 20 inches (the fertilizer will cause mean plant growth to increase)

If the p-value of the test is significantly less then it will reject the null hypothesis and as a result, the fertilizer leads to increased plant growth.

In clinical trials, hypothesis tests are often used to determine whether some new treatment, drug, procedure, etc. cause improved outcomes in patients and thus elevate the treatment levels. Suppose a doctor wishes to test a new drug that reduces blood pressure in obese patients. To test its effectiveness, the doctor may measure the blood pressure of 40 patients before and after using the new drug for a month. Then a hypothesis test is performed using the following hypothesis. H 0 : μafter = μbefore  means the  mean blood pressure is the same before and after using the drug

H A : μafter < μbefore (the mean blood pressure is less after using the drug)

Hypothesis tests are often conducted in business organizations to determine whether a new advertising campaign will get more leads and cause an increase in sales. Suppose, the executive team of an organization believes that spending more on digital advertising leads to increased sales. To test this, the organization may increase money spent on digital advertising during a 3 month period and collect data to see if overall sales have increased or not. A hypothesis test can be performed using the following hypothesis.

H 0 : μafter = μbefore signifies  the  mean sales is the same before and after spending more on advertising

H A : μafter > μbefore (the mean sales increased spending more on advertising)

Apart from just the above 3 examples, hypothesis testing is widely used in research centers, chemical laboratories, stock market predictions, the Investment industry and several business organizations.

Decision errors.

There are two types of decision errors that can happen when doing a hypothesis test. They are described below.

Type I error: This happens when one rejects the null hypothesis despite the null hypothesis being true. The probability of making a type I error is equal to the alpha level significantly

Type II error. This happens when one fails to reject the null hypothesis when it is actually false. The probability of committing a type II error is the power of the test. 

Importance of Hypothesis testing in real life and examples

The main aim of a hypothesis statement in statistics is to determine whether or not some hypothesis of a population parameter is true or false. It is an essential part of the procedures in statistics. 

A hypothesis test evaluates two mutually exclusive statements and determines which statement is best supported by sample data. A discovery or a new finding holds a significant value when supported by a hypothesis test. Because the hypothesis test provides sufficient evidence to check the credibility of the given data.

To conduct a hypothesis test in the practical world, researchers obtain a random sample from the population and perform a hypothesis test on the sample data using Null and Alternative hypotheses.                          

Difference between Null Hypothesis and Alternative Hypothesis.

FAQs on Null Hypothesis

1. Why is a null hypothesis important and can you accept a null hypothesis?

The purpose and the importance of the null hypothesis and alternative hypothesis are that they provide an approximate description of any given phenomena. The purpose of the null hypothesis is to provide the researcher or an investigator with the relational statement that is directly tested in any research study. 

A null hypothesis is a hypothesis that is never acceptable. We either reject them or we fail to reject them. Hence, failing to reject the null hypothesis does not mean that we have shown that there is no difference in accepting the null hypothesis.

2. State one example where a Null hypothesis is applied in a practical life scenario?

The concept of null hypothesis has a variety of practical applications. Let us take the example of stock investment, buying shares of a company. The annual return of company stock when invested for a long duration is assumed to be 7.5%. 

Now to test if the assumption is true or false, we consider the null hypothesis to be, ”the mean annual return for XYZ company shares is not 7.5%”. To test this hypothesis, we first accept the null hypothesis.

We then check the history of the stock performance of the last 5 years of XYZ company and then calculate the mean of the annual return. The result is then compared to the assumed annual return of 7.5%. 

If it turns out that the average annual return of the last 5 years is 7.5%, then the null hypothesis is rejected. And the alternative hypothesis is accepted.

3. What is the purpose of Hypothesis testing?

Hypothesis testing is a statistical process of testing an assumption regarding a particular phenomenon or parameter. Often a good theory can make accurate predictions. However, for an analyst, hypothesis testing is a rigorous way of backing up his prediction with statistical analysis to support the predictions. It is also helpful to determine sufficient statistical evidence that can favor a certain hypothesis about the specified parameter.

In simple words, hypothesis testing is a systematic approach to assessing theories through observations and then determining whether the stated statement is true or false.

Stats without Tears Statistics Symbol Sheet

Updated 5 Nov 2020 ( What’s New? ) Copyright © 2002–2024 by Stan Brown, BrownMath.com

Here are symbols for various sample statistics and the corresponding population parameters. They are not repeated in the list below.

μ and σ can take subscripts to show what you are taking the mean or standard deviation of. For instance, σ x̅ (“sigma sub x-bar”) is the standard deviation of sample means, or standard error of the mean.

Roman Letters

• b = y intercept of a line. Defined here in Chapter 4. (Some statistics books use b 0 .)
• BD or BPD = binomial probability distribution. Defined here in Chapter 6.
• CI = confidence interval. Defined here in Chapter 9.
• CLT = Central Limit Theorem. Defined here in Chapter 8.
• d = difference between paired data. Defined here in Chapter 11.
• df or ν “nu” = degrees of freedom in a Student’s t or χ² distribution. Defined here in Chapter 9. Defined here in Chapter 12.
• DPD = discrete probability distribution. Defined here in Chapter 6.
• E = margin of error, a/k/a maximum error of the estimate. Defined here in Chapter 9.
• f = frequency. Defined here in Chapter 2.
• f / n = relative frequency. Defined here in Chapter 2.
• HT = hypothesis test. Defined here in Chapter 10.
• H o = null hypothesis. Defined here in Chapter 10.
• H 1 or H a = alternative hypothesis. Defined here in Chapter 10.
• IQR = interquartile range, Q 3 −Q 1 . Defined here in Chapter 3.
• m = slope of a line. Defined here in Chapter 4. (The TI-83 uses a and some statistics books use b 1 .)
• M or Med = median of a sample. Defined here in Chapter 3.
• n = sample size, number of data points. Defined here in Chapter 2. Also, number of trials in a probability experiment with a binomial model. Defined here in Chapter 6.
• N = population size.
• ND = normal distribution, whose graph is a bell-shaped curve; also “normally distributed”. Defined here in Chapter 7.

In geometric and binomial probability distributions, p is the probability of “success” ( defined here in Chapter 6) on any one trial and q  = (1− p ) is the probability of “failure” (the only other possibility) on any one trial.

In hypothesis testing, p is the calculated p-value ( defined here in Chapter 10), the probability that rejecting the null hypothesis would be a wrong decision.

In tests of population proportions, p stands for population proportion and p̂ for sample proportion (see table above).

• P( A ) = the probability of event A .
• P( A C ) or P(not  A ) = the probability that A does not happen. Defined here in Chapter 5.

Caution! The order of A and B may seem backward to you at first.

• P80 or P 80 = 80th percentile ( Pk or P k = k -th percentile) Defined here in Chapter 3.
• q = probability of failure on any one trial in binomial or geometric distribution, equal to (1− p ) where p is the probability of success on any one trial. Defined here in Chapter 6.
• Q1 or Q 1 = first quartile ( Q3 or Q 3 = third quartile) Defined here in Chapter 3.
• r = linear correlation coefficient of a sample. Defined here in Chapter 4.
• R ² = coefficient of determination. Defined here in Chapter 4.
• s = standard deviation of a sample. Defined here in Chapter 3.
• SD (or s.d.) = standard deviation. Defined here in Chapter 3.
• SEM = standard error of the mean (symbol is σ x̅ ). Defined here in Chapter 8.
• SEP = standard error of the proportion (symbol is σ p̂ ). Defined here in Chapter 8.
• X (capital X ) = a variable.
• x (lower-case x ) = one data value (“raw score”). As a column heading, x means a series of data values.
• x̅ “x-bar” = mean of a sample. Defined here in Chapter 3.
• x̃ “x-tilde” = median of a sample. Defined here in Chapter 3.
• ŷ “y-hat” = predicted average y value for a given x , found by using the regression equation. Defined here in Chapter 4.
• z = standard score or z-score. Defined here in Chapter 3.
• z ( area ) or z area = the z-score, such that that much of the area under the normal curve lies to the right of that z . This is not a multiplication! (See The z Function .)

Greek Letters

• α “alpha” = significance level in hypothesis test, or acceptable probability of a Type I error (probability you can live with). Defined here in Chapter 10. 1−α = confidence level.
• β “beta” = in a hypothesis test, the acceptable probability of a Type II error; 1−β is called the power of the test.
• μ mu, pronounced “mew” = mean of a population. Defined here in Chapter 3.
• ν nu: see df , above.
• ρ rho, pronounced “roe” = linear correlation coefficient of a population.
• σ “sigma” = standard deviation of a population. Defined here in Chapter 3.
• σ x̅ “sigma-sub-x-bar”; see SEM above.
• σ p̂ “sigma-sub-p-hat”; see SEP above.
• ∑ “sigma” = summation. (This is upper-case sigma. Lower-case sigma, σ, means standard deviation of a population; see the table near the start of this page .) See ∑ Means Add ’em Up in Chapter 1.
• χ² “chi-squared” = distribution for multinomial experiments and contingency tables. Defined here in Chapter 12.

What’s New?

• 5 Nov 2020 : Convert document to HTML5, and italicize the variables.
• 14 Feb 2018 : Add x̃ for the median, as suggested by reader “Trone”.
• (intervening changes suppressed)
• 27 Sept 2002 : New article.

Updates and new info: https://BrownMath.com/swt/

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