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• Null and Alternative Hypotheses | Definitions & Examples

Null & Alternative Hypotheses | Definitions, Templates & Examples

Published on May 6, 2022 by Shaun Turney . Revised on June 22, 2023.

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

• Null hypothesis ( H 0 ): There’s no effect in the population .
• Alternative hypothesis ( H a or H 1 ) : There’s an effect in the population.

Table of contents

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

The null and alternative hypotheses offer competing answers to your research question . When the research question asks “Does the independent variable affect the dependent variable?”:

• The null hypothesis ( H 0 ) answers “No, there’s no effect in the population.”
• The alternative hypothesis ( H a ) answers “Yes, there is an effect in the population.”

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

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

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

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

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

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

You can never know with complete certainty whether there is an effect in the population. Some percentage of the time, your inference about the population will 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 a type II error.

Examples of null hypotheses

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

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

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

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

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

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

Examples of alternative hypotheses

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

Null and alternative hypotheses are similar in some ways:

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

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

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

General template sentences

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

Does independent variable affect dependent variable ?

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

Test-specific template sentences

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

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

If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.

• Normal distribution
• Descriptive statistics
• Measures of central tendency
• Correlation coefficient

Methodology

• Cluster sampling
• Stratified sampling
• Types of interviews
• Cohort study
• Thematic analysis

Research bias

• Implicit bias
• Cognitive bias
• Survivorship bias
• Availability heuristic
• Nonresponse bias
• Regression to the mean

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is used by scientists to test specific predictions, called hypotheses , by calculating how likely it is that a pattern or relationship between variables could have arisen by chance.

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

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

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

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

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

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Module 9: Hypothesis Testing With One Sample

Null and alternative hypotheses, learning outcomes.

• Describe hypothesis testing in general and in practice

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

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

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

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

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

Mathematical Symbols Used in  H 0 and H a :

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

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

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

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

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

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

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

H 0 : μ = 2.0

H a : μ ≠ 2.0

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

• H 0 : μ = 66
• H a : μ ≠ 66

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

H 0 : μ ≥ 5

H a : μ < 5

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

• H 0 : μ ≥ 45
• H a : μ < 45

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

H 0 : p ≤ 0.066

H a : p > 0.066

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

• H 0 : p = 0.40
• H a : p > 0.40

Concept Review

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

Formula Review

H 0 and H a are contradictory.

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

About the null and alternative hypotheses

The null and alternative hypotheses are two mutually exclusive statements about a population. A hypothesis test uses sample data to determine whether to reject the null hypothesis.

One-sided and two-sided hypotheses

Examples of two-sided and one-sided hypotheses.

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Null vs. Alternative Hypothesis

04.28.2023 • 5 min read

Sarah Thomas

Subject Matter Expert

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

In This Article

What Is a Null Hypothesis?

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

Main Differences Between Null & Alternative Hypothesis

Similarities Between Null & Alternative Hypothesis

Hypothesis Testing & Errors

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

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

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

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

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

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

Null Hypothesis

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

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

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

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

Alternative Hypothesis

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

Examples of Null and Alternative Hypotheses

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

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

Example 1: Two-Tailed T-test

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

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

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

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

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

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

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

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

You will run into either 2 outcomes:

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

Reject the null hypothesis in favor of the alternative.

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

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

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

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

Main Differences Between Null and Alternative Hypothesis

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

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

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

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

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

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

Similarities Between Null and Alternative Hypothesis

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

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

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

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

Hypothesis Testing and Errors

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

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

Type I Errors

Type II Errors

Type I Error (ɑ)

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

Type II Error (ꞵ)

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

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Statistics By Jim

Making statistics intuitive

Alternative hypothesis

By Jim Frost

The alternative hypothesis is one of two mutually exclusive hypotheses in a hypothesis test. The alternative hypothesis states that a population parameter does not equal a specified value. Typically, this value is the null hypothesis value associated with no effect , such as zero. If your sample contains sufficient evidence, you can reject the null hypothesis and favor the alternative hypothesis. The alternative hypothesis is often denoted as H 1 or H A .

If you are performing a two-tailed hypothesis test, the alternative hypothesis states that the population parameter does not equal the null hypothesis value. For example, when the alternative hypothesis is H A : μ ≠ 0, the test can detect differences both greater than and less than the null value.

A one-tailed alternative hypothesis can test for a difference only in one direction. For example, H A : μ > 0 can only test for differences that are greater than zero.

• How Hypothesis Tests Work: Significance Levels (Alpha) and P values
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Null Hypothesis and Alternative Hypothesis

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

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

The Null Hypothesis

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

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

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

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

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

The Alternative Hypothesis

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

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

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

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

• Null hypothesis: “ x is equal to y .” Alternative hypothesis “ x is not equal to y .”
• Null hypothesis: “ x is at least y .” Alternative hypothesis “ x is less than y .”
• Null hypothesis: “ x is at most y .” Alternative hypothesis “ x is greater than y .”
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Choosing $H_0$ and $H_a$ in hypothesis testing

There seems to be some ambiguity or contradiction in how to correctly choose the null and alternative hypotheses, both online and in my instructor's notes. I'm trying to figure out if this stems merely from my lack of understanding or if there actually is a disagreement in the scientific community at large. I've seen the following two ideas on choosing $H_0$ and $H_a$

The null hypothesis is the status quo, the state of things already accepted and/or shown to be true by previous data. We assume it to be true and need convincing evidence to reject it. The alternative hypothesis is the one being proposed based on data from the experiment in question, and is assumed to be false unless the data supporting it can convincingly show otherwise.

The null hypothesis is always the one that includes the equality, and the alternative hypothesis is the complement to it. It doesn't matter whether the equality is the status quo or is being claimed by the researcher, it is always $H_0$ .

An example I made up myself for demonstrative purposes, I'm not looking for an actual solution. Only interested in the following hypotheses:

A researcher believes that children in economically disadvantaged areas are more likely to be raised in single-parent homes. He surveys 1000 children from such an area and finds that 317 of them are raised in a single-parent home. Can we conclude with 95% confidence that 30% or more of the children in economically disadvantaged areas are raised in single-parent homes?

What would be the $H_0$ and $H_a$ in this case and why? My professor provided the correct answer (for an equivalent question but with different numbers) to be

$H_0$ : $p >= 0.3$ ; $H_a$ : $p < 0.3$

With the rationale that H0 must include the equality, which in this case is greater or equal to 30% . Her solution than failed to reject the null hypothesis and concluded that the researcher's claim is therefore correct. To me, this seems like assuming the claim to be true and giving it the benefit of the doubt, which is the opposite of what I thought was the correct approach.

A professor in this related question Difference between "at least" and "more than" in hypothesis testing? seemingly took the same approach.

I wish I could talk to my professor about this, but unfortunately, there's a significant language barrier.

• hypothesis-testing

3 Answers 3

Your null hypothesis is $H_0:p=0.3$

The alternative hypothesis is $H_1:p>0.3$

You need to calculate $$p(X\geq317)$$ using $X\sim Bin(1000,0.3)$

Can you finish?

Just to clarify:

• The null hypothesis always has an equal sign and never an inequality symbol
• In this particular example we conclude that $317$ is not in the critical region.

We conclude that in accepting the null hypothesis there is insufficient evidence that the probability is more than $30$%

• $\begingroup$ The question is purely hypothetical, this isn't an actual homework assignment. I'm trying to understand the overall strategy of picking the null and alternative hypotheses. If you check the related question, you'll see that a professor would have picked H1 : p < 0.3 instead. Why? $\endgroup$ –  Egor Commented Apr 9, 2018 at 17:35
• $\begingroup$ It wouldn't make sense to choose this alternative hypothesis, since 317 is greater than the mean, we should be looking in the upper part of the distribution, not the lower $\endgroup$ –  David Quinn Commented Apr 9, 2018 at 17:38
• $\begingroup$ please see my additional comments to my previous answer. $\endgroup$ –  David Quinn Commented Apr 9, 2018 at 18:08
• $\begingroup$ The way my professor explained it in class, we have two hypotheses: p >= 0.3 (since the question states "30% or more") and p < 0.3 as the complement. We must pick the one that has an equality as the null hypothesis, and the other one as the alternative. $\endgroup$ –  Egor Commented Apr 9, 2018 at 18:13
• $\begingroup$ The null hypothesis always states a particular value, in this case $0.3$. The alternative hypothesis is always either $<$ or $>$ or $\neq$ $\endgroup$ –  David Quinn Commented Apr 9, 2018 at 18:16

Both ideas of the null and alternative hypothesis are true. The null hypothesis must always include an equals sign, whether it be $\geq\text{, } \leq\text{, or just}=$. Usually, however, it's just $=$. The alternative hypothesis is what we wish to show.

The null hypothesis in this case is that the proportion of children in economically disadvantaged areas raised in single-parent homes is $30$%.

The alternative hypothesis is that the proportion of children in economically disadvantaged areas raised in single-parent homes is greater than $30$%.

More formally

$$H_0 : p=0.3$$

$$H_a : p \gt 0.3$$

There are two ways you can test this hypothesis if you so wish. Letting $X$ be the number of children raised in single-parent homes, you can use normal approximation to the binomial:

$$P(X\geq317)=1-P(X\lt317)=1-\Phi\left(\frac{316.5-300}{\sqrt{1000\cdot0.3\cdot0.7}}\right)$$

where I used a continuity correction

In R statistical software

You could also, using software, find the exact probability using the standard binomial distribution:

$$P(X\geq317)=\sum_{k=317}^{1000} {1000 \choose k}\cdot0.3^k\cdot0.7^{1000-k}$$

Since $n$ is large, the normal approximation does very well.

At $\alpha=0.05$ we fail to reject the null hypothesis.

• 1 $\begingroup$ You choice is also the one I would have gone with, since for a continuous distribution >= 30% and >30% are the same thing. However, both my professor, and the professor in the linked related question, picked H0 : p = 0.3; H1 : p < 0.3 . That's what I'm trying to understand. $\endgroup$ –  Egor Commented Apr 9, 2018 at 17:42

You always have to choose $H_a$ so that the sample’s estimation fulfills $H_a$.

The reason is that otherwise the rejection rule will always vote for $H_0$ as in the incorrect choice of your professor.

In your case you want to test a probability against $0.3$, the sample’s estimation was $0.37$, hence $H_a\colon p>0.3$ as $0.37>0.3$. And it does in no way matter where the equal-sign occurs as long as you’re dealing with continuous random variables.

• 1 $\begingroup$ Why should we choose the alternative according to the observede data? $\endgroup$ –  Thomas Commented Jun 20, 2022 at 6:36

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9.1 Null and Alternative Hypotheses

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

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

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

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

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

Mathematical Symbols Used in H 0 and H a :

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

Example 9.1

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

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

Example 9.2

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

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

• H 0 : μ __ 66
• H a : μ __ 66

Example 9.3

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

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

• H 0 : μ __ 45
• H a : μ __ 45

Example 9.4

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

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

• H 0 : p __ 0.40
• H a : p __ 0.40

Collaborative Exercise

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

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Alternative Hypothesis: Definition, Types and Examples

In statistical hypothesis testing, the alternative hypothesis is an important proposition in the hypothesis test. The goal of the hypothesis test is to demonstrate that in the given condition, there is sufficient evidence supporting the credibility of the alternative hypothesis instead of the default assumption made by the null hypothesis.

Alternative Hypotheses

Both hypotheses include statements with the same purpose of providing the researcher with a basic guideline. The researcher uses the statement from each hypothesis to guide their research. In statistics, alternative hypothesis is often denoted as H a or H 1 .

Table of Content

What is a Hypothesis?

Alternative hypothesis, types of alternative hypothesis, difference between null and alternative hypothesis, formulating an alternative hypothesis, example of alternative hypothesis, application of alternative hypothesis.

“A hypothesis is a statement of a relationship between two or more variables.” It is a working statement or theory that is based on insufficient evidence.

While experimenting, researchers often make a claim, that they can test. These claims are often based on the relationship between two or more variables. “What causes what?” and “Up to what extent?” are a few of the questions that a hypothesis focuses on answering. The hypothesis can be true or false, based on complete evidence.

While there are different hypotheses, we discuss only null and alternate hypotheses. The null hypothesis, denoted H o , is the default position where variables do not have a relation with each other. That means the null hypothesis is assumed true until evidence indicates otherwise. The alternative hypothesis, denoted H 1 , on the other hand, opposes the null hypothesis. It assumes a relation between the variables and serves as evidence to reject the null hypothesis.

Example of Hypothesis:

Mean age of all college students is 20.4 years. (simple hypothesis).

An Alternative Hypothesis is a claim or a complement to the null hypothesis. If the null hypothesis predicts a statement to be true, the Alternative Hypothesis predicts it to be false. Let’s say the null hypothesis states there is no difference between height and shoe size then the alternative hypothesis will oppose the claim by stating that there is a relation.

We see that the null hypothesis assumes no relationship between the variables whereas an alternative hypothesis proposes a significant relation between variables. An alternative theory is the one tested by the researcher and if the researcher gathers enough data to support it, then the alternative hypothesis replaces the null hypothesis.

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

There are a few types of alternative hypothesis that we will see:

1. One-tailed test H 1 : A one-tailed alternative hypothesis focuses on only one region of rejection of the sampling distribution. The region of rejection can be upper or lower.

• Upper-tailed test H 1 : Population characteristic > Hypothesized value
• Lower-tailed test H 1 : Population characteristic < Hypothesized value

2. Two-tailed test H 1 : A two-tailed alternative hypothesis is concerned with both regions of rejection of the sampling distribution.

3. Non-directional test H 1 : A non-directional alternative hypothesis is not concerned with either region of rejection; rather, it is only concerned that null hypothesis is not true.

4. Point test H 1 : Point alternative hypotheses occur when the hypothesis test is framed so that the population distribution under the alternative hypothesis is a fully defined distribution, with no unknown parameters; such hypotheses are usually of no practical interest but are fundamental to theoretical considerations of statistical inference and are the basis of the Neyman–Pearson lemma.

the differences between Null Hypothesis and Alternative Hypothesis is explained in the table below:

Formulating an alternative hypothesis means identifying the relationships, effects or condition being studied. Based on the data we conclude that there is a different inference from the null-hypothesis being considered.

• Understand the null hypothesis.
• Consider the alternate hypothesis
• Choose the type of alternate hypothesis (one-tailed or two-tailed)

Alternative hypothesis must be true when the null hypothesis is false. When trying to identify the information need for alternate hypothesis statement, look for the following phrases:

• “Is it reasonable to conclude…”
• “Is there enough evidence to substantiate…”
• “Does the evidence suggest…”
• “Has there been a significant…”

When alternative hypotheses in mathematical terms, they always include an inequality ( usually ≠, but sometimes < or >) . When writing the alternate hypothesis, make sure it never includes an “=” symbol.

To help you write your hypotheses, you can use the template sentences below.

Does independent variable affect dependent variable?

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

Various examples of Alternative Hypothesis includes:

Two-Tailed Example

• Research Question : Do home games affect a team’s performance?
• Null-Hypothesis: Home games do not affect a team’s performance.
• Alternative Hypothesis: Home games have an effect on team’s performance.
• Research Question: Does sleeping less lead to depression?
• Null-Hypothesis: Sleeping less does not have an effect on depression.
• Alternative Hypothesis : Sleeping less has an effect on depression.

One-Tailed Example

• Research Question: Are candidates with experience likely to get a job?
• Null-Hypothesis: Experience does not matter in getting a job.
• Alternative Hypothesis: Candidates with work experience are more likely to receive an interview.
• Alternative Hypothesis : Teams with home advantage are more likely to win a match.

Some applications of Alternative Hypothesis includes:

• Rejecting Null-Hypothesis : A researcher performs additional research to find flaws in the null hypothesis. Following the research, which uses the alternative hypothesis as a guide, they may decide whether they have enough evidence to reject the null hypothesis.
• Guideline for Research : An alternative and null hypothesis include statements with the same purpose of providing the researcher with a basic guideline. The researcher uses the statement from each hypothesis to guide their research.
• New Theories : Alternative hypotheses can provide the opportunity to discover new theories that a researcher can use to disprove an existing theory that may not have been backed up by evidence.

We defined the relationship that exist between null-hypothesis and alternative hypothesis. While the null hypothesis is always a default assumption about our test data, the alternative hypothesis puts in all the effort to make sure the null hypothesis is disproved.

Null-hypothesis always explores new relationships between the independent variables to find potential outcomes from our test data. We should note that for every null hypothesis, one or more alternate hypotheses can be developed.

Also Check:

Mathematics Maths Formulas Branches of Mathematics

FAQs on Alternative Hypothesis

What is hypothesis.

A hypothesis is a statement of a relationship between two or more variables.” It is a working statement or theory that is based on insufficient evidence.

What is an Alternative Hypothesis?

Alternative hypothesis, denoted by H 1 , opposes the null-hypothesis. It assumes a relation between the variables and serves as an evidence to reject the null-hypothesis.

What is the Difference between Null-Hypothesis and Alternative Hypothesis?

Null hypothesis is the default claim that assumes no relationship between variables while alternative hypothesis is the opposite claim which considers statistical significance between the variables.

What is Alternative and Experimental Hypothesis?

Null hypothesis (H 0 ) states there is no effect or difference, while the alternative hypothesis (H 1 or H a ) asserts the presence of an effect, difference, or relationship between variables. In hypothesis testing, we seek evidence to either reject the null hypothesis in favor of the alternative hypothesis or fail to do so.

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What is the alternate hypothesis of the null hypothesis β_1=β_2=β_3=0?

Usually with simple hypotheses I will have something like

$$H_0: \beta_1 = 0 | H_A: \beta_1 \ne 0$$

But suppose I have a null hypotheis

$$H_0: \beta_1 = \beta_2 = \beta_3 = 0$$

What is the alternative hypothesis? Is there an assumed one or could there be multiple plausible one's and it is up to the tester to specify?

My econometrics professor is super hand wavy and just said $\text{Not} H_0$ was the alternate. But that seems ridiculous to me.

Possibility 1:

$$H_A: \beta_1 = \beta_2 = \beta_3 \ne 0$$ Possibility 2:

$$H_A: \beta_1 \ne 0, \beta_2 \ne 0, \beta_3 \ne 0$$

There are obviously more possibilities, but these are enough to illustrate my point.

But these are like completely different statements. Does this mean I have to specify the alternate hypothesis and there isn't a given/assumed one unlike simple hypotheses? Does this just mean my professor did a bad job?

• hypothesis-testing

• 1 $\begingroup$ I think you missed a few possibilities there: eg $\beta_1 = \beta_2 = 0, \beta_3 \ne 0$ $\endgroup$ –  Kitter Catter Commented May 26, 2016 at 1:59
• $\begingroup$ Yes, you have to specify the alternate. Indeed, the complement of the null is determined by the set of all distributions you are considering, so your professor merely pushed the answer one step back without saying anything informative: you could have replied, "complement of the null in which space of distributions ?" The dialog at stats.stackexchange.com/a/130772/919 illustrates some of the thought processes that go into specifying a null and alternate hypothesis. It purposely uses a setting in which the alternate is unconventional. $\endgroup$ –  whuber ♦ Commented May 26, 2016 at 2:07

3 Answers 3

The general alternative to complete equality (a point null) is "at least one beta is not equal to zero", which can be expressed in all manner of alternative ways.

If you have a more specific alternative than that you should give it specifically, and if at all possible choose a test statistic that relates to that particular alternative instead of a much more general one (this will improve power).

• $\begingroup$ +1 The reference to power is key: it is a hint as to why the "obvious" logical answer (the alternative is the logical negation of the null) which--even were it well-defined (which it is not, at least until the full set of distributions is specified)--can be exceptionally poor because it can lead to ineffective tests. $\endgroup$ –  whuber ♦ Commented May 26, 2016 at 2:55
• 1 $\begingroup$ @whuber I don't understand your comment. What is exceptionally poor about the negation of the null? $\endgroup$ –  Stan Shunpike Commented May 26, 2016 at 3:39
• $\begingroup$ Stan: Until the full set of distributions--that is, the complete underlying parametric model--has been described, negating the null is at best ambiguous. This is a crucial error. Consider a classical t-test with $H_0:\mu = 0$. When the set of distributions is parameterized by $\mu\in[0,\infty)$, the alternative is one-sided , whereas when $\mu\in(-\infty,\infty)$ the alternative is two-sided. Thus, those who provide merely logical arguments, and suggest thereby that $H_A$ is uniquely determined by $H_0$, are guilty of ignoring such essential distinctions. $\endgroup$ –  whuber ♦ Commented May 26, 2016 at 15:13

The alternative hypothesis is: $\beta_{1} \ne 0$ OR $\beta_{2} \ne 0$ OR $\beta_{3} \ne 0$ OR ($\beta_{1} \ne 0$ AND $\beta_{2} \ne 0$) OR ($\beta_{0} \ne 0$ AND $\beta_{3} \ne 0$) OR ($\beta_{2} \ne 0$ AND $\beta_{3} \ne 0$) OR ( $\beta_{2} \ne 0$ AND $\beta_{2} \ne 0$ AND $\beta_{3} \ne 0$).

More simply: at least one of the $\beta$s is not equal to $0$ .

• 3 $\begingroup$ Perhaps you're including conditions 4 through 7 to try to be explicit/complete, but from a propositional logic perspective, they're entirely superfluous. $\endgroup$ –  Matthew Gunn Commented May 26, 2016 at 5:00
• 1 $\begingroup$ @MatthewGunn From a substantive research perspective, they motivate subsequent post hoc tests. ;) $\endgroup$ –  Alexis Commented May 26, 2016 at 20:00

The alternative hypothesis is β1≠0 OR β2≠0 OR β3≠0, via De Morgan's Laws. Proving any of those three conditions would disprove the null hypothesis.

The breakdown β1≠β2 OR β2≠β3 OR β3≠0 is mathematically equivalent but might be easier to show significance or design an experiment for.

• $\begingroup$ Yes, $\neg\left( (\beta_1 = 0) \wedge (\beta_2 = 0) \wedge (\beta_3 = 0)\right) =( \beta_1 \neq 0) \vee (\beta_2 \neq 0) \vee (\beta_3 \neq 0)$. The two are entirely equivalent, and in my opinion, there's no reason to prefer one way of writing it over the other. $\endgroup$ –  Matthew Gunn Commented May 26, 2016 at 5:11

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