null hypothesis ho or h0

What is The Null Hypothesis & When Do You Reject The Null Hypothesis

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BA (Hons) Psychology, Princeton University

Julia Simkus is a graduate of Princeton University with a Bachelor of Arts in Psychology. She is currently studying for a Master's Degree in Counseling for Mental Health and Wellness in September 2023. Julia's research has been published in peer reviewed journals.

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On This Page:

A null hypothesis is a statistical concept suggesting no significant difference or relationship between measured variables. It’s the default assumption unless empirical evidence proves otherwise.

The null hypothesis states no relationship exists between the two variables being studied (i.e., one variable does not affect the other).

The null hypothesis is the statement that a researcher or an investigator wants to disprove.

Testing the null hypothesis can tell you whether your results are due to the effects of manipulating the dependent variable or due to random chance.

How to Write a Null Hypothesis

Null hypotheses (H0) start as research questions that the investigator rephrases as statements indicating no effect or relationship between the independent and dependent variables.

It is a default position that your research aims to challenge or confirm.

For example, if studying the impact of exercise on weight loss, your null hypothesis might be:

There is no significant difference in weight loss between individuals who exercise daily and those who do not.

Examples of Null Hypotheses

When do we reject the null hypothesis .

We reject the null hypothesis when the data provide strong enough evidence to conclude that it is likely incorrect. This often occurs when the p-value (probability of observing the data given the null hypothesis is true) is below a predetermined significance level.

If the collected data does not meet the expectation of the null hypothesis, a researcher can conclude that the data lacks sufficient evidence to back up the null hypothesis, and thus the null hypothesis is rejected.

Rejecting the null hypothesis means that a relationship does exist between a set of variables and the effect is statistically significant ( p > 0.05).

If the data collected from the random sample is not statistically significance , then the null hypothesis will be accepted, and the researchers can conclude that there is no relationship between the variables.

You need to perform a statistical test on your data in order to evaluate how consistent it is with the null hypothesis. A p-value is one statistical measurement used to validate a hypothesis against observed data.

Calculating the p-value is a critical part of null-hypothesis significance testing because it quantifies how strongly the sample data contradicts the null hypothesis.

The level of statistical significance is often expressed as a p -value between 0 and 1. The smaller the p-value, the stronger the evidence that you should reject the null hypothesis.

Probability and statistical significance in ab testing. Statistical significance in a b experiments

Usually, a researcher uses a confidence level of 95% or 99% (p-value of 0.05 or 0.01) as general guidelines to decide if you should reject or keep the null.

When your p-value is less than or equal to your significance level, you reject the null hypothesis.

In other words, smaller p-values are taken as stronger evidence against the null hypothesis. Conversely, when the p-value is greater than your significance level, you fail to reject the null hypothesis.

In this case, the sample data provides insufficient data to conclude that the effect exists in the population.

Because you can never know with complete certainty whether there is an effect in the population, your inferences about a population will sometimes be incorrect.

When you incorrectly reject the null hypothesis, it’s called a type I error. When you incorrectly fail to reject it, it’s called a type II error.

Why Do We Never Accept The Null Hypothesis?

The reason we do not say “accept the null” is because we are always assuming the null hypothesis is true and then conducting a study to see if there is evidence against it. And, even if we don’t find evidence against it, a null hypothesis is not accepted.

A lack of evidence only means that you haven’t proven that something exists. It does not prove that something doesn’t exist.

It is risky to conclude that the null hypothesis is true merely because we did not find evidence to reject it. It is always possible that researchers elsewhere have disproved the null hypothesis, so we cannot accept it as true, but instead, we state that we failed to reject the null.

One can either reject the null hypothesis, or fail to reject it, but can never accept it.

Why Do We Use The Null Hypothesis?

We can never prove with 100% certainty that a hypothesis is true; We can only collect evidence that supports a theory. However, testing a hypothesis can set the stage for rejecting or accepting this hypothesis within a certain confidence level.

The null hypothesis is useful because it can tell us whether the results of our study are due to random chance or the manipulation of a variable (with a certain level of confidence).

A null hypothesis is rejected if the measured data is significantly unlikely to have occurred and a null hypothesis is accepted if the observed outcome is consistent with the position held by the null hypothesis.

Rejecting the null hypothesis sets the stage for further experimentation to see if a relationship between two variables exists.

Hypothesis testing is a critical part of the scientific method as it helps decide whether the results of a research study support a particular theory about a given population. Hypothesis testing is a systematic way of backing up researchers’ predictions with statistical analysis.

It helps provide sufficient statistical evidence that either favors or rejects a certain hypothesis about the population parameter.

Purpose of a Null Hypothesis

The primary purpose of the null hypothesis is to disprove an assumption.
Whether rejected or accepted, the null hypothesis can help further progress a theory in many scientific cases.
A null hypothesis can be used to ascertain how consistent the outcomes of multiple studies are.

Do you always need both a Null Hypothesis and an Alternative Hypothesis?

The null (H0) and alternative (Ha or H1) hypotheses are two competing claims that describe the effect of the independent variable on the dependent variable. They are mutually exclusive, which means that only one of the two hypotheses can be true.

While the null hypothesis states that there is no effect in the population, an alternative hypothesis states that there is statistical significance between two variables.

The goal of hypothesis testing is to make inferences about a population based on a sample. In order to undertake hypothesis testing, you must express your research hypothesis as a null and alternative hypothesis. Both hypotheses are required to cover every possible outcome of the study.

What is the difference between a null hypothesis and an alternative hypothesis?

The alternative hypothesis is the complement to the null hypothesis. The null hypothesis states that there is no effect or no relationship between variables, while the alternative hypothesis claims that there is an effect or relationship in the population.

It is the claim that you expect or hope will be true. The null hypothesis and the alternative hypothesis are always mutually exclusive, meaning that only one can be true at a time.

What are some problems with the null hypothesis?

One major problem with the null hypothesis is that researchers typically will assume that accepting the null is a failure of the experiment. However, accepting or rejecting any hypothesis is a positive result. Even if the null is not refuted, the researchers will still learn something new.

Why can a null hypothesis not be accepted?

We can either reject or fail to reject a null hypothesis, but never accept it. If your test fails to detect an effect, this is not proof that the effect doesn’t exist. It just means that your sample did not have enough evidence to conclude that it exists.

We can’t accept a null hypothesis because a lack of evidence does not prove something that does not exist. Instead, we fail to reject it.

Failing to reject the null indicates that the sample did not provide sufficient enough evidence to conclude that an effect exists.

If the p-value is greater than the significance level, then you fail to reject the null hypothesis.

Is a null hypothesis directional or non-directional?

A hypothesis test can either contain an alternative directional hypothesis or a non-directional alternative hypothesis. A directional hypothesis is one that contains the less than (“<“) or greater than (“>”) sign.

A nondirectional hypothesis contains the not equal sign (“≠”). However, a null hypothesis is neither directional nor non-directional.

A null hypothesis is a prediction that there will be no change, relationship, or difference between two variables.

The directional hypothesis or nondirectional hypothesis would then be considered alternative hypotheses to the null hypothesis.

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Krueger, J. (2001). Null hypothesis significance testing: On the survival of a flawed method. American Psychologist , 56 (1), 16.

Masson, M. E. (2011). A tutorial on a practical Bayesian alternative to null-hypothesis significance testing. Behavior research methods , 43 , 679-690.

Nickerson, R. S. (2000). Null hypothesis significance testing: a review of an old and continuing controversy. Psychological methods , 5 (2), 241.

Rozeboom, W. W. (1960). The fallacy of the null-hypothesis significance test. Psychological bulletin , 57 (5), 416.

Module 9: Hypothesis Testing With One Sample

Null and alternative hypotheses, learning outcomes.

Describe hypothesis testing in general and in practice

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

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

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

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

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

Mathematical Symbols Used in H 0 and H a :

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

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

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

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

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

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

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

H 0 : μ = 2.0

H a : μ ≠ 2.0

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

H 0 : μ = 66
H a : μ ≠ 66

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

H 0 : μ ≥ 5

H a : μ < 5

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

H 0 : μ ≥ 45
H a : μ < 45

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

H 0 : p ≤ 0.066

H a : p > 0.066

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

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

Concept Review

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

Formula Review

H 0 and H a are contradictory.

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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: 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: 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 :

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 H a : 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 = 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 fewer than five years to graduate from college, on the average. The null and alternative hypotheses are the following: H 0 : μ ≥ 5 H a : μ < 5

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. H 0 : p ≤ 0.066 H a : p > 0.066

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.

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Exploring the Null Hypothesis: Definition and Purpose

Updated: July 5, 2023 by Ken Feldman

Hypothesis testing is a branch of statistics in which, using data from a sample, an inference is made about a population parameter or a population probability distribution .

First, a hypothesis statement and assumption is made about the population parameter or probability distribution. This initial statement is called the Null Hypothesis and is denoted by H o. An alternative or alternate hypothesis (denoted Ha ), is then stated which will be the opposite of the Null Hypothesis.

The hypothesis testing process and analysis involves using sample data to determine whether or not you can be statistically confident that you can reject or fail to reject the H o. If the H o is rejected, the statistical conclusion is that the alternative or alternate hypothesis Ha is true.

Overview: What is the Null Hypothesis (Ho)?

Hypothesis testing applies to all forms of statistical inquiry. For example, it can be used to determine whether there are differences between population parameters or an understanding about slopes of regression lines or equality of probability distributions.

In all cases, the first thing you do is state the Null and Alternate Hypotheses. The word Null in the context of hypothesis testing means “nothing” or “zero.”

As an example, if we wanted to test whether there was a difference in two population means based on the calculations from two samples, we would state the Null Hypothesis in the form of:

Ho: mu1 = mu2 or mu1- mu2 = 0

In other words, there is no difference, or the difference is zero. Note that the notation is in the form of a population parameter, not a sample statistic.

Since you are using sample data to make your inferences about the population, it’s possible you’ll make an error. In the case of the Null Hypothesis, we can make one of two errors.

Type 1 , or alpha error: An alpha error is when you mistakenly reject the Null and believe that something significant happened. In other words, you believe that the means of the two populations are different when they aren’t.
Type 2, or beta error: A beta error is when you fail to reject the null when you should have. In this case, you missed something significant and failed to take action.

A classic example is when you get the results back from your doctor after taking a blood test. If the doctor says you have an infection when you really don’t, that is an alpha error. That is thinking that there is something significant going on when there isn’t. We also call that a false positive. The doctor rejected the null that “there was zero infection” and missed the call.

On the other hand, if the doctor told you that everything was OK when you really did have an infection, then he made a beta, or type 2, error. He failed to reject the Null Hypothesis when he should have. That is called a false negative.

The decision to reject or not to reject the Null Hypothesis is based on three numbers.

Alpha, which you get to choose. Alpha is the risk you are willing to assume of falsely rejecting the Null. The typical values for alpha are 1%, 5%, or 10%. Depending on the importance of the conclusion, you only want to falsely claim a difference when there is none, 1%, 5%, or 10% of the time.
Beta, which is typically 20%. This means you’re willing to be wrong 20% of the time in failing to reject the null when you should have.
P-value, which is calculated from the data. The p-value is the actual risk you have in being wrong if you reject the null. You would like that to be low.

Your decision as to what to do about the null is made by comparing the alpha value (your assumed risk) with the p-value (actual risk). If the actual risk is lower than your assumed risk, you can feel comfortable in rejecting the null and claiming something has happened. But, if the actual risk is higher than your assumed risk you will be taking a bigger risk than you want by rejecting the null.

RELATED: NULL VS. ALTERNATIVE HYPOTHESIS

3 benefits of the null hypothesis .

The stating and testing of the null hypothesis is the foundation of hypothesis testing. By doing so, you set the parameters for your statistical inference.

1. Statistical assurance of determining differences between population parameters

Just looking at the mathematical difference between the means of two samples and making a decision is woefully inadequate. By statistically testing the null hypothesis, you will have more confidence in any inferences you want to make about populations based on your samples.

2. Statistically based estimation of the probability of a population distribution

Many statistical tests require assumptions of specific distributions. Many of these tests assume that the population follows the normal distribution . If it doesn’t, the test may be invalid.

3. Assess the strength of your conclusions as to what to do with the null hypothesis

Hypothesis testing calculations will provide some relative strength to your decisions as to whether you reject or fail to reject the null hypothesis.

Why is the Null Hypothesis important to understand?

The interpretation of the statistics relative to the null hypothesis is what’s important.

1. Properly write the null hypothesis to properly capture what you are seeking to prove

The null is always written in the same format. That is, the lack of difference or some other condition. The alternative hypothesis can be written in three formats depending on what you want to prove.

2. Frame your statement and select an appropriate alpha risk

You don’t want to place too big of a hurdle or burden on your decision-making relative to action on the null hypothesis by selecting an alpha value that is too high or too low.

3. There are decision errors when deciding on how to respond to the Null Hypothesis

Since your decision relative to rejecting or not rejecting the null is based on statistical calculations, it is important to understand how that decision works.

An industry example of using the Null Hypothesis

The new director of marketing just completed the rollout of a new marketing campaign targeting the Hispanic market. Early indications showed that the campaign was successful in increasing sales in the Hispanic market.

He came to that conclusion by comparing a sample of sales prior to the campaign and current sales after implementation of the campaign. He was anxious to proudly tell his boss how successful the campaign was. But, he decided to first check with his Lean Six Sigma Black Belt to see whether she agreed with his conclusion.

The Black Belt first asked the director his tolerance for risk of being wrong by telling the boss the campaign was successful when in fact, it wasn’t. That was the alpha value. The Director picked 5% since he was new and didn’t want to make a false claim so early in his career. He also picked 20% as his beta value.

When the Black Belt was done analyzing the data, she found out that the p-value was 15%. That meant if the director told the VP the campaign worked, there was a 15% chance he would be wrong and that the campaign probably needed some revising. Since he was only willing to be wrong 5% of the time, the decision was to not reject the null since his 5% assumed risk was less than the 15% actual risk.

3 best practices when thinking about the Null Hypothesis

Using hypothesis testing to help make better data-driven decisions requires that you properly address the Null Hypothesis.

1. Always use the proper nomenclature when stating the Null Hypothesis

The null will always be in the form of decisions regarding the population, not the sample.

2. The Null Hypothesis will always be written as the absence of some parameter or process characteristic

The writing of the Alternate Hypothesis can vary, so be sure you understand exactly what condition you are testing against.

3. Pick a reasonable alpha risk so you’re not always failing to reject the Null Hypothesis

Being too cautious will lead you to make beta errors, and you’ll never learn anything about your population data.

Frequently Asked Questions (FAQ) about the Null Hypothesis

What form should the null hypothesis be written in.

The Null Hypothesis should always be in the form of no difference or zero and always refer to the state of the population, not the sample.

What is an alpha error?

An alpha error, or Type 1 error, is rejecting the Null Hypothesis and claiming a significant event has occurred when, in fact, that is not true and the Null should not have been rejected.

How do I use the alpha error and p-value to decide on what decision I should make about the Null Hypothesis?

The most common way of answering this is, “If the p-value is low (less than the alpha), the Null should be rejected. If the p-value is high (greater than the alpha) then the Null should not be rejected.”

Becoming familiar with the Null Hypothesis (Ho)

The proper writing of the Null Hypothesis is the basis for applying hypothesis testing to help you make better data-driven decisions. The format of the Null will always be in the form of zero, or the non-existence of some condition. It will always refer to a population parameter and not the sample you use to do your hypothesis testing calculations.

Be aware of the two types of errors you can make when deciding on what to do with the Null. Select reasonable risks values for your alpha and beta risks. By comparing your alpha risk with the calculated risk computed from the data, you will have sufficient information to make a wise decision as to whether you should reject the Null Hypothesis or not.

About the Author

Ken Feldman

Hypothesis Testing (cont...)

Hypothesis testing, the null and alternative hypothesis.

In order to undertake hypothesis testing you need to express your research hypothesis as a null and alternative hypothesis. The null hypothesis and alternative hypothesis are statements regarding the differences or effects that occur in the population. You will use your sample to test which statement (i.e., the null hypothesis or alternative hypothesis) is most likely (although technically, you test the evidence against the null hypothesis). So, with respect to our teaching example, the null and alternative hypothesis will reflect statements about all statistics students on graduate management courses.

The null hypothesis is essentially the "devil's advocate" position. That is, it assumes that whatever you are trying to prove did not happen ( hint: it usually states that something equals zero). For example, the two different teaching methods did not result in different exam performances (i.e., zero difference). Another example might be that there is no relationship between anxiety and athletic performance (i.e., the slope is zero). The alternative hypothesis states the opposite and is usually the hypothesis you are trying to prove (e.g., the two different teaching methods did result in different exam performances). Initially, you can state these hypotheses in more general terms (e.g., using terms like "effect", "relationship", etc.), as shown below for the teaching methods example:

Depending on how you want to "summarize" the exam performances will determine how you might want to write a more specific null and alternative hypothesis. For example, you could compare the mean exam performance of each group (i.e., the "seminar" group and the "lectures-only" group). This is what we will demonstrate here, but other options include comparing the distributions , medians , amongst other things. As such, we can state:

Now that you have identified the null and alternative hypotheses, you need to find evidence and develop a strategy for declaring your "support" for either the null or alternative hypothesis. We can do this using some statistical theory and some arbitrary cut-off points. Both these issues are dealt with next.

Significance levels

The level of statistical significance is often expressed as the so-called p -value . Depending on the statistical test you have chosen, you will calculate a probability (i.e., the p -value) of observing your sample results (or more extreme) given that the null hypothesis is true . Another way of phrasing this is to consider the probability that a difference in a mean score (or other statistic) could have arisen based on the assumption that there really is no difference. Let us consider this statement with respect to our example where we are interested in the difference in mean exam performance between two different teaching methods. If there really is no difference between the two teaching methods in the population (i.e., given that the null hypothesis is true), how likely would it be to see a difference in the mean exam performance between the two teaching methods as large as (or larger than) that which has been observed in your sample?

So, you might get a p -value such as 0.03 (i.e., p = .03). This means that there is a 3% chance of finding a difference as large as (or larger than) the one in your study given that the null hypothesis is true. However, you want to know whether this is "statistically significant". Typically, if there was a 5% or less chance (5 times in 100 or less) that the difference in the mean exam performance between the two teaching methods (or whatever statistic you are using) is as different as observed given the null hypothesis is true, you would reject the null hypothesis and accept the alternative hypothesis. Alternately, if the chance was greater than 5% (5 times in 100 or more), you would fail to reject the null hypothesis and would not accept the alternative hypothesis. As such, in this example where p = .03, we would reject the null hypothesis and accept the alternative hypothesis. We reject it because at a significance level of 0.03 (i.e., less than a 5% chance), the result we obtained could happen too frequently for us to be confident that it was the two teaching methods that had an effect on exam performance.

Whilst there is relatively little justification why a significance level of 0.05 is used rather than 0.01 or 0.10, for example, it is widely used in academic research. However, if you want to be particularly confident in your results, you can set a more stringent level of 0.01 (a 1% chance or less; 1 in 100 chance or less).

One- and two-tailed predictions

When considering whether we reject the null hypothesis and accept the alternative hypothesis, we need to consider the direction of the alternative hypothesis statement. For example, the alternative hypothesis that was stated earlier is:

The alternative hypothesis tells us two things. First, what predictions did we make about the effect of the independent variable(s) on the dependent variable(s)? Second, what was the predicted direction of this effect? Let's use our example to highlight these two points.

Sarah predicted that her teaching method (independent variable: teaching method), whereby she not only required her students to attend lectures, but also seminars, would have a positive effect (that is, increased) students' performance (dependent variable: exam marks). If an alternative hypothesis has a direction (and this is how you want to test it), the hypothesis is one-tailed. That is, it predicts direction of the effect. If the alternative hypothesis has stated that the effect was expected to be negative, this is also a one-tailed hypothesis.

Alternatively, a two-tailed prediction means that we do not make a choice over the direction that the effect of the experiment takes. Rather, it simply implies that the effect could be negative or positive. If Sarah had made a two-tailed prediction, the alternative hypothesis might have been:

In other words, we simply take out the word "positive", which implies the direction of our effect. In our example, making a two-tailed prediction may seem strange. After all, it would be logical to expect that "extra" tuition (going to seminar classes as well as lectures) would either have a positive effect on students' performance or no effect at all, but certainly not a negative effect. However, this is just our opinion (and hope) and certainly does not mean that we will get the effect we expect. Generally speaking, making a one-tail prediction (i.e., and testing for it this way) is frowned upon as it usually reflects the hope of a researcher rather than any certainty that it will happen. Notable exceptions to this rule are when there is only one possible way in which a change could occur. This can happen, for example, when biological activity/presence in measured. That is, a protein might be "dormant" and the stimulus you are using can only possibly "wake it up" (i.e., it cannot possibly reduce the activity of a "dormant" protein). In addition, for some statistical tests, one-tailed tests are not possible.

Rejecting or failing to reject the null hypothesis

Let's return finally to the question of whether we reject or fail to reject the null hypothesis.

If our statistical analysis shows that the significance level is below the cut-off value we have set (e.g., either 0.05 or 0.01), we reject the null hypothesis and accept the alternative hypothesis. Alternatively, if the significance level is above the cut-off value, we fail to reject the null hypothesis and cannot accept the alternative hypothesis. You should note that you cannot accept the null hypothesis, but only find evidence against it.

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Hypothesis Testing with One Sample

Null and Alternative Hypotheses

OpenStaxCollege

[latexpage]

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

H 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 :

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.

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

If α > p -value, then reject H 0 .

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

p < 0.30
p > 0.30

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

p > 0.20
p < 0.20

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

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

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

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6a.1 - introduction to hypothesis testing, basic terms section .

The first step in hypothesis testing is to set up two competing hypotheses. The hypotheses are the most important aspect. If the hypotheses are incorrect, your conclusion will also be incorrect.

The two hypotheses are named the null hypothesis and the alternative hypothesis.

The goal of hypothesis testing is to see if there is enough evidence against the null hypothesis. In other words, to see if there is enough evidence to reject the null hypothesis. If there is not enough evidence, then we fail to reject the null hypothesis.

Consider the following example where we set up these hypotheses.

Example 6-1 Section

A man, Mr. Orangejuice, goes to trial and is tried for the murder of his ex-wife. He is either guilty or innocent. Set up the null and alternative hypotheses for this example.

Putting this in a hypothesis testing framework, the hypotheses being tested are:

The man is guilty
The man is innocent

Let's set up the null and alternative hypotheses.

\(H_0\colon \) Mr. Orangejuice is innocent

\(H_a\colon \) Mr. Orangejuice is guilty

Remember that we assume the null hypothesis is true and try to see if we have evidence against the null. Therefore, it makes sense in this example to assume the man is innocent and test to see if there is evidence that he is guilty.

The Logic of Hypothesis Testing Section

We want to know the answer to a research question. We determine our null and alternative hypotheses. Now it is time to make a decision.

The decision is either going to be...

reject the null hypothesis or...
fail to reject the null hypothesis.

Consider the following table. The table shows the decision/conclusion of the hypothesis test and the unknown "reality", or truth. We do not know if the null is true or if it is false. If the null is false and we reject it, then we made the correct decision. If the null hypothesis is true and we fail to reject it, then we made the correct decision.

So what happens when we do not make the correct decision?

When doing hypothesis testing, two types of mistakes may be made and we call them Type I error and Type II error. If we reject the null hypothesis when it is true, then we made a type I error. If the null hypothesis is false and we failed to reject it, we made another error called a Type II error.

Types of errors

The “reality”, or truth, about the null hypothesis is unknown and therefore we do not know if we have made the correct decision or if we committed an error. We can, however, define the likelihood of these events.

\(\alpha\) and \(\beta\) are probabilities of committing an error so we want these values to be low. However, we cannot decrease both. As \(\alpha\) decreases, \(\beta\) increases.

Example 6-1 Cont'd... Section

A man, Mr. Orangejuice, goes to trial and is tried for the murder of his ex-wife. He is either guilty or not guilty. We found before that...

\( H_0\colon \) Mr. Orangejuice is innocent
\( H_a\colon \) Mr. Orangejuice is guilty

Interpret Type I error, \(\alpha \), Type II error, \(\beta \).

As you can see here, the Type I error (putting an innocent man in jail) is the more serious error. Ethically, it is more serious to put an innocent man in jail than to let a guilty man go free. So to minimize the probability of a type I error we would choose a smaller significance level.

Try it! Section

An inspector has to choose between certifying a building as safe or saying that the building is not safe. There are two hypotheses:

Building is safe
Building is not safe

Set up the null and alternative hypotheses. Interpret Type I and Type II error.

\( H_0\colon\) Building is not safe vs \(H_a\colon \) Building is safe

Power and \(\beta \) are complements of each other. Therefore, they have an inverse relationship, i.e. as one increases, the other decreases.

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Null Hypothesis

Hypothesis Testing Formula
SQLite IS NULL
MySQL NOT NULL Constraint
Real-life Applications of Hypothesis Testing
Null in JavaScript
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Hypothesis in Machine Learning
Understanding Hypothesis Testing
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Null Hypothesis , often denoted as H 0, is a foundational concept in statistical hypothesis testing. It represents an assumption that no significant difference, effect, or relationship exists between variables within a population. It serves as a baseline assumption, positing no observed change or effect occurring. The null is t he truth or falsity of an idea in analysis.

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

Table of Content

What is Null Hypothesis?

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

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

Null Hypothesis Meaning

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

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

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

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

Mean Comparison (Two-sample t-test)

H 0 : μ 1 = μ 2

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

Proportion Comparison

H 0 : p 1 − p 2 = 0

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

Equality in Variance (F-test in ANOVA)

H 0 : σ 1 = σ 2

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

Independence (Chi-square Test of Independence):

H 0 : Variables are independent

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

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

Equality Null Hypothesis (Simple Null Hypothesis)

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

Non-Inferiority Null Hypothesis

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

Superiority Null Hypothesis

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

Independence Null Hypothesis

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

Homogeneity Null Hypothesis

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

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

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

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

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

Null Hypothesis Rejection

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

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

When is Null Hypothesis Rejected?

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

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

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

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

Difference Between Null Hypothesis and Alternative Hypothesis

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

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

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

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

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

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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Statistics Made Easy

Understanding the Null Hypothesis for Linear Regression

Linear regression is a technique we can use to understand the relationship between one or more predictor variables and a response variable .

If we only have one predictor variable and one response variable, we can use simple linear regression , which uses the following formula to estimate the relationship between the variables:

ŷ = β 0 + β 1 x

ŷ: The estimated response value.
β 0 : The average value of y when x is zero.
β 1 : The average change in y associated with a one unit increase in x.
x: The value of the predictor variable.

Simple linear regression uses the following null and alternative hypotheses:

H 0 : β 1 = 0
H A : β 1 ≠ 0

The null hypothesis states that the coefficient β 1 is equal to zero. In other words, there is no statistically significant relationship between the predictor variable, x, and the response variable, y.

The alternative hypothesis states that β 1 is not equal to zero. In other words, there is a statistically significant relationship between x and y.

If we have multiple predictor variables and one response variable, we can use multiple linear regression , which uses the following formula to estimate the relationship between the variables:

ŷ = β 0 + β 1 x 1 + β 2 x 2 + … + β k x k

β 0 : The average value of y when all predictor variables are equal to zero.
β i : The average change in y associated with a one unit increase in x i .
x i : The value of the predictor variable x i .

Multiple linear regression uses the following null and alternative hypotheses:

H 0 : β 1 = β 2 = … = β k = 0
H A : β 1 = β 2 = … = β k ≠ 0

The null hypothesis states that all coefficients in the model are equal to zero. In other words, none of the predictor variables have a statistically significant relationship with the response variable, y.

The alternative hypothesis states that not every coefficient is simultaneously equal to zero.

The following examples show how to decide to reject or fail to reject the null hypothesis in both simple linear regression and multiple linear regression models.

Example 1: Simple Linear Regression

Suppose a professor would like to use the number of hours studied to predict the exam score that students will receive in his class. He collects data for 20 students and fits a simple linear regression model.

The following screenshot shows the output of the regression model:

Output of simple linear regression in Excel

The fitted simple linear regression model is:

Exam Score = 67.1617 + 5.2503*(hours studied)

To determine if there is a statistically significant relationship between hours studied and exam score, we need to analyze the overall F value of the model and the corresponding p-value:

Overall F-Value: 47.9952
P-value: 0.000

Since this p-value is less than .05, we can reject the null hypothesis. In other words, there is a statistically significant relationship between hours studied and exam score received.

Example 2: Multiple Linear Regression

Suppose a professor would like to use the number of hours studied and the number of prep exams taken to predict the exam score that students will receive in his class. He collects data for 20 students and fits a multiple linear regression model.

Multiple linear regression output in Excel

The fitted multiple linear regression model is:

Exam Score = 67.67 + 5.56*(hours studied) – 0.60*(prep exams taken)

To determine if there is a jointly statistically significant relationship between the two predictor variables and the response variable, we need to analyze the overall F value of the model and the corresponding p-value:

Overall F-Value: 23.46
P-value: 0.00

Since this p-value is less than .05, we can reject the null hypothesis. In other words, hours studied and prep exams taken have a jointly statistically significant relationship with exam score.

Note: Although the p-value for prep exams taken (p = 0.52) is not significant, prep exams combined with hours studied has a significant relationship with exam score.

Additional Resources

Understanding the F-Test of Overall Significance in Regression How to Read and Interpret a Regression Table How to Report Regression Results How to Perform Simple Linear Regression in Excel How to Perform Multiple Linear Regression in Excel

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11.1: Testing the Hypothesis that β = 0

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The correlation coefficient, \(r\), tells us about the strength and direction of the linear relationship between \(x\) and \(y\). However, the reliability of the linear model also depends on how many observed data points are in the sample. We need to look at both the value of the correlation coefficient \(r\) and the sample size \(n\), together. We perform a hypothesis test of the "significance of the correlation coefficient" to decide whether the linear relationship in the sample data is strong enough to use to model the relationship in the population.

The sample data are used to compute \(r\), the correlation coefficient for the sample. If we had data for the entire population, we could find the population correlation coefficient. But because we have only sample data, we cannot calculate the population correlation coefficient. The sample correlation coefficient, \(r\), is our estimate of the unknown population correlation coefficient.

The symbol for the population correlation coefficient is \(\rho\), the Greek letter "rho."
\(\rho =\) population correlation coefficient (unknown)
\(r =\) sample correlation coefficient (known; calculated from sample data)

The hypothesis test lets us decide whether the value of the population correlation coefficient \(\rho\) is "close to zero" or "significantly different from zero". We decide this based on the sample correlation coefficient \(r\) and the sample size \(n\).

If the test concludes that the correlation coefficient is significantly different from zero, we say that the correlation coefficient is "significant."

Conclusion: There is sufficient evidence to conclude that there is a significant linear relationship between \(x\) and \(y\) because the correlation coefficient is significantly different from zero.
What the conclusion means: There is a significant linear relationship between \(x\) and \(y\). We can use the regression line to model the linear relationship between \(x\) and \(y\) in the population.

If the test concludes that the correlation coefficient is not significantly different from zero (it is close to zero), we say that correlation coefficient is "not significant".

Conclusion: "There is insufficient evidence to conclude that there is a significant linear relationship between \(x\) and \(y\) because the correlation coefficient is not significantly different from zero."
What the conclusion means: There is not a significant linear relationship between \(x\) and \(y\). Therefore, we CANNOT use the regression line to model a linear relationship between \(x\) and \(y\) in the population.
If \(r\) is significant and the scatter plot shows a linear trend, the line can be used to predict the value of \(y\) for values of \(x\) that are within the domain of observed \(x\) values.
If \(r\) is not significant OR if the scatter plot does not show a linear trend, the line should not be used for prediction.
If \(r\) is significant and if the scatter plot shows a linear trend, the line may NOT be appropriate or reliable for prediction OUTSIDE the domain of observed \(x\) values in the data.

PERFORMING THE HYPOTHESIS TEST

Null Hypothesis: \(H_{0}: \rho = 0\)
Alternate Hypothesis: \(H_{a}: \rho \neq 0\)

WHAT THE HYPOTHESES MEAN IN WORDS:

Null Hypothesis \(H_{0}\) : The population correlation coefficient IS NOT significantly different from zero. There IS NOT a significant linear relationship(correlation) between \(x\) and \(y\) in the population.
Alternate Hypothesis \(H_{a}\) : The population correlation coefficient IS significantly DIFFERENT FROM zero. There IS A SIGNIFICANT LINEAR RELATIONSHIP (correlation) between \(x\) and \(y\) in the population.

DRAWING A CONCLUSION:There are two methods of making the decision. The two methods are equivalent and give the same result.

Method 1: Using the \(p\text{-value}\)
Method 2: Using a table of critical values

In this chapter of this textbook, we will always use a significance level of 5%, \(\alpha = 0.05\)

Using the \(p\text{-value}\) method, you could choose any appropriate significance level you want; you are not limited to using \(\alpha = 0.05\). But the table of critical values provided in this textbook assumes that we are using a significance level of 5%, \(\alpha = 0.05\). (If we wanted to use a different significance level than 5% with the critical value method, we would need different tables of critical values that are not provided in this textbook.)

METHOD 1: Using a \(p\text{-value}\) to make a decision

To calculate the \(p\text{-value}\) using LinRegTTEST:

On the LinRegTTEST input screen, on the line prompt for \(\beta\) or \(\rho\), highlight "\(\neq 0\)"

The output screen shows the \(p\text{-value}\) on the line that reads "\(p =\)".

(Most computer statistical software can calculate the \(p\text{-value}\).)

If the \(p\text{-value}\) is less than the significance level ( \(\alpha = 0.05\) ):

Decision: Reject the null hypothesis.
Conclusion: "There is sufficient evidence to conclude that there is a significant linear relationship between \(x\) and \(y\) because the correlation coefficient is significantly different from zero."

If the \(p\text{-value}\) is NOT less than the significance level ( \(\alpha = 0.05\) )

Decision: DO NOT REJECT the null hypothesis.
Conclusion: "There is insufficient evidence to conclude that there is a significant linear relationship between \(x\) and \(y\) because the correlation coefficient is NOT significantly different from zero."

Calculation Notes:

You will use technology to calculate the \(p\text{-value}\). The following describes the calculations to compute the test statistics and the \(p\text{-value}\):
The \(p\text{-value}\) is calculated using a \(t\)-distribution with \(n - 2\) degrees of freedom.
The formula for the test statistic is \(t = \frac{r\sqrt{n-2}}{\sqrt{1-r^{2}}}\). The value of the test statistic, \(t\), is shown in the computer or calculator output along with the \(p\text{-value}\). The test statistic \(t\) has the same sign as the correlation coefficient \(r\).
The \(p\text{-value}\) is the combined area in both tails.

An alternative way to calculate the \(p\text{-value}\) ( \(p\) ) given by LinRegTTest is the command 2*tcdf(abs(t),10^99, n-2) in 2nd DISTR.

THIRD-EXAM vs FINAL-EXAM EXAMPLE: \(p\text{-value}\) method

Consider the third exam/final exam example.
The line of best fit is: \(\hat{y} = -173.51 + 4.83x\) with \(r = 0.6631\) and there are \(n = 11\) data points.
Can the regression line be used for prediction? Given a third exam score ( \(x\) value), can we use the line to predict the final exam score (predicted \(y\) value)?
\(H_{0}: \rho = 0\)
\(H_{a}: \rho \neq 0\)
\(\alpha = 0.05\)
The \(p\text{-value}\) is 0.026 (from LinRegTTest on your calculator or from computer software).
The \(p\text{-value}\), 0.026, is less than the significance level of \(\alpha = 0.05\).
Decision: Reject the Null Hypothesis \(H_{0}\)
Conclusion: There is sufficient evidence to conclude that there is a significant linear relationship between the third exam score (\(x\)) and the final exam score (\(y\)) because the correlation coefficient is significantly different from zero.

Because \(r\) is significant and the scatter plot shows a linear trend, the regression line can be used to predict final exam scores.

METHOD 2: Using a table of Critical Values to make a decision

The 95% Critical Values of the Sample Correlation Coefficient Table can be used to give you a good idea of whether the computed value of \(r\) is significant or not . Compare \(r\) to the appropriate critical value in the table. If \(r\) is not between the positive and negative critical values, then the correlation coefficient is significant. If \(r\) is significant, then you may want to use the line for prediction.

Example \(\PageIndex{1}\)

Suppose you computed \(r = 0.801\) using \(n = 10\) data points. \(df = n - 2 = 10 - 2 = 8\). The critical values associated with \(df = 8\) are \(-0.632\) and \(+0.632\). If \(r <\) negative critical value or \(r >\) positive critical value, then \(r\) is significant. Since \(r = 0.801\) and \(0.801 > 0.632\), \(r\) is significant and the line may be used for prediction. If you view this example on a number line, it will help you.

Exercise \(\PageIndex{1}\)

For a given line of best fit, you computed that \(r = 0.6501\) using \(n = 12\) data points and the critical value is 0.576. Can the line be used for prediction? Why or why not?

If the scatter plot looks linear then, yes, the line can be used for prediction, because \(r >\) the positive critical value.

Example \(\PageIndex{2}\)

Suppose you computed \(r = –0.624\) with 14 data points. \(df = 14 – 2 = 12\). The critical values are \(-0.532\) and \(0.532\). Since \(-0.624 < -0.532\), \(r\) is significant and the line can be used for prediction

Exercise \(\PageIndex{2}\)

For a given line of best fit, you compute that \(r = 0.5204\) using \(n = 9\) data points, and the critical value is \(0.666\). Can the line be used for prediction? Why or why not?

No, the line cannot be used for prediction, because \(r <\) the positive critical value.

Example \(\PageIndex{3}\)

Suppose you computed \(r = 0.776\) and \(n = 6\). \(df = 6 - 2 = 4\). The critical values are \(-0.811\) and \(0.811\). Since \(-0.811 < 0.776 < 0.811\), \(r\) is not significant, and the line should not be used for prediction.

Exercise \(\PageIndex{3}\)

For a given line of best fit, you compute that \(r = -0.7204\) using \(n = 8\) data points, and the critical value is \(= 0.707\). Can the line be used for prediction? Why or why not?

Yes, the line can be used for prediction, because \(r <\) the negative critical value.

THIRD-EXAM vs FINAL-EXAM EXAMPLE: critical value method

Consider the third exam/final exam example. The line of best fit is: \(\hat{y} = -173.51 + 4.83x\) with \(r = 0.6631\) and there are \(n = 11\) data points. Can the regression line be used for prediction? Given a third-exam score ( \(x\) value), can we use the line to predict the final exam score (predicted \(y\) value)?

Use the "95% Critical Value" table for \(r\) with \(df = n - 2 = 11 - 2 = 9\).
The critical values are \(-0.602\) and \(+0.602\)
Since \(0.6631 > 0.602\), \(r\) is significant.
Conclusion:There is sufficient evidence to conclude that there is a significant linear relationship between the third exam score (\(x\)) and the final exam score (\(y\)) because the correlation coefficient is significantly different from zero.

Example \(\PageIndex{4}\)

Suppose you computed the following correlation coefficients. Using the table at the end of the chapter, determine if \(r\) is significant and the line of best fit associated with each r can be used to predict a \(y\) value. If it helps, draw a number line.

\(r = –0.567\) and the sample size, \(n\), is \(19\). The \(df = n - 2 = 17\). The critical value is \(-0.456\). \(-0.567 < -0.456\) so \(r\) is significant.
\(r = 0.708\) and the sample size, \(n\), is \(9\). The \(df = n - 2 = 7\). The critical value is \(0.666\). \(0.708 > 0.666\) so \(r\) is significant.
\(r = 0.134\) and the sample size, \(n\), is \(14\). The \(df = 14 - 2 = 12\). The critical value is \(0.532\). \(0.134\) is between \(-0.532\) and \(0.532\) so \(r\) is not significant.
\(r = 0\) and the sample size, \(n\), is five. No matter what the \(dfs\) are, \(r = 0\) is between the two critical values so \(r\) is not significant.

Exercise \(\PageIndex{4}\)

For a given line of best fit, you compute that \(r = 0\) using \(n = 100\) data points. Can the line be used for prediction? Why or why not?

No, the line cannot be used for prediction no matter what the sample size is.

Assumptions in Testing the Significance of the Correlation Coefficient

Testing the significance of the correlation coefficient requires that certain assumptions about the data are satisfied. The premise of this test is that the data are a sample of observed points taken from a larger population. We have not examined the entire population because it is not possible or feasible to do so. We are examining the sample to draw a conclusion about whether the linear relationship that we see between \(x\) and \(y\) in the sample data provides strong enough evidence so that we can conclude that there is a linear relationship between \(x\) and \(y\) in the population.

The regression line equation that we calculate from the sample data gives the best-fit line for our particular sample. We want to use this best-fit line for the sample as an estimate of the best-fit line for the population. Examining the scatter plot and testing the significance of the correlation coefficient helps us determine if it is appropriate to do this.

The assumptions underlying the test of significance are:

There is a linear relationship in the population that models the average value of \(y\) for varying values of \(x\). In other words, the expected value of \(y\) for each particular value lies on a straight line in the population. (We do not know the equation for the line for the population. Our regression line from the sample is our best estimate of this line in the population.)
The \(y\) values for any particular \(x\) value are normally distributed about the line. This implies that there are more \(y\) values scattered closer to the line than are scattered farther away. Assumption (1) implies that these normal distributions are centered on the line: the means of these normal distributions of \(y\) values lie on the line.
The standard deviations of the population \(y\) values about the line are equal for each value of \(x\). In other words, each of these normal distributions of \(y\) values has the same shape and spread about the line.
The residual errors are mutually independent (no pattern).
The data are produced from a well-designed, random sample or randomized experiment.

Linear regression is a procedure for fitting a straight line of the form \(\hat{y} = a + bx\) to data. The conditions for regression are:

Linear In the population, there is a linear relationship that models the average value of \(y\) for different values of \(x\).
Independent The residuals are assumed to be independent.
Normal The \(y\) values are distributed normally for any value of \(x\).
Equal variance The standard deviation of the \(y\) values is equal for each \(x\) value.
Random The data are produced from a well-designed random sample or randomized experiment.

The slope \(b\) and intercept \(a\) of the least-squares line estimate the slope \(\beta\) and intercept \(\alpha\) of the population (true) regression line. To estimate the population standard deviation of \(y\), \(\sigma\), use the standard deviation of the residuals, \(s\). \(s = \sqrt{\frac{SEE}{n-2}}\). The variable \(\rho\) (rho) is the population correlation coefficient. To test the null hypothesis \(H_{0}: \rho =\) hypothesized value , use a linear regression t-test. The most common null hypothesis is \(H_{0}: \rho = 0\) which indicates there is no linear relationship between \(x\) and \(y\) in the population. The TI-83, 83+, 84, 84+ calculator function LinRegTTest can perform this test (STATS TESTS LinRegTTest).

Formula Review

Least Squares Line or Line of Best Fit:

\[\hat{y} = a + bx\]

\[a = y\text{-intercept}\]

\[b = \text{slope}\]

Standard deviation of the residuals:

\[s = \sqrt{\frac{SEE}{n-2}}\]

\[SSE = \text{sum of squared errors}\]

\[n = \text{the number of data points}\]

Math Article

Null Hypothesis

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

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

Table of contents:

Comparison with Alternative Hypothesis

Null Hypothesis Definition

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

Null Hypothesis Symbol

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

Null Hypothesis Principle

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

Null Hypothesis Formula

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

The formula for the null hypothesis is:

H 0 : p = p 0

The formula for the alternative hypothesis is:

H a = p >p 0 , < p 0 ≠ p 0

The formula for the test static is:

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

Also, read:

Types of Null Hypothesis

There are different types of hypothesis. They are:

Simple Hypothesis

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

Composite Hypothesis

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

Exact Hypothesis

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

Inexact Hypothesis

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

Null Hypothesis Rejection

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

How do you Find the Null Hypothesis?

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

When is Null Hypothesis Rejected?

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

Null Hypothesis and Alternative Hypothesis

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

Null Hypothesis Examples

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

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

Few more examples are:

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

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

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

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

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

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

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

Ans: Age has no effect on Mathematical skills.

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

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

Frequently Asked Questions on Null Hypothesis

What is meant by the null hypothesis.

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

What are the benefits of hypothesis testing?

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

When a null hypothesis is accepted and rejected?

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

Why is the null hypothesis important?

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

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

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

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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 ...
When Do You Reject the Null Hypothesis? (3 Examples)
A hypothesis test is a formal statistical test we use to reject or fail to reject a statistical hypothesis. We always use the following steps to perform a hypothesis test: Step 1: State the null and alternative hypotheses. The null hypothesis, denoted as H0, is the hypothesis that the sample data occurs purely from chance.
What Is The Null Hypothesis & When To Reject It
When your p-value is less than or equal to your significance level, you reject the null hypothesis. In other words, smaller p-values are taken as stronger evidence against the null hypothesis. Conversely, when the p-value is greater than your significance level, you fail to reject the null hypothesis. In this case, the sample data provides ...
Null Hypothesis: Definition, Rejecting & Examples
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 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.
Null hypothesis
Basic definitions. The null hypothesis and the alternative hypothesis are types of conjectures used in statistical tests to make statistical inferences, which are formal methods of reaching conclusions and separating scientific claims from statistical noise.. The statement being tested in a test of statistical significance is called the null hypothesis. . The test of significance is designed ...
Null and Alternative Hypotheses
H0: 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. 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 ...
9.1 Null and Alternative Hypotheses
They are called the null hypothesis and the alternative hypothesis. These hypotheses contain opposing viewpoints. H0, 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.
Exploring the Null Hypothesis: Definition and Purpose
The word Null in the context of hypothesis testing means "nothing" or "zero.". As an example, if we wanted to test whether there was a difference in two population means based on the calculations from two samples, we would state the Null Hypothesis in the form of: Ho: mu1 = mu2 or mu1- mu2 = 0. In other words, there is no difference, or ...
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.
Hypothesis Testing
Table of contents. Step 1: State your null and alternate hypothesis. Step 2: Collect data. Step 3: Perform a statistical test. Step 4: Decide whether to reject or fail to reject your null hypothesis. Step 5: Present your findings. Other interesting articles. Frequently asked questions about hypothesis testing.
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.
How to Write a Null Hypothesis (5 Examples)
Whenever we perform a hypothesis test, we always write a null hypothesis and an alternative hypothesis, which take the following forms: H0 (Null Hypothesis): Population parameter =, ≤, ≥ some value. HA (Alternative Hypothesis): Population parameter <, >, ≠ some value. Note that the null hypothesis always contains the equal sign.
Hypothesis Testing
Let's return finally to the question of whether we reject or fail to reject the null hypothesis. If our statistical analysis shows that the significance level is below the cut-off value we have set (e.g., either 0.05 or 0.01), we reject the null hypothesis and accept the alternative hypothesis. Alternatively, if the significance level is above ...
Null and Alternative Hypotheses
H0: 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. Ha: The alternative hypothesis: It is a claim about the population that is.
6a.1
The first step in hypothesis testing is to set up two competing hypotheses. The hypotheses are the most important aspect. If the hypotheses are incorrect, your conclusion will also be incorrect. The two hypotheses are named the null hypothesis and the alternative hypothesis. The null hypothesis is typically denoted as H 0.
PDF Introduction to Hypothesis Testing
For example, one hypothesis might claim that the wages of men and women are equal, while the alternative might claim that men make more than women. C. The hypothesis actually to be tested is usually given the symbol H0, and is commonly referred to as the null hypothesis. As is explained more below, the null hypothesis is
PDF The Null and the Alternative Hypotheses
In hypothesis testing there are two mutually exclusive hypotheses; the Null Hypothesis (H0) and the Alternative Hypothesis (H1). One of these is the claim to be tested and based on the sampling results (which infers a similar measurement in the population), the claim will either be supported or not. The claim might be that the population ...
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. It serves as a baseline assumption, positing no observed change or effect occurring.
Null Hypothesis (H0)
The value α, called the level of significance of the test, is usually set in advance, with commonly chosen values being α = 0.10, 0.05, and 0.01. The classical procedure for testing a null hypothesis is to fix a small significance level α and then require that the probability of rejecting H 0 when H 0 is true is less than or equal to α.
Understanding the Null Hypothesis for Linear Regression
xi: The value of the predictor variable xi. Multiple linear regression uses the following null and alternative hypotheses: H0: β1 = β2 = … = βk = 0. HA: β1 = β2 = … = βk ≠ 0. The null hypothesis states that all coefficients in the model are equal to zero. In other words, none of the predictor variables have a statistically ...
11.1: Testing the Hypothesis that β = 0
METHOD 1: Using a p-value to make a decision. To calculate the p-value using LinRegTTEST: On the LinRegTTEST input screen, on the line prompt for β or ρ, highlight " ≠ 0 ". The output screen shows the p-value on the line that reads " p = ". (Most computer statistical software can calculate the p-value .)
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.
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.

What is The Null Hypothesis & When Do You Reject The Null Hypothesis

How to Write a Null Hypothesis

For example, if studying the impact of exercise on weight loss, your null hypothesis might be:

Examples of Null Hypotheses

Why Do We Never Accept The Null Hypothesis?

Why Do We Use The Null Hypothesis?

Purpose of a Null Hypothesis

Do you always need both a Null Hypothesis and an Alternative Hypothesis?

What is the difference between a null hypothesis and an alternative hypothesis?

What are some problems with the null hypothesis?

Why can a null hypothesis not be accepted?

Is a null hypothesis directional or non-directional?

Module 9: Hypothesis Testing With One Sample

Concept Review

Formula Review

9.1 Null and Alternative Hypotheses

Example 9.1

Example 9.2

Example 9.3

Example 9.4

Collaborative Exercise

Exploring the Null Hypothesis: Definition and Purpose

Overview: What is the Null Hypothesis (Ho)?

RELATED: NULL VS. ALTERNATIVE HYPOTHESIS

1. Statistical assurance of determining differences between population parameters

2. Statistically based estimation of the probability of a population distribution

3. Assess the strength of your conclusions as to what to do with the null hypothesis

Why is the Null Hypothesis important to understand?

1. Properly write the null hypothesis to properly capture what you are seeking to prove

2. Frame your statement and select an appropriate alpha risk

3. There are decision errors when deciding on how to respond to the Null Hypothesis

An industry example of using the Null Hypothesis

3 best practices when thinking about the Null Hypothesis

1. Always use the proper nomenclature when stating the Null Hypothesis

2. The Null Hypothesis will always be written as the absence of some parameter or process characteristic

3. Pick a reasonable alpha risk so you’re not always failing to reject the Null Hypothesis

Frequently Asked Questions (FAQ) about the Null Hypothesis

What is an alpha error?

How do I use the alpha error and p-value to decide on what decision I should make about the Null Hypothesis?

Becoming familiar with the Null Hypothesis (Ho)

About the Author

Ken Feldman

Hypothesis Testing (cont...)

Significance levels

One- and two-tailed predictions

Rejecting or failing to reject the null hypothesis

Null and Alternative Hypotheses

Chapter Review

Formula Review

User Preferences

Keyboard Shortcuts

Example 6-1 Section

The Logic of Hypothesis Testing Section

Types of errors

Example 6-1 Cont'd... Section

Try it! Section

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

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Summary – Null Hypothesis and Alternative Hypothesis

FAQs on Null Hypothesis

How to Form a Null Hypothesis?

When Do we reject the Null Hypothesis?

What is a Null Hypothesis in Research?

What Are Alternative and Null Hypotheses?

What Does it Mean to Reject the Null Hypothesis?

How to Find Null Hypothesis?