null hypothesis symbol u

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 :


equal (=)	not equal (≠) greater than (>) less than (<)
greater than or equal to (≥)	less than (<)
less than or equal to (≤)	more than (>)

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

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

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

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

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

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

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

H 0 : μ = 2.0

H a : μ ≠ 2.0

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

H 0 : μ = 66
H a : μ ≠ 66

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

H 0 : μ ≥ 5

H a : μ < 5

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

H 0 : μ ≥ 45
H a : μ < 45

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

H 0 : p ≤ 0.066

H a : p > 0.066

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

H 0 : p = 0.40
H a : p > 0.40

Concept Review

In a hypothesis test , sample data is evaluated in order to arrive at a decision about some type of claim. If certain conditions about the sample are satisfied, then the claim can be evaluated for a population. In a hypothesis test, we: Evaluate the null hypothesis , typically denoted with H 0 . The null is not rejected unless the hypothesis test shows otherwise. The null statement must always contain some form of equality (=, ≤ or ≥) Always write the alternative hypothesis , typically denoted with H a or H 1 , using less than, greater than, or not equals symbols, i.e., (≠, >, or <). If we reject the null hypothesis, then we can assume there is enough evidence to support the alternative hypothesis. Never state that a claim is proven true or false. Keep in mind the underlying fact that hypothesis testing is based on probability laws; therefore, we can talk only in terms of non-absolute certainties.

Formula Review

H 0 and H a are contradictory.

OpenStax, Statistics, Null and Alternative Hypotheses. Provided by : OpenStax. Located at : http://cnx.org/contents/[email protected]:58/Introductory_Statistics . License : CC BY: Attribution
Introductory Statistics . Authored by : Barbara Illowski, Susan Dean. Provided by : Open Stax. Located at : http://cnx.org/contents/[email protected] . License : CC BY: Attribution . License Terms : Download for free at http://cnx.org/contents/[email protected]
Simple hypothesis testing | Probability and Statistics | Khan Academy. Authored by : Khan Academy. Located at : https://youtu.be/5D1gV37bKXY . License : All Rights Reserved . License Terms : Standard YouTube License

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Statistics By Jim

Making statistics intuitive

Null Hypothesis: Definition, Rejecting & Examples

By Jim Frost 6 Comments

What is a Null Hypothesis?

The null hypothesis in statistics states that there is no difference between groups or no relationship between variables. It is one of two mutually exclusive hypotheses about a population in a hypothesis test.

Photograph of Rodin's statue, The Thinker who is pondering the null hypothesis.

Null Hypothesis H 0 : No effect exists in the population.
Alternative Hypothesis H A : The effect exists in the population.

In every study or experiment, researchers assess an effect or relationship. This effect can be the effectiveness of a new drug, building material, or other intervention that has benefits. There is a benefit or connection that the researchers hope to identify. Unfortunately, no effect may exist. In statistics, we call this lack of an effect the null hypothesis. Researchers assume that this notion of no effect is correct until they have enough evidence to suggest otherwise, similar to how a trial presumes innocence.

In this context, the analysts don’t necessarily believe the null hypothesis is correct. In fact, they typically want to reject it because that leads to more exciting finds about an effect or relationship. The new vaccine works!

You can think of it as the default theory that requires sufficiently strong evidence to reject. Like a prosecutor, researchers must collect sufficient evidence to overturn the presumption of no effect. Investigators must work hard to set up a study and a data collection system to obtain evidence that can reject the null hypothesis.

Related post : What is an Effect in Statistics?

Null Hypothesis Examples

Null hypotheses start as research questions that the investigator rephrases as a statement indicating there is no effect or relationship.


Does the vaccine prevent infections?	The vaccine does not affect the infection rate.
Does the new additive increase product strength?	The additive does not affect mean product strength.
Does the exercise intervention increase bone mineral density?	The intervention does not affect bone mineral density.
As screen time increases, does test performance decrease?	There is no relationship between screen time and test performance.

After reading these examples, you might think they’re a bit boring and pointless. However, the key is to remember that the null hypothesis defines the condition that the researchers need to discredit before suggesting an effect exists.

Let’s see how you reject the null hypothesis and get to those more exciting findings!

When to Reject the Null Hypothesis

So, you want to reject the null hypothesis, but how and when can you do that? To start, you’ll need to perform a statistical test on your data. The following is an overview of performing a study that uses a hypothesis test.

The first step is to devise a research question and the appropriate null hypothesis. After that, the investigators need to formulate an experimental design and data collection procedures that will allow them to gather data that can answer the research question. Then they collect the data. For more information about designing a scientific study that uses statistics, read my post 5 Steps for Conducting Studies with Statistics .

After data collection is complete, statistics and hypothesis testing enter the picture. Hypothesis testing takes your sample data and evaluates how consistent they are with the null hypothesis. The p-value is a crucial part of the statistical results because it quantifies how strongly the sample data contradict the null hypothesis.

When the sample data provide sufficient evidence, you can reject the null hypothesis. In a hypothesis test, this process involves comparing the p-value to your significance level .

Rejecting the Null Hypothesis

Reject the null hypothesis when the p-value is less than or equal to your significance level. Your sample data favor the alternative hypothesis, which suggests that the effect exists in the population. For a mnemonic device, remember—when the p-value is low, the null must go!

When you can reject the null hypothesis, your results are statistically significant. Learn more about Statistical Significance: Definition & Meaning .

Failing to Reject the Null Hypothesis

Conversely, when the p-value is greater than your significance level, you fail to reject the null hypothesis. The sample data provides insufficient data to conclude that the effect exists in the population. When the p-value is high, the null must fly!

Note that failing to reject the null is not the same as proving it. For more information about the difference, read my post about Failing to Reject the Null .

That’s a very general look at the process. But I hope you can see how the path to more exciting findings depends on being able to rule out the less exciting null hypothesis that states there’s nothing to see here!

Let’s move on to learning how to write the null hypothesis for different types of effects, relationships, and tests.

Related posts : How Hypothesis Tests Work and Interpreting P-values

How to Write a Null Hypothesis

The null hypothesis varies by the type of statistic and hypothesis test. Remember that inferential statistics use samples to draw conclusions about populations. Consequently, when you write a null hypothesis, it must make a claim about the relevant population parameter . Further, that claim usually indicates that the effect does not exist in the population. Below are typical examples of writing a null hypothesis for various parameters and hypothesis tests.

Related posts : Descriptive vs. Inferential Statistics and Populations, Parameters, and Samples in Inferential Statistics

Group Means

T-tests and ANOVA assess the differences between group means. For these tests, the null hypothesis states that there is no difference between group means in the population. In other words, the experimental conditions that define the groups do not affect the mean outcome. Mu (µ) is the population parameter for the mean, and you’ll need to include it in the statement for this type of study.

For example, an experiment compares the mean bone density changes for a new osteoporosis medication. The control group does not receive the medicine, while the treatment group does. The null states that the mean bone density changes for the control and treatment groups are equal.

Null Hypothesis H 0 : Group means are equal in the population: µ 1 = µ 2 , or µ 1 – µ 2 = 0
Alternative Hypothesis H A : Group means are not equal in the population: µ 1 ≠ µ 2 , or µ 1 – µ 2 ≠ 0.

Group Proportions

Proportions tests assess the differences between group proportions. For these tests, the null hypothesis states that there is no difference between group proportions. Again, the experimental conditions did not affect the proportion of events in the groups. P is the population proportion parameter that you’ll need to include.

For example, a vaccine experiment compares the infection rate in the treatment group to the control group. The treatment group receives the vaccine, while the control group does not. The null states that the infection rates for the control and treatment groups are equal.

Null Hypothesis H 0 : Group proportions are equal in the population: p 1 = p 2 .
Alternative Hypothesis H A : Group proportions are not equal in the population: p 1 ≠ p 2 .

Correlation and Regression Coefficients

Some studies assess the relationship between two continuous variables rather than differences between groups.

In these studies, analysts often use either correlation or regression analysis . For these tests, the null states that there is no relationship between the variables. Specifically, it says that the correlation or regression coefficient is zero. As one variable increases, there is no tendency for the other variable to increase or decrease. Rho (ρ) is the population correlation parameter and beta (β) is the regression coefficient parameter.

For example, a study assesses the relationship between screen time and test performance. The null states that there is no correlation between this pair of variables. As screen time increases, test performance does not tend to increase or decrease.

Null Hypothesis H 0 : The correlation in the population is zero: ρ = 0.
Alternative Hypothesis H A : The correlation in the population is not zero: ρ ≠ 0.

For all these cases, the analysts define the hypotheses before the study. After collecting the data, they perform a hypothesis test to determine whether they can reject the null hypothesis.

The preceding examples are all for two-tailed hypothesis tests. To learn about one-tailed tests and how to write a null hypothesis for them, read my post One-Tailed vs. Two-Tailed Tests .

Related post : Understanding Correlation

Neyman, J; Pearson, E. S. (January 1, 1933). On the Problem of the most Efficient Tests of Statistical Hypotheses . Philosophical Transactions of the Royal Society A . 231 (694–706): 289–337.

Reader Interactions

January 11, 2024 at 2:57 pm

Thanks for the reply.

January 10, 2024 at 1:23 pm

Hi Jim, In your comment you state that equivalence test null and alternate hypotheses are reversed. For hypothesis tests of data fits to a probability distribution, the null hypothesis is that the probability distribution fits the data. Is this correct?

January 10, 2024 at 2:15 pm

Those two separate things, equivalence testing and normality tests. But, yes, you’re correct for both.

Hypotheses are switched for equivalence testing. You need to “work” (i.e., collect a large sample of good quality data) to be able to reject the null that the groups are different to be able to conclude they’re the same.

With typical hypothesis tests, if you have low quality data and a low sample size, you’ll fail to reject the null that they’re the same, concluding they’re equivalent. But that’s more a statement about the low quality and small sample size than anything to do with the groups being equal.

So, equivalence testing make you work to obtain a finding that the groups are the same (at least within some amount you define as a trivial difference).

For normality testing, and other distribution tests, the null states that the data follow the distribution (normal or whatever). If you reject the null, you have sufficient evidence to conclude that your sample data don’t follow the probability distribution. That’s a rare case where you hope to fail to reject the null. And it suffers from the problem I describe above where you might fail to reject the null simply because you have a small sample size. In that case, you’d conclude the data follow the probability distribution but it’s more that you don’t have enough data for the test to register the deviation. In this scenario, if you had a larger sample size, you’d reject the null and conclude it doesn’t follow that distribution.

I don’t know of any equivalence testing type approach for distribution fit tests where you’d need to work to show the data follow a distribution, although I haven’t looked for one either!

February 20, 2022 at 9:26 pm

Is a null hypothesis regularly (always) stated in the negative? “there is no” or “does not”

February 23, 2022 at 9:21 pm

Typically, the null hypothesis includes an equal sign. The null hypothesis states that the population parameter equals a particular value. That value is usually one that represents no effect. In the case of a one-sided hypothesis test, the null still contains an equal sign but it’s “greater than or equal to” or “less than or equal to.” If you wanted to translate the null hypothesis from its native mathematical expression, you could use the expression “there is no effect.” But the mathematical form more specifically states what it’s testing.

It’s the alternative hypothesis that typically contains does not equal.

There are some exceptions. For example, in an equivalence test where the researchers want to show that two things are equal, the null hypothesis states that they’re not equal.

In short, the null hypothesis states the condition that the researchers hope to reject. They need to work hard to set up an experiment and data collection that’ll gather enough evidence to be able to reject the null condition.

February 15, 2022 at 9:32 am

Dear sir I always read your notes on Research methods.. Kindly tell is there any available Book on all these..wonderfull Urgent

Comments and Questions Cancel reply

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.


1	The null hypothesis is a statement. There exists no relation between two variables	Alternative hypothesis a statement, there exists some relationship between two measured phenomenon
2	Denoted by H	Denoted by H
3	The observations of this hypothesis are the result of chance	The observations of this hypothesis are the result of real effect
4	The mathematical formulation of the null hypothesis is an equal sign	The mathematical formulation alternative hypothesis is an inequality sign such as greater than, less than, etc.

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.

Put your understanding of this concept to test by answering a few MCQs. Click ‘Start Quiz’ to begin!

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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

Want to create or adapt books like this? Learn more about how Pressbooks supports open publishing practices.

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 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 .

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 :


equal (=)	not equal (≠) greater than (>) less than (<)
greater than or equal to (≥)	less than (<)
less than or equal to (≤)	more than (>)

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

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 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 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

Formula Review

H 0 and H a are contradictory.

has:	equal (=)	greater than or equal to (≥)	less than or equal to (≤)
has:	not equal (≠) greater than (>) less than (<)	less than (<)	greater than (>)

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 Hypothesis

Null hypothesis is used to make decisions based on data and by using statistical tests. Null hypothesis is represented using H o and it states that there is no difference between the characteristics of two samples. Null hypothesis is generally a statement of no difference. The rejection of null hypothesis is equivalent to the acceptance of the alternate hypothesis.

Let us learn more about null hypotheses, tests for null hypotheses, the difference between null hypothesis and alternate hypothesis, with the help of examples, FAQs.

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What Is Null Hypothesis?

Null hypothesis states that there is no significant difference between the observed characteristics across two sample sets. Null hypothesis states the observed population parameters or variables is the same across the samples. The null hypothesis states that there is no relationship between the sample parameters, the independent variable, and the dependent variable. The term null hypothesis is used in instances to mean that there is no differences in the two means, or that the difference is not so significant.

If the experimental outcome is the same as the theoretical outcome then the null hypothesis holds good. But if there are any differences in the observed parameters across the samples then the null hypothesis is rejected, and we consider an alternate hypothesis. The rejection of the null hypothesis does not mean that there were flaws in the basic experimentation, but it sets the stage for further research. Generally, the strength of the evidence is tested against the null hypothesis.

Null hypothesis and alternate hypothesis are the two approaches used across statistics. The alternate hypothesis states that there is a significant difference between the parameters across the samples. The alternate hypothesis is the inverse of null hypothesis. An important reason to reject the null hypothesis and consider the alternate hypothesis is due to experimental or sampling errors.

Tests For Null Hypothesis

The two important approaches of statistical interference of null hypothesis are significance testing and hypothesis testing. The null hypothesis is a theoretical hypothesis and is based on insufficient evidence, which requires further testing to prove if it is true or false.

Significance Testing

The aim of significance testing is to provide evidence to reject the null hypothesis. If the difference is strong enough then reject the null hypothesis and accept the alternate hypothesis. The testing is designed to test the strength of the evidence against the hypothesis. The four important steps of significance testing are as follows.

First state the null and alternate hypotheses.
Calculate the test statistics.
Find the p-value.
Test the p-value with the α and decide if the null hypothesis should be rejected or accepted.

If the p-value is lesser than the significance level α, then the null hypothesis is rejected. And if the p-value is greater than the significance level α, then the null hypothesis is accepted.

Hypothesis Testing

Hypothesis testing takes the parameters from the sample and makes a derivation about the population. A hypothesis is an educated guess about a sample, which can be tested either through an experiment or an observation. Initially, a tentative assumption is made about the sample in the form of a null hypothesis.

There are four steps to perform hypothesis testing. They are:

Identify the null hypothesis.
Define the null hypothesis statement.
Choose the test to be performed.
Accept the null hypothesis or the alternate hypothesis.

There are often errors in the process of testing the hypothesis. The two important errors observed in hypothesis testing is as follows.

Type - I error is rejecting the null hypothesis when the null hypothesis is actually true.
Type - II error is accepting the null hypothesis when the null hypothesis is actually false.

Difference Between Null Hypothesis And Alternate Hypothesis

The difference between null hypothesis and alternate hypothesis can be understood through the following points.

The opposite of the null hypothesis is the alternate hypothesis and it is the claim which is being proved by research to be true.
The null hypothesis states that the two samples of the population are the same, and the alternate hypothesis states that there is a significant difference between the two samples of the population.
The null hypothesis is designated as H o and the alternate hypothesis is designated as H a .
For the null hypothesis, the same means are assumed to be equal, and we have H 0 : µ 1 = µ 2. And for the alternate hypothesis, the sample means are unequal, and we have H a : µ 1 ≠ µ 2.
The observed population parameters and variables are the same across the samples, for a null hypothesis, but in an alternate hypothesis, there is a significant difference between the observed parameters and variables across the samples.

☛ Related Topics

The following topics help in a better understanding of the null hypothesis.

Probability and Statistics
Basic Statistics Formula
Sample Space

Examples on Null Hypothesis

Example 1: A medical experiment and trial is conducted to check if a particular drug can serve as the vaccine for Covid-19, and can prevent from occurrence of Corona. Write the null hypothesis and the alternate hypothesis for this situation.

The given situation refers to a possible new drug and its effectiveness of being a vaccine for Covid-19 or not. The null hypothesis (H o ) and alternate hypothesis (H a ) for this medical experiment is as follows.

H 0 : The use of the new drug is not helpful for the prevention of Covid-19.
H a : The use of the new drug serves as a vaccine and helps for the prevention of Covid-19.

Example 2: The teacher has prepared a set of important questions and informs the student that preparing these questions helps in scoring more than 60% marks in the board exams. Write the null hypothesis and the alternate hypothesis for this situation.

The given situation refers to the teacher who has claimed that her important questions helps to score more than 60% marks in the board exams. The null hypothesis(H o ) and alternate hypothesis(H a ) for this situation is as follows.

H o : The important questions given by the teacher does not really help the students to get a score of more than 60% in the board exams.
H a : The important questions given by the teacher is helpful for the students to score more than 60% marks in the board exams.

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Practice Questions on Null Hypothesis

Faqs on null hypothesis, what is null hypothesis in maths.

Null hypothesis is used in statistics and it states if there is any significant difference between the two samples. The acceptance of null hypothesis mean that there is no significant difference between the two samples. And the rejection of null hypothesis means that the two samples are different, and we need to accept the alternate hypothesis. The null hypothesis statement is represented as H 0 and the alternate hypothesis is represented as H a .

How Do You Test Null Hypothesis?

The null hypothesis is broadly tested using two methods. The null hypothesis can be tested using significance testing and hypothesis testing.Broadly the test for null hypothesis is performed across four stages. First the null hypothesis is identified, secondly the null hypothesis is defined. Next a suitable test is used to test the hypothesis, and finally either the null hypothesis or the alternate hypothesis is accepted.

How To Accept or Reject Null Hypothesis?

The null hypothesis is accepted or rejected based on the result of the hypothesis testing. The p value is found and the significance level is defined. If the p-value is lesser than the significance level α, then the null hypothesis is rejected. And if the p-value is greater than the significance level α, then the null hypothesis is accepted.

What Is the Difference Between Null Hypothesis And Alternate Hypothesis?

The null hypothesis states that there is no significant difference between the two samples, and the alternate hypothesis states that there is a significant difference between the two samples. The null hypothesis is referred using H o and the alternate hypothesis is referred using H a . As per null hypothesis the observed variables and parameters are the same across the samples, but as per alternate hypothesis there is a significant difference between the observed variables and parameters across the samples.

What Is Null Hypothesis Example?

A few quick examples of null hypothesis are as follows.

The salary of a person is independent of his profession, is an example of null hypothesis. And the salary is dependent on the profession of a person, is an alternate hypothesis.
The performance of the students in Maths from two different classes is a null hypothesis. And the performance of the students from each of the classes is different, is an example of alternate hypothesis.
The nutrient content of mango and a mango milk shake is equal and it can be taken as a null hypothesis. The test to prove the different nutrient content of the two is referred to as alternate hypothesis.

Null Hypothesis and Alternative Hypothesis

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

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

The Null Hypothesis

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

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

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

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

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

The Alternative Hypothesis

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

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

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

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

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

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

Featured Posts

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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Null Hypothesis

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.


Criteria	Null Hypothesis	Alternative Hypothesis
Definition	Assumes no effect or difference	Asserts a specific effect or difference
Symbol	H	H (or Ha)
Formulation	States equality or absence of parameter	States a specific value or relationship
Testing Outcome	Rejected if evidence of a significant effect	Accepted if evidence supports the hypothesis

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.

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.

Parameter	Null Hypothesis	Alternative Hypothesis
Observation	Concludes that the results are observed as a result of chance.	Concludes that the results are observed due to some real causes.
Symbol	Denoted by H0	Denoted by HA
Definition	States that two factors or groups are unrelated and there is no difference between certain characteristics of a population.	States that there is a relation between the two variables or groups under consideration.
Nature	Researchers usually try to disprove the null hypothesis.	Researchers try to prove an alternative hypothesis.
Acceptance	If the obtained p-value is greater than the level of significance, then the null hypothesis is accepted.	If the obtained p-value is smaller than the level of significance, then an alternative hypothesis is accepted.

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.

Frequently asked questions

What symbols are used to represent null hypotheses.

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 ≤).

Frequently asked questions: Statistics

As the degrees of freedom increase, Student’s t distribution becomes less leptokurtic , meaning that the probability of extreme values decreases. The distribution becomes more and more similar to a standard normal distribution .

The three categories of kurtosis are:

Mesokurtosis : An excess kurtosis of 0. Normal distributions are mesokurtic.
Platykurtosis : A negative excess kurtosis. Platykurtic distributions are thin-tailed, meaning that they have few outliers .
Leptokurtosis : A positive excess kurtosis. Leptokurtic distributions are fat-tailed, meaning that they have many outliers.

Probability distributions belong to two broad categories: discrete probability distributions and continuous probability distributions . Within each category, there are many types of probability distributions.

Probability is the relative frequency over an infinite number of trials.

For example, the probability of a coin landing on heads is .5, meaning that if you flip the coin an infinite number of times, it will land on heads half the time.

Since doing something an infinite number of times is impossible, relative frequency is often used as an estimate of probability. If you flip a coin 1000 times and get 507 heads, the relative frequency, .507, is a good estimate of the probability.

Categorical variables can be described by a frequency distribution. Quantitative variables can also be described by a frequency distribution, but first they need to be grouped into interval classes .

A histogram is an effective way to tell if a frequency distribution appears to have a normal distribution .

Plot a histogram and look at the shape of the bars. If the bars roughly follow a symmetrical bell or hill shape, like the example below, then the distribution is approximately normally distributed.

Frequency-distribution-Normal-distribution

You can use the CHISQ.INV.RT() function to find a chi-square critical value in Excel.

For example, to calculate the chi-square critical value for a test with df = 22 and α = .05, click any blank cell and type:

=CHISQ.INV.RT(0.05,22)

You can use the qchisq() function to find a chi-square critical value in R.

For example, to calculate the chi-square critical value for a test with df = 22 and α = .05:

qchisq(p = .05, df = 22, lower.tail = FALSE)

You can use the chisq.test() function to perform a chi-square test of independence in R. Give the contingency table as a matrix for the “x” argument. For example:

m = matrix(data = c(89, 84, 86, 9, 8, 24), nrow = 3, ncol = 2)

chisq.test(x = m)

You can use the CHISQ.TEST() function to perform a chi-square test of independence in Excel. It takes two arguments, CHISQ.TEST(observed_range, expected_range), and returns the p value.

Chi-square goodness of fit tests are often used in genetics. One common application is to check if two genes are linked (i.e., if the assortment is independent). When genes are linked, the allele inherited for one gene affects the allele inherited for another gene.

Suppose that you want to know if the genes for pea texture (R = round, r = wrinkled) and color (Y = yellow, y = green) are linked. You perform a dihybrid cross between two heterozygous ( RY / ry ) pea plants. The hypotheses you’re testing with your experiment are:

This would suggest that the genes are unlinked.
This would suggest that the genes are linked.

You observe 100 peas:

78 round and yellow peas
6 round and green peas
4 wrinkled and yellow peas
12 wrinkled and green peas

Step 1: Calculate the expected frequencies

To calculate the expected values, you can make a Punnett square. If the two genes are unlinked, the probability of each genotypic combination is equal.


RRYY	RrYy	RRYy	RrYY
RrYy	rryy	Rryy	rrYy
RRYy	Rryy	RRyy	RrYy
RrYY	rrYy	RrYy	rrYY

The expected phenotypic ratios are therefore 9 round and yellow: 3 round and green: 3 wrinkled and yellow: 1 wrinkled and green.

From this, you can calculate the expected phenotypic frequencies for 100 peas:


Round and yellow	78	100 * (9/16) = 56.25
Round and green	6	100 * (3/16) = 18.75
Wrinkled and yellow	4	100 * (3/16) = 18.75
Wrinkled and green	12	100 * (1/16) = 6.21

Step 2: Calculate chi-square

				−	−
Round and yellow	78	56.25	21.75	473.06	8.41
Round and green	6	18.75	−12.75	162.56	8.67
Wrinkled and yellow	4	18.75	−14.75	217.56	11.6
Wrinkled and green	12	6.21	5.79	33.52	5.4

Χ 2 = 8.41 + 8.67 + 11.6 + 5.4 = 34.08

Step 3: Find the critical chi-square value

Since there are four groups (round and yellow, round and green, wrinkled and yellow, wrinkled and green), there are three degrees of freedom .

For a test of significance at α = .05 and df = 3, the Χ 2 critical value is 7.82.

Step 4: Compare the chi-square value to the critical value

Χ 2 = 34.08

Critical value = 7.82

The Χ 2 value is greater than the critical value .

Step 5: Decide whether the reject the null hypothesis

The Χ 2 value is greater than the critical value, so we reject the null hypothesis that the population of offspring have an equal probability of inheriting all possible genotypic combinations. There is a significant difference between the observed and expected genotypic frequencies ( p < .05).

The data supports the alternative hypothesis that the offspring do not have an equal probability of inheriting all possible genotypic combinations, which suggests that the genes are linked

You can use the chisq.test() function to perform a chi-square goodness of fit test in R. Give the observed values in the “x” argument, give the expected values in the “p” argument, and set “rescale.p” to true. For example:

chisq.test(x = c(22,30,23), p = c(25,25,25), rescale.p = TRUE)

You can use the CHISQ.TEST() function to perform a chi-square goodness of fit test in Excel. It takes two arguments, CHISQ.TEST(observed_range, expected_range), and returns the p value .

Both correlations and chi-square tests can test for relationships between two variables. However, a correlation is used when you have two quantitative variables and a chi-square test of independence is used when you have two categorical variables.

Both chi-square tests and t tests can test for differences between two groups. However, a t test is used when you have a dependent quantitative variable and an independent categorical variable (with two groups). A chi-square test of independence is used when you have two categorical variables.

The two main chi-square tests are the chi-square goodness of fit test and the chi-square test of independence .

A chi-square distribution is a continuous probability distribution . The shape of a chi-square distribution depends on its degrees of freedom , k . The mean of a chi-square distribution is equal to its degrees of freedom ( k ) and the variance is 2 k . The range is 0 to ∞.

As the degrees of freedom ( k ) increases, the chi-square distribution goes from a downward curve to a hump shape. As the degrees of freedom increases further, the hump goes from being strongly right-skewed to being approximately normal.

To find the quartiles of a probability distribution, you can use the distribution’s quantile function.

You can use the quantile() function to find quartiles in R. If your data is called “data”, then “quantile(data, prob=c(.25,.5,.75), type=1)” will return the three quartiles.

You can use the QUARTILE() function to find quartiles in Excel. If your data is in column A, then click any blank cell and type “=QUARTILE(A:A,1)” for the first quartile, “=QUARTILE(A:A,2)” for the second quartile, and “=QUARTILE(A:A,3)” for the third quartile.

You can use the PEARSON() function to calculate the Pearson correlation coefficient in Excel. If your variables are in columns A and B, then click any blank cell and type “PEARSON(A:A,B:B)”.

There is no function to directly test the significance of the correlation.

You can use the cor() function to calculate the Pearson correlation coefficient in R. To test the significance of the correlation, you can use the cor.test() function.

You should use the Pearson correlation coefficient when (1) the relationship is linear and (2) both variables are quantitative and (3) normally distributed and (4) have no outliers.

The Pearson correlation coefficient ( r ) is the most common way of measuring a linear correlation. It is a number between –1 and 1 that measures the strength and direction of the relationship between two variables.

This table summarizes the most important differences between normal distributions and Poisson distributions :

Characteristic	Normal	Poisson
	Continuous
	Mean (µ) and standard deviation (σ)	Lambda (λ)
Shape	Bell-shaped	Depends on λ
	Symmetrical	Asymmetrical (right-skewed). As λ increases, the asymmetry decreases.
Range	−∞ to ∞	0 to ∞

When the mean of a Poisson distribution is large (>10), it can be approximated by a normal distribution.

In the Poisson distribution formula, lambda (λ) is the mean number of events within a given interval of time or space. For example, λ = 0.748 floods per year.

The e in the Poisson distribution formula stands for the number 2.718. This number is called Euler’s constant. You can simply substitute e with 2.718 when you’re calculating a Poisson probability. Euler’s constant is a very useful number and is especially important in calculus.

The three types of skewness are:

Right skew (also called positive skew ) . A right-skewed distribution is longer on the right side of its peak than on its left.
Left skew (also called negative skew). A left-skewed distribution is longer on the left side of its peak than on its right.
Zero skew. It is symmetrical and its left and right sides are mirror images.

Skewness and kurtosis are both important measures of a distribution’s shape.

Skewness measures the asymmetry of a distribution.
Kurtosis measures the heaviness of a distribution’s tails relative to a normal distribution .

Difference between skewness and kurtosis

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.

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 >).

The t distribution was first described by statistician William Sealy Gosset under the pseudonym “Student.”

To calculate a confidence interval of a mean using the critical value of t , follow these four steps:

Choose the significance level based on your desired confidence level. The most common confidence level is 95%, which corresponds to α = .05 in the two-tailed t table .
Find the critical value of t in the two-tailed t table.
Multiply the critical value of t by s / √ n .
Add this value to the mean to calculate the upper limit of the confidence interval, and subtract this value from the mean to calculate the lower limit.

To test a hypothesis using the critical value of t , follow these four steps:

Calculate the t value for your sample.
Find the critical value of t in the t table .
Determine if the (absolute) t value is greater than the critical value of t .
Reject the null hypothesis if the sample’s t value is greater than the critical value of t . Otherwise, don’t reject the null hypothesis .

You can use the T.INV() function to find the critical value of t for one-tailed tests in Excel, and you can use the T.INV.2T() function for two-tailed tests.

You can use the qt() function to find the critical value of t in R. The function gives the critical value of t for the one-tailed test. If you want the critical value of t for a two-tailed test, divide the significance level by two.

You can use the RSQ() function to calculate R² in Excel. If your dependent variable is in column A and your independent variable is in column B, then click any blank cell and type “RSQ(A:A,B:B)”.

You can use the summary() function to view the R² of a linear model in R. You will see the “R-squared” near the bottom of the output.

There are two formulas you can use to calculate the coefficient of determination (R²) of a simple linear regression .

The coefficient of determination (R²) is a number between 0 and 1 that measures how well a statistical model predicts an outcome. You can interpret the R² as the proportion of variation in the dependent variable that is predicted by the statistical model.

There are three main types of missing data .

Missing completely at random (MCAR) data are randomly distributed across the variable and unrelated to other variables .

Missing at random (MAR) data are not randomly distributed but they are accounted for by other observed variables.

Missing not at random (MNAR) data systematically differ from the observed values.

To tidy up your missing data , your options usually include accepting, removing, or recreating the missing data.

Acceptance: You leave your data as is
Listwise or pairwise deletion: You delete all cases (participants) with missing data from analyses
Imputation: You use other data to fill in the missing data

Missing data are important because, depending on the type, they can sometimes bias your results. This means your results may not be generalizable outside of your study because your data come from an unrepresentative sample .

Missing data , or missing values, occur when you don’t have data stored for certain variables or participants.

In any dataset, there’s usually some missing data. In quantitative research , missing values appear as blank cells in your spreadsheet.

There are two steps to calculating the geometric mean :

Multiply all values together to get their product.
Find the n th root of the product ( n is the number of values).

Before calculating the geometric mean, note that:

The geometric mean can only be found for positive values.
If any value in the data set is zero, the geometric mean is zero.

The arithmetic mean is the most commonly used type of mean and is often referred to simply as “the mean.” While the arithmetic mean is based on adding and dividing values, the geometric mean multiplies and finds the root of values.

Even though the geometric mean is a less common measure of central tendency , it’s more accurate than the arithmetic mean for percentage change and positively skewed data. The geometric mean is often reported for financial indices and population growth rates.

The geometric mean is an average that multiplies all values and finds a root of the number. For a dataset with n numbers, you find the n th root of their product.

Outliers are extreme values that differ from most values in the dataset. You find outliers at the extreme ends of your dataset.

It’s best to remove outliers only when you have a sound reason for doing so.

Some outliers represent natural variations in the population , and they should be left as is in your dataset. These are called true outliers.

Other outliers are problematic and should be removed because they represent measurement errors , data entry or processing errors, or poor sampling.

You can choose from four main ways to detect outliers :

Sorting your values from low to high and checking minimum and maximum values
Visualizing your data with a box plot and looking for outliers
Using the interquartile range to create fences for your data
Using statistical procedures to identify extreme values

Outliers can have a big impact on your statistical analyses and skew the results of any hypothesis test if they are inaccurate.

These extreme values can impact your statistical power as well, making it hard to detect a true effect if there is one.

No, the steepness or slope of the line isn’t related to the correlation coefficient value. The correlation coefficient only tells you how closely your data fit on a line, so two datasets with the same correlation coefficient can have very different slopes.

To find the slope of the line, you’ll need to perform a regression analysis .

Correlation coefficients always range between -1 and 1.

The sign of the coefficient tells you the direction of the relationship: a positive value means the variables change together in the same direction, while a negative value means they change together in opposite directions.

The absolute value of a number is equal to the number without its sign. The absolute value of a correlation coefficient tells you the magnitude of the correlation: the greater the absolute value, the stronger the correlation.

These are the assumptions your data must meet if you want to use Pearson’s r :

Both variables are on an interval or ratio level of measurement
Data from both variables follow normal distributions
Your data have no outliers
Your data is from a random or representative sample
You expect a linear relationship between the two variables

A correlation coefficient is a single number that describes the strength and direction of the relationship between your variables.

Different types of correlation coefficients might be appropriate for your data based on their levels of measurement and distributions . The Pearson product-moment correlation coefficient (Pearson’s r ) is commonly used to assess a linear relationship between two quantitative variables.

There are various ways to improve power:

Increase the potential effect size by manipulating your independent variable more strongly,
Increase sample size,
Increase the significance level (alpha),
Reduce measurement error by increasing the precision and accuracy of your measurement devices and procedures,
Use a one-tailed test instead of a two-tailed test for t tests and z tests.

A power analysis is a calculation that helps you determine a minimum sample size for your study. It’s made up of four main components. If you know or have estimates for any three of these, you can calculate the fourth component.

Statistical power : the likelihood that a test will detect an effect of a certain size if there is one, usually set at 80% or higher.
Sample size : the minimum number of observations needed to observe an effect of a certain size with a given power level.
Significance level (alpha) : the maximum risk of rejecting a true null hypothesis that you are willing to take, usually set at 5%.
Expected effect size : a standardized way of expressing the magnitude of the expected result of your study, usually based on similar studies or a pilot study.

Null and alternative hypotheses are used in statistical hypothesis testing . The null hypothesis of a test always predicts no effect or no relationship between variables, while the alternative hypothesis states your research prediction of an effect or relationship.

Statistical analysis is the main method for analyzing quantitative research data . It uses probabilities and models to test predictions about a population from sample data.

The risk of making a Type II error is inversely related to the statistical power of a test. Power is the extent to which a test can correctly detect a real effect when there is one.

To (indirectly) reduce the risk of a Type II error, you can increase the sample size or the significance level to increase statistical power.

The risk of making a Type I error is the significance level (or alpha) that you choose. That’s a value that you set at the beginning of your study to assess the statistical probability of obtaining your results ( p value ).

The significance level is usually set at 0.05 or 5%. This means that your results only have a 5% chance of occurring, or less, if the null hypothesis is actually true.

To reduce the Type I error probability, you can set a lower significance level.

In statistics, a Type I error means rejecting the null hypothesis when it’s actually true, while a Type II error means failing to reject the null hypothesis when it’s actually false.

In statistics, power refers to the likelihood of a hypothesis test detecting a true effect if there is one. A statistically powerful test is more likely to reject a false negative (a Type II error).

If you don’t ensure enough power in your study, you may not be able to detect a statistically significant result even when it has practical significance. Your study might not have the ability to answer your research question.

While statistical significance shows that an effect exists in a study, practical significance shows that the effect is large enough to be meaningful in the real world.

Statistical significance is denoted by p -values whereas practical significance is represented by effect sizes .

There are dozens of measures of effect sizes . The most common effect sizes are Cohen’s d and Pearson’s r . Cohen’s d measures the size of the difference between two groups while Pearson’s r measures the strength of the relationship between two variables .

Effect size tells you how meaningful the relationship between variables or the difference between groups is.

A large effect size means that a research finding has practical significance, while a small effect size indicates limited practical applications.

Using descriptive and inferential statistics , you can make two types of estimates about the population : point estimates and interval estimates.

A point estimate is a single value estimate of a parameter . For instance, a sample mean is a point estimate of a population mean.
An interval estimate gives you a range of values where the parameter is expected to lie. A confidence interval is the most common type of interval estimate.

Both types of estimates are important for gathering a clear idea of where a parameter is likely to lie.

Standard error and standard deviation are both measures of variability . The standard deviation reflects variability within a sample, while the standard error estimates the variability across samples of a population.

The standard error of the mean , or simply standard error , indicates how different the population mean is likely to be from a sample mean. It tells you how much the sample mean would vary if you were to repeat a study using new samples from within a single population.

To figure out whether a given number is a parameter or a statistic , ask yourself the following:

Does the number describe a whole, complete population where every member can be reached for data collection ?
Is it possible to collect data for this number from every member of the population in a reasonable time frame?

If the answer is yes to both questions, the number is likely to be a parameter. For small populations, data can be collected from the whole population and summarized in parameters.

If the answer is no to either of the questions, then the number is more likely to be a statistic.

The arithmetic mean is the most commonly used mean. It’s often simply called the mean or the average. But there are some other types of means you can calculate depending on your research purposes:

Weighted mean: some values contribute more to the mean than others.
Geometric mean : values are multiplied rather than summed up.
Harmonic mean: reciprocals of values are used instead of the values themselves.

You can find the mean , or average, of a data set in two simple steps:

Find the sum of the values by adding them all up.
Divide the sum by the number of values in the data set.

This method is the same whether you are dealing with sample or population data or positive or negative numbers.

The median is the most informative measure of central tendency for skewed distributions or distributions with outliers. For example, the median is often used as a measure of central tendency for income distributions, which are generally highly skewed.

Because the median only uses one or two values, it’s unaffected by extreme outliers or non-symmetric distributions of scores. In contrast, the mean and mode can vary in skewed distributions.

To find the median , first order your data. Then calculate the middle position based on n , the number of values in your data set.

$\dfrac{(n+1)}{2}$

A data set can often have no mode, one mode or more than one mode – it all depends on how many different values repeat most frequently.

Your data can be:

without any mode
unimodal, with one mode,
bimodal, with two modes,
trimodal, with three modes, or
multimodal, with four or more modes.

To find the mode :

If your data is numerical or quantitative, order the values from low to high.
If it is categorical, sort the values by group, in any order.

Then you simply need to identify the most frequently occurring value.

The interquartile range is the best measure of variability for skewed distributions or data sets with outliers. Because it’s based on values that come from the middle half of the distribution, it’s unlikely to be influenced by outliers .

The two most common methods for calculating interquartile range are the exclusive and inclusive methods.

The exclusive method excludes the median when identifying Q1 and Q3, while the inclusive method includes the median as a value in the data set in identifying the quartiles.

For each of these methods, you’ll need different procedures for finding the median, Q1 and Q3 depending on whether your sample size is even- or odd-numbered. The exclusive method works best for even-numbered sample sizes, while the inclusive method is often used with odd-numbered sample sizes.

While the range gives you the spread of the whole data set, the interquartile range gives you the spread of the middle half of a data set.

Homoscedasticity, or homogeneity of variances, is an assumption of equal or similar variances in different groups being compared.

This is an important assumption of parametric statistical tests because they are sensitive to any dissimilarities. Uneven variances in samples result in biased and skewed test results.

Statistical tests such as variance tests or the analysis of variance (ANOVA) use sample variance to assess group differences of populations. They use the variances of the samples to assess whether the populations they come from significantly differ from each other.

Variance is the average squared deviations from the mean, while standard deviation is the square root of this number. Both measures reflect variability in a distribution, but their units differ:

Standard deviation is expressed in the same units as the original values (e.g., minutes or meters).
Variance is expressed in much larger units (e.g., meters squared).

Although the units of variance are harder to intuitively understand, variance is important in statistical tests .

The empirical rule, or the 68-95-99.7 rule, tells you where most of the values lie in a normal distribution :

Around 68% of values are within 1 standard deviation of the mean.
Around 95% of values are within 2 standard deviations of the mean.
Around 99.7% of values are within 3 standard deviations of the mean.

The empirical rule is a quick way to get an overview of your data and check for any outliers or extreme values that don’t follow this pattern.

In a normal distribution , data are symmetrically distributed with no skew. Most values cluster around a central region, with values tapering off as they go further away from the center.

The measures of central tendency (mean, mode, and median) are exactly the same in a normal distribution.

The standard deviation is the average amount of variability in your data set. It tells you, on average, how far each score lies from the mean .

In normal distributions, a high standard deviation means that values are generally far from the mean, while a low standard deviation indicates that values are clustered close to the mean.

No. Because the range formula subtracts the lowest number from the highest number, the range is always zero or a positive number.

In statistics, the range is the spread of your data from the lowest to the highest value in the distribution. It is the simplest measure of variability .

While central tendency tells you where most of your data points lie, variability summarizes how far apart your points from each other.

Data sets can have the same central tendency but different levels of variability or vice versa . Together, they give you a complete picture of your data.

Variability is most commonly measured with the following descriptive statistics :

Range : the difference between the highest and lowest values
Interquartile range : the range of the middle half of a distribution
Standard deviation : average distance from the mean
Variance : average of squared distances from the mean

Variability tells you how far apart points lie from each other and from the center of a distribution or a data set.

Variability is also referred to as spread, scatter or dispersion.

While interval and ratio data can both be categorized, ranked, and have equal spacing between adjacent values, only ratio scales have a true zero.

For example, temperature in Celsius or Fahrenheit is at an interval scale because zero is not the lowest possible temperature. In the Kelvin scale, a ratio scale, zero represents a total lack of thermal energy.

A critical value is the value of the test statistic which defines the upper and lower bounds of a confidence interval , or which defines the threshold of statistical significance in a statistical test. It describes how far from the mean of the distribution you have to go to cover a certain amount of the total variation in the data (i.e. 90%, 95%, 99%).

If you are constructing a 95% confidence interval and are using a threshold of statistical significance of p = 0.05, then your critical value will be identical in both cases.

The t -distribution gives more probability to observations in the tails of the distribution than the standard normal distribution (a.k.a. the z -distribution).

In this way, the t -distribution is more conservative than the standard normal distribution: to reach the same level of confidence or statistical significance , you will need to include a wider range of the data.

A t -score (a.k.a. a t -value) is equivalent to the number of standard deviations away from the mean of the t -distribution .

The t -score is the test statistic used in t -tests and regression tests. It can also be used to describe how far from the mean an observation is when the data follow a t -distribution.

The t -distribution is a way of describing a set of observations where most observations fall close to the mean , and the rest of the observations make up the tails on either side. It is a type of normal distribution used for smaller sample sizes, where the variance in the data is unknown.

The t -distribution forms a bell curve when plotted on a graph. It can be described mathematically using the mean and the standard deviation .

In statistics, ordinal and nominal variables are both considered categorical variables .

Even though ordinal data can sometimes be numerical, not all mathematical operations can be performed on them.

Ordinal data has two characteristics:

The data can be classified into different categories within a variable.
The categories have a natural ranked order.

However, unlike with interval data, the distances between the categories are uneven or unknown.

Nominal and ordinal are two of the four levels of measurement . Nominal level data can only be classified, while ordinal level data can be classified and ordered.

Nominal data is data that can be labelled or classified into mutually exclusive categories within a variable. These categories cannot be ordered in a meaningful way.

For example, for the nominal variable of preferred mode of transportation, you may have the categories of car, bus, train, tram or bicycle.

If your confidence interval for a difference between groups includes zero, that means that if you run your experiment again you have a good chance of finding no difference between groups.

If your confidence interval for a correlation or regression includes zero, that means that if you run your experiment again there is a good chance of finding no correlation in your data.

In both of these cases, you will also find a high p -value when you run your statistical test, meaning that your results could have occurred under the null hypothesis of no relationship between variables or no difference between groups.

If you want to calculate a confidence interval around the mean of data that is not normally distributed , you have two choices:

Find a distribution that matches the shape of your data and use that distribution to calculate the confidence interval.
Perform a transformation on your data to make it fit a normal distribution, and then find the confidence interval for the transformed data.

The standard normal distribution , also called the z -distribution, is a special normal distribution where the mean is 0 and the standard deviation is 1.

Any normal distribution can be converted into the standard normal distribution by turning the individual values into z -scores. In a z -distribution, z -scores tell you how many standard deviations away from the mean each value lies.

The z -score and t -score (aka z -value and t -value) show how many standard deviations away from the mean of the distribution you are, assuming your data follow a z -distribution or a t -distribution .

These scores are used in statistical tests to show how far from the mean of the predicted distribution your statistical estimate is. If your test produces a z -score of 2.5, this means that your estimate is 2.5 standard deviations from the predicted mean.

The predicted mean and distribution of your estimate are generated by the null hypothesis of the statistical test you are using. The more standard deviations away from the predicted mean your estimate is, the less likely it is that the estimate could have occurred under the null hypothesis .

To calculate the confidence interval , you need to know:

The point estimate you are constructing the confidence interval for
The critical values for the test statistic
The standard deviation of the sample
The sample size

Then you can plug these components into the confidence interval formula that corresponds to your data. The formula depends on the type of estimate (e.g. a mean or a proportion) and on the distribution of your data.

The confidence level is the percentage of times you expect to get close to the same estimate if you run your experiment again or resample the population in the same way.

The confidence interval consists of the upper and lower bounds of the estimate you expect to find at a given level of confidence.

For example, if you are estimating a 95% confidence interval around the mean proportion of female babies born every year based on a random sample of babies, you might find an upper bound of 0.56 and a lower bound of 0.48. These are the upper and lower bounds of the confidence interval. The confidence level is 95%.

The mean is the most frequently used measure of central tendency because it uses all values in the data set to give you an average.

For data from skewed distributions, the median is better than the mean because it isn’t influenced by extremely large values.

The mode is the only measure you can use for nominal or categorical data that can’t be ordered.

The measures of central tendency you can use depends on the level of measurement of your data.

For a nominal level, you can only use the mode to find the most frequent value.
For an ordinal level or ranked data, you can also use the median to find the value in the middle of your data set.
For interval or ratio levels, in addition to the mode and median, you can use the mean to find the average value.

Measures of central tendency help you find the middle, or the average, of a data set.

The 3 most common measures of central tendency are the mean, median and mode.

The mode is the most frequent value.
The median is the middle number in an ordered data set.
The mean is the sum of all values divided by the total number of values.

Some variables have fixed levels. For example, gender and ethnicity are always nominal level data because they cannot be ranked.

However, for other variables, you can choose the level of measurement . For example, income is a variable that can be recorded on an ordinal or a ratio scale:

At an ordinal level , you could create 5 income groupings and code the incomes that fall within them from 1–5.
At a ratio level , you would record exact numbers for income.

If you have a choice, the ratio level is always preferable because you can analyze data in more ways. The higher the level of measurement, the more precise your data is.

The level at which you measure a variable determines how you can analyze your data.

Depending on the level of measurement , you can perform different descriptive statistics to get an overall summary of your data and inferential statistics to see if your results support or refute your hypothesis .

Levels of measurement tell you how precisely variables are recorded. There are 4 levels of measurement, which can be ranked from low to high:

Nominal : the data can only be categorized.
Ordinal : the data can be categorized and ranked.
Interval : the data can be categorized and ranked, and evenly spaced.
Ratio : the data can be categorized, ranked, evenly spaced and has a natural zero.

No. The p -value only tells you how likely the data you have observed is to have occurred under the null hypothesis .

If the p -value is below your threshold of significance (typically p < 0.05), then you can reject the null hypothesis, but this does not necessarily mean that your alternative hypothesis is true.

The alpha value, or the threshold for statistical significance , is arbitrary – which value you use depends on your field of study.

In most cases, researchers use an alpha of 0.05, which means that there is a less than 5% chance that the data being tested could have occurred under the null hypothesis.

P -values are usually automatically calculated by the program you use to perform your statistical test. They can also be estimated using p -value tables for the relevant test statistic .

P -values are calculated from the null distribution of the test statistic. They tell you how often a test statistic is expected to occur under the null hypothesis of the statistical test, based on where it falls in the null distribution.

If the test statistic is far from the mean of the null distribution, then the p -value will be small, showing that the test statistic is not likely to have occurred under the null hypothesis.

A p -value , or probability value, is a number describing how likely it is that your data would have occurred under the null hypothesis of your statistical test .

The test statistic you use will be determined by the statistical test.

You can choose the right statistical test by looking at what type of data you have collected and what type of relationship you want to test.

The test statistic will change based on the number of observations in your data, how variable your observations are, and how strong the underlying patterns in the data are.

For example, if one data set has higher variability while another has lower variability, the first data set will produce a test statistic closer to the null hypothesis , even if the true correlation between two variables is the same in either data set.

The formula for the test statistic depends on the statistical test being used.

Generally, the test statistic is calculated as the pattern in your data (i.e. the correlation between variables or difference between groups) divided by the variance in the data (i.e. the standard deviation ).

Univariate statistics summarize only one variable at a time.
Bivariate statistics compare two variables .
Multivariate statistics compare more than two variables .

The 3 main types of descriptive statistics concern the frequency distribution, central tendency, and variability of a dataset.

Distribution refers to the frequencies of different responses.
Measures of central tendency give you the average for each response.
Measures of variability show you the spread or dispersion of your dataset.

Descriptive statistics summarize the characteristics of a data set. Inferential statistics allow you to test a hypothesis or assess whether your data is generalizable to the broader population.

In statistics, model selection is a process researchers use to compare the relative value of different statistical models and determine which one is the best fit for the observed data.

The Akaike information criterion is one of the most common methods of model selection. AIC weights the ability of the model to predict the observed data against the number of parameters the model requires to reach that level of precision.

AIC model selection can help researchers find a model that explains the observed variation in their data while avoiding overfitting.

In statistics, a model is the collection of one or more independent variables and their predicted interactions that researchers use to try to explain variation in their dependent variable.

You can test a model using a statistical test . To compare how well different models fit your data, you can use Akaike’s information criterion for model selection.

The Akaike information criterion is calculated from the maximum log-likelihood of the model and the number of parameters (K) used to reach that likelihood. The AIC function is 2K – 2(log-likelihood) .

Lower AIC values indicate a better-fit model, and a model with a delta-AIC (the difference between the two AIC values being compared) of more than -2 is considered significantly better than the model it is being compared to.

The Akaike information criterion is a mathematical test used to evaluate how well a model fits the data it is meant to describe. It penalizes models which use more independent variables (parameters) as a way to avoid over-fitting.

AIC is most often used to compare the relative goodness-of-fit among different models under consideration and to then choose the model that best fits the data.

A factorial ANOVA is any ANOVA that uses more than one categorical independent variable . A two-way ANOVA is a type of factorial ANOVA.

Some examples of factorial ANOVAs include:

Testing the combined effects of vaccination (vaccinated or not vaccinated) and health status (healthy or pre-existing condition) on the rate of flu infection in a population.
Testing the effects of marital status (married, single, divorced, widowed), job status (employed, self-employed, unemployed, retired), and family history (no family history, some family history) on the incidence of depression in a population.
Testing the effects of feed type (type A, B, or C) and barn crowding (not crowded, somewhat crowded, very crowded) on the final weight of chickens in a commercial farming operation.

In ANOVA, the null hypothesis is that there is no difference among group means. If any group differs significantly from the overall group mean, then the ANOVA will report a statistically significant result.

Significant differences among group means are calculated using the F statistic, which is the ratio of the mean sum of squares (the variance explained by the independent variable) to the mean square error (the variance left over).

If the F statistic is higher than the critical value (the value of F that corresponds with your alpha value, usually 0.05), then the difference among groups is deemed statistically significant.

The only difference between one-way and two-way ANOVA is the number of independent variables . A one-way ANOVA has one independent variable, while a two-way ANOVA has two.

One-way ANOVA : Testing the relationship between shoe brand (Nike, Adidas, Saucony, Hoka) and race finish times in a marathon.
Two-way ANOVA : Testing the relationship between shoe brand (Nike, Adidas, Saucony, Hoka), runner age group (junior, senior, master’s), and race finishing times in a marathon.

All ANOVAs are designed to test for differences among three or more groups. If you are only testing for a difference between two groups, use a t-test instead.

Multiple linear regression is a regression model that estimates the relationship between a quantitative dependent variable and two or more independent variables using a straight line.

Linear regression most often uses mean-square error (MSE) to calculate the error of the model. MSE is calculated by:

measuring the distance of the observed y-values from the predicted y-values at each value of x;
squaring each of these distances;
calculating the mean of each of the squared distances.

Linear regression fits a line to the data by finding the regression coefficient that results in the smallest MSE.

Simple linear regression is a regression model that estimates the relationship between one independent variable and one dependent variable using a straight line. Both variables should be quantitative.

For example, the relationship between temperature and the expansion of mercury in a thermometer can be modeled using a straight line: as temperature increases, the mercury expands. This linear relationship is so certain that we can use mercury thermometers to measure temperature.

A regression model is a statistical model that estimates the relationship between one dependent variable and one or more independent variables using a line (or a plane in the case of two or more independent variables).

A regression model can be used when the dependent variable is quantitative, except in the case of logistic regression, where the dependent variable is binary.

A t-test should not be used to measure differences among more than two groups, because the error structure for a t-test will underestimate the actual error when many groups are being compared.

If you want to compare the means of several groups at once, it’s best to use another statistical test such as ANOVA or a post-hoc test.

A one-sample t-test is used to compare a single population to a standard value (for example, to determine whether the average lifespan of a specific town is different from the country average).

A paired t-test is used to compare a single population before and after some experimental intervention or at two different points in time (for example, measuring student performance on a test before and after being taught the material).

A t-test measures the difference in group means divided by the pooled standard error of the two group means.

In this way, it calculates a number (the t-value) illustrating the magnitude of the difference between the two group means being compared, and estimates the likelihood that this difference exists purely by chance (p-value).

Your choice of t-test depends on whether you are studying one group or two groups, and whether you care about the direction of the difference in group means.

If you are studying one group, use a paired t-test to compare the group mean over time or after an intervention, or use a one-sample t-test to compare the group mean to a standard value. If you are studying two groups, use a two-sample t-test .

If you want to know only whether a difference exists, use a two-tailed test . If you want to know if one group mean is greater or less than the other, use a left-tailed or right-tailed one-tailed test .

A t-test is a statistical test that compares the means of two samples . It is used in hypothesis testing , with a null hypothesis that the difference in group means is zero and an alternate hypothesis that the difference in group means is different from zero.

Statistical significance is a term used by researchers to state that it is unlikely their observations could have occurred under the null hypothesis of a statistical test . Significance is usually denoted by a p -value , or probability value.

Statistical significance is arbitrary – it depends on the threshold, or alpha value, chosen by the researcher. The most common threshold is p < 0.05, which means that the data is likely to occur less than 5% of the time under the null hypothesis .

When the p -value falls below the chosen alpha value, then we say the result of the test is statistically significant.

A test statistic is a number calculated by a statistical test . It describes how far your observed data is from the null hypothesis of no relationship between variables or no difference among sample groups.

The test statistic tells you how different two or more groups are from the overall population mean , or how different a linear slope is from the slope predicted by a null hypothesis . Different test statistics are used in different statistical tests.

Statistical tests commonly assume that:

the data are normally distributed
the groups that are being compared have similar variance
the data are independent

If your data does not meet these assumptions you might still be able to use a nonparametric statistical test , which have fewer requirements but also make weaker inferences.

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9.1 Null and Alternative Hypotheses

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

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

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

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

Mathematical Symbols Used in H 0 and H a :


equal (=)	not equal (≠) greater than (>) less than (<)
greater than or equal to (≥)	less than (<)
less than or equal to (≤)	more than (>)

Example 9.1

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

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

Example 9.2

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

H 0 : μ __ 66
H a : μ __ 66

Example 9.3

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

H 0 : μ __ 45
H a : μ __ 45

Example 9.4

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

H 0 : p __ 0.40
H a : p __ 0.40

Collaborative Exercise

This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission.

Want to cite, share, or modify this book? This book uses the Creative Commons Attribution License and you must attribute OpenStax.

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

Authors: Barbara Illowsky, Susan Dean
Publisher/website: OpenStax
Book title: Introductory Statistics 2e
Publication date: Dec 13, 2023
Location: Houston, Texas
Book URL: https://openstax.org/books/introductory-statistics-2e/pages/1-introduction
Section URL: https://openstax.org/books/introductory-statistics-2e/pages/9-1-null-and-alternative-hypotheses

© Dec 6, 2023 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.

9.1 Null and Alternative Hypotheses

Introduction.

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 sample means or proportions or no difference between a sample mean or proportion and a population mean or proportion. In other words, the difference equals 0.

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 .

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 :


equal (=)	not equal (≠) greater than (>) less than (<)
greater than or equal to (≥)	less than (<)
less than or equal to (≤)	more than (>)

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 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 percent 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 percent. 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 the following:

H 0 : μ __ 66
H a : μ __ 66

Example 9.3

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

H 0 : μ __ 45
H a : μ __ 45

Example 9.4

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

On a state driver’s test, about 40 percent pass the test on the first try. We want to test if more than 40 percent 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.

Copy and paste the link code above.

How to Insert the Null Hypothesis & Alternate Hypothesis Symbols in Microsoft Word

Although the symbols for the null hypothesis and alternative hypothesis -- sometimes called the alternate hypothesis -- do not exist as special characters in Microsoft Word, they are easily created with subscripts.The alternate hypothesis is symbolically represented by a capitalized "H," followed by a subscript "1," although some researchers prefer an "a." The null hypothesis is represented by a capitalized "H," followed by a subscript "0" or "o." The accepted practice in the scientific community is to use two hypotheses when testing the relationship between two events. The alternative hypothesis states that the two events are related. However, scientists have found that testing for a direct correlation can cause bias in the testing procedure. To avoid this bias, scientists test a null hypothesis that states there is no correlation. By disproving the null hypothesis, you imply a correlation in the alternate hypothesis. A similar system is used in the United States legal system where a defendant is found "not guilty," rather than being found "innocent."

Open your document in Microsoft Word and click wherever you want the hypothesis symbols to appear.

Video of the Day

Type a capital "H" on your keyboard.

Click the subscript button, located in the "Font" group of the "Home" tab. This button's icon looks like an "x" with a subscript "2." Alternatively, hold the "Ctrl" key and press "=".

Type a "0" to create a null hypothesis symbol or "1" to create an alternative hypothesis symbol. Alternatively, type an "o" or "a" to represent the null and alternative hypotheses, respectively, although these symbols are not as frequently used.

Press the subscript button again to exit this formatting mode.

University of New England: Null and Alternative Hypothesis

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JSmol Viewer

Applying the laplace transform procedure, testing exponentiality against the nbru mgf class.

1. Introduction

2. moment-generating function class uses new better than renewal, 3. testing exponentiality versus nbru mgf class for complete data, 3.1. test statistic empirical for nbru mgf, 3.2. pitman’s asymptotic efficiency, 4. critical points of the monte carlo null distribution, 5. the estimated power, 6. applications for complete data.

0.315	0.496	0.616	1.135	1.208	1.263	1.414	2.025	2.036
2.162	2.211	2.370	2.532	2.693	2.805	2.910	2.912	3.192
3.263	3.348	3.348	3.427	3.499	3.534	3.767	3.751	3.858
3.986	4.049	4.244	4.323	4.381	4.392	4.397	4.647	4.753
4.929	4.973	5.074	4.381

1.80	2.89	2.93	3.03	3.15	3.43	3.48	3.57
3.85	3.92	3.98	4.06	4.11	4.13	4.16	4.23
4.34	4.37	4.53	4.62	4.65	4.84	4.91	4.99
5.17

0.205	0.363	0.407	0.770	0.720	0.782	1.178	1.255	1.592
1.635	2.310

0.019	0.129	0.159	0.203	0.485	0.636	0.748	0.781
0.869	1.175	1.206	1.219	1.219	1.282	1.356	1.362
1.458	1.564	1.586	1.592	1.781	1.923	1.959	2.134
2.413	2.466	2.548	2.652	2.951	3.038	3.600	3.655
3.745	4.203	4.690	4.888	5.143	5.167	5.603	5.633
6.192	6.655	6.874

7. Conclusions

Author contributions, data availability statement, acknowledgments, conflicts of interest.

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Click here to enlarge figure

Distribution	β	s = 0.58	s = 0.59	s = 0.6	s = 0.7	s = 0.8	s = 0.9
LFR	1	1.7817	1.4408	1.2897	1.0789	1.2268	1.6276
	2	1.1115	1.1730	1.2425	2.5188	6.6169	30.933
	3	0.9666	1.0273	1.0947	2.3120	6.2829	30.698
	4	0.8793	0.9374	1.0017	2.1595	5.9807	29.925
	5	0.8156	0.8712	0.9326	2.0386	5.7253	29.189
Makeham	1	4.3495	3.4915	3.1033	2.4676	2.6535	3.3851
	2	1.2162	1.2416	1.2718	1.7986	2.9998	6.8103
	3	1.0070	1.0344	1.0652	1.5593	2.6799	6.3496
	4	0.9055	0.9330	0.9632	1.4374	2.5160	6.1036
	5	0.8364	0.8634	0.8929	1.3508	2.3971	5.9251
Weibull	1	1.5097	1.2191	1.0887	0.8530	0.7948	0.6836
	2	0.1815	0.2075	0.2356	0.6892	1.8969	6.8722
	3	0.1242	0.1479	0.1734	0.5945	1.7671	6.7673
	4	0.1028	0.1247	0.1484	0.5441	1.6603	6.5830
	5	0.0902	0.1107	0.1329	0.5081	1.5838	6.4152

Test	LFR	Makeham	Weibull
Abu-Yousef et al. (2022) [ ]	12.8986	0.7462	3.532
Abu-Yousef et al. (2022) [ ]	1.3000	0.5800	---------
Hassan et al. (2021) [ ]	8.30159	2.90418	2.2092
Abu-Yousef et al. (2020) [ ]	5.665	0.8765	2.1679
Mahmoud et al. (2019) [ ]	0.9150	0.1720	0.6180
El. Arishy et al. (2019) [ ]	0.95658	0.20395	0.73343
	30.9330	6.8103	6.8722

PARE	LFR	Makeham	Weibull
e ( [ ])	2.398	9.127	3.532
e ( [ ])	23.795	11.742	-------
e ( [ ])	3.726	2.345	3.111
e ( [ ])	5.460	7.770	3.170
e ( [ ])	33.807	39.595	11.120
e ( [ ])	32.337	33.392	9.370

n	90%	95%	99%
5	6.832471	9.111409	20.881602
10	3.767876	5.492639	16.612772
11	3.487769	5.054081	15.462071
15	2.767289	4.163040	12.836489
20	2.164377	3.087190	9.591994
25	1.841200	2.661309	8.859419
30	1.576625	2.301766	7.199750
35	1.366794	1.959073	5.227894
40	1.220647	1.811924	6.043173
43	1.127370	1.700669	5.076861
46	1.057042	1.515008	5.439105
50	0.997464	1.479134	4.638035

n	90%	95%	99%
5	1.425526	1.654881	2.512848
10	0.688724	0.800892	1.272633
11	0.622419	0.708224	1.075039
15	0.463981	0.550187	1.034439
20	0.352117	0.415252	0.704349
25	0.282936	0.337393	0.573153
30	0.235479	0.276930	0.457279
35	0.204216	0.244069	0.409616
40	0.178988	0.213249	0.340378
43	0.165415	0.195653	0.328344
46	0.154761	0.181204	0.276341
50	0.145253	0.169742	0.279583

n	θ	LFR	Makeham	Weibull
10	1	0.1027	0.3316	0.4630
	2	0.4202	0.9797	0.6214
	3	0.6839	1.000	0.7641
	4	0.8395	1.000	0.9673
	5	0.9728	1.000	0.9754
20	1	0.1056	0.3687	0.5571
	2	0.4590	0.9985	0.6431
	3	0.7679	1.000	0.7883
	4	0.8893	1.000	0.9780
	5	0.9750	1.000	0.9832
30	1	0.1077	0.4772	0.5597
	2	0.5616	0.9992	0.6782
	3	0.8066	1.000	0.8542
	4	0.8905	1.000	0.9812
	5	0.9834	1.000	0.9931

n	θ	LFR	Makeham	Weibull
10	1	0.1965	0.2621	0.5335
	2	0.4099	0.9977	0.7452
	3	0.6220	1.000	0.8682
	4	0.7775	1.000	0.9786
	5	0.9902	1.000	0.9860
20	1	0.3105	0.4932	0.6900
	2	0.7065	1.000	0.7743
	3	0.9212	1.000	0.9123
	4	0.9827	1.000	0.9892
	5	0.9970	1.000	0.9976
30	1	04193	0.6597	0.7342
	2	0.8653	1.000	0.8547
	3	0.9870	1.000	0.9678
	4	0.9996	1.000	0.9901
	5	1.000	1.000	0.9987

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Hassan, N.A.; Said, M.M.; Attwa, R.A.E.-W.; Radwan, T. Applying the Laplace Transform Procedure, Testing Exponentiality against the NBRU mgf Class. Mathematics 2024 , 12 , 2045. https://doi.org/10.3390/math12132045

Hassan NA, Said MM, Attwa RAE-W, Radwan T. Applying the Laplace Transform Procedure, Testing Exponentiality against the NBRU mgf Class. Mathematics . 2024; 12(13):2045. https://doi.org/10.3390/math12132045

Hassan, Naglaa A., Mayar M. Said, Rasha Abd El-Wahab Attwa, and Taha Radwan. 2024. "Applying the Laplace Transform Procedure, Testing Exponentiality against the NBRU mgf Class" Mathematics 12, no. 13: 2045. https://doi.org/10.3390/math12132045

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Null And Alternative Hypothesis
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Null hypothesis vs alternative hypothesis

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Null & Alternative Hypotheses
The null and alternative hypotheses offer competing answers to your research question. When the research question asks "Does the independent variable affect the dependent variable?": The null hypothesis ( H0) answers "No, there's no effect in the population.". The alternative hypothesis ( Ha) answers "Yes, there is an effect in the ...
Null hypothesis
In scientific research, the null hypothesis (often denoted H 0) ... The statement that is being tested against the null hypothesis is the alternative hypothesis. Symbols may include H 1 and H a. A statistical significance test starts with a random sample from a population. If the sample data are consistent with the null hypothesis, then you do ...
9.1 Null and Alternative Hypotheses
The actual test begins by considering two hypotheses.They are called the null hypothesis and the alternative hypothesis.These hypotheses contain opposing viewpoints. H 0, the —null hypothesis: a statement of no difference between sample means or proportions or no difference between a sample mean or proportion and a population mean or proportion. In other words, the difference equals 0.
9.1: Null and Alternative Hypotheses
Example 9.1.3 9.1. 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: H0: μ ≥ 5 H 0: μ ≥ 5. Ha: μ < 5 H a: μ < 5. Exercise 9.1.3 9.1. 3. We want to test if it takes fewer than 45 minutes to teach a lesson plan.
Null and Alternative Hypotheses
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.
Null Hypothesis: Definition, Rejecting & Examples
It is one of two mutually exclusive hypotheses about a population in a hypothesis test. When your sample contains sufficient evidence, you can reject the null and conclude that the effect is statistically significant. Statisticians often denote the null hypothesis as H 0 or H A. Null Hypothesis H0: No effect exists in the population.
Null Hypothesis
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.
Examples of null and alternative hypotheses
It is the opposite of your research hypothesis. The alternative hypothesis--that is, the research hypothesis--is the idea, phenomenon, observation that you want to prove. If you suspect that girls take longer to get ready for school than boys, then: Alternative: girls time > boys time. Null: girls time <= boys time.
10.2: Null and Alternative Hypotheses
The alternative hypothesis ( Ha H a) is a claim about the population that is contradictory to H0 H 0 and what we conclude when we reject H0 H 0. Since the null and alternative hypotheses are contradictory, you must examine evidence to decide if you have enough evidence to reject the null hypothesis or not. The evidence is in the form of sample ...
Null and Alternative Hypotheses
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.
Null Hypothesis
Null Hypothesis. Null hypothesis is used to make decisions based on data and by using statistical tests. Null hypothesis is represented using H o and it states that there is no difference between the characteristics of two samples. Null hypothesis is generally a statement of no difference. The rejection of null hypothesis is equivalent to the ...
How to Insert the Null Hypothesis Symbol in Microsoft Word
Typing the Symbol. To type the null hypothesis symbol, type the letter "H" and then click the subscript icon in the Font section of the Home tab. Your cursor will appear smaller, and you can now type the numeral "0." When you press the space bar, your font will change back to your default font size and you can continue typing.
Null Hypothesis and Alternative Hypothesis
Alternative hypothesis " x is not equal to y .". Null hypothesis: " x is at least y .". Alternative hypothesis " x is less than y .". Null hypothesis: " x is at most y .". Alternative hypothesis " x is greater than y .". Here are the differences between the null and alternative hypotheses and how to distinguish between them.
PDF The Null and the Alternative Hypotheses
They want to test what proportion of the parts do not meet the specifications. Since they claim that the proportion is less than 2%, the symbol for the Alternative Hypothesis will be <. As is the usual practice, an equal symbol is used for the Null Hypothesis. H0: p = 0.02 H1: p < 0.02 (This is the claim). This instructional aid was prepared by ...
How to Write a Null Hypothesis (5 Examples)
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.
The Null Hypothesis, the Alternative Hypothesis, and the Alpha level
The decision ALWAYS reflects the null hypothesis, and yes, that is the proper wording. It may seem a little bit backwards to make our decision in terms of the null hypothesis since that hypothesis is based on the notion that what we believe to be true isn't really true! Our legal system provides a good analogy for the logic of this approach.
Null Hypothesis
Null hypothesis, often denoted as H0, 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. Learn more about Null Hypothesis, its formula, symbol and example in this article
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 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.
Null hypothesis
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.
What symbols are used to represent null hypotheses?
The null hypothesis is often abbreviated as H0 and always includes an equality symbol (usually =, but sometimes ≥ or ≤). FAQ About us . Our editors; Apply as editor ... What symbols are used to represent null hypotheses? The null hypothesis is often abbreviated as H 0. When the null hypothesis is written using mathematical symbols, it ...
9.1 Null and Alternative Hypotheses
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.
9.1 Null and Alternative Hypotheses
Ha — The alternative hypothesis: It is a claim about the population that is contradictory to H0 and what we conclude when we reject H0. 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.
How to Insert the Null Hypothesis & Alternate Hypothesis Symbols in
Step 4. Type a "0" to create a null hypothesis symbol or "1" to create an alternative hypothesis symbol. Alternatively, type an "o" or "a" to represent the null and alternative hypotheses, respectively, although these symbols are not as frequently used. Advertisement.
Applying the Laplace Transform Procedure, Testing Exponentiality ...
This paper addresses a hypothesis testing problem for comparing exponentially distributed data against a new class termed "New Better than Renewal Used in Moment Generating Function" (NBRUmgf). A measure of departure from exponentiality is constructed using the Laplace transform, followed by the development of a U-statistic-based test for the hypothesis. Additionally, a test based on the ...

Module 9: Hypothesis Testing With One Sample

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What is a Null Hypothesis?

Null Hypothesis Examples

When to Reject the Null Hypothesis

Rejecting the Null Hypothesis

Failing to Reject the Null Hypothesis

How to Write a Null Hypothesis

Group Means

Group Proportions

Correlation and Regression Coefficients

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Null and Alternative Hypotheses

Chapter Review

Formula Review

Null Hypothesis

What Is Null Hypothesis?

Tests For Null Hypothesis

Significance Testing

Difference Between Null Hypothesis And Alternate Hypothesis

Examples on Null Hypothesis

Practice Questions on Null Hypothesis

How Do You Test Null Hypothesis?

How To Accept or Reject Null Hypothesis?

What Is the Difference Between Null Hypothesis And Alternate Hypothesis?

What Is Null Hypothesis Example?

Null Hypothesis and Alternative Hypothesis

The Null Hypothesis

The Alternative Hypothesis

How to Write a Null Hypothesis (5 Examples)

Example 1: Weight of Turtles

Example 2: Height of Males

Example 3: Graduation Rates

Example 4: Burger Weights

Example 5: Citizen Support

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Null Hypothesis

What is Null Hypothesis?

Null Hypothesis Meaning

Mean Comparison (Two-sample t-test)

Proportion Comparison

Equality in Variance (F-test in ANOVA)

Independence (Chi-square Test of Independence):

Equality Null Hypothesis (Simple Null Hypothesis)

Non-Inferiority Null Hypothesis

Superiority Null Hypothesis

Independence Null Hypothesis

Homogeneity Null Hypothesis

Null Hypothesis Rejection

When is Null Hypothesis Rejected?

Difference Between Null Hypothesis and Alternative Hypothesis