Null & Alternative Hypotheses | Definitions, Templates & Examples

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

  • Null hypothesis (H0): There’s no effect in the population.
  • Alternative hypothesis (Ha or H1): There’s an effect in the population.

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

Answering your research question with 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 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.

What is a null hypothesis?

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.

Example: Population on trial
Think of a statistical test as being like a legal trial. The population is accused of the “crime” of having an effect, and the sample is the criminal evidence. In the United States and many other countries, a person accused of a crime is assumed to be innocent until proven guilty. Similarly, we start by assuming the population is “innocent” of having an effect.

In other words, the null hypothesis (i.e., that there is no effect) is assumed to be true until the sample provides enough evidence to reject it.

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.

Research question Null hypothesis (H0)
General Test-specific
Does tooth flossing affect the number of cavities? Tooth flossing has no effect on the number of cavities. t test:

The mean number of cavities per person does not differ between the flossing group (µ1) and the non-flossing group (µ2) in the population; µ1 = µ2.

Does the amount of text highlighted in the textbook affect exam scores? The amount of text highlighted in the textbook has no effect on exam scores. Linear regression:

There is no relationship between the amount of text highlighted and exam scores in the population; β1 = 0.

Does daily meditation decrease the incidence of depression? Daily meditation does not decrease the incidence of depression.* Two-proportions z test:

The proportion of people with depression in the daily-meditation group (p1) is greater than or equal to the no-meditation group (p2) in the population; p1p2.

*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 p1 = p2.

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What is an alternative hypothesis?

The alternative hypothesis (Ha) 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.

Tip
Be careful with your words when you report the results of a statistical test in a research paper or thesis. If you reject the null hypothesis, you can say that the alternative hypothesis is supported. On the other hand, if you fail to reject the null hypothesis, then you can say that the alternative hypothesis is not supported. Never say that you’ve proven or disproven a hypothesis.

Alternative hypotheses often include phrases such as “an effect,” “a difference,” or “a relationship.” When null 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.

Research question Alternative hypothesis (Ha)
General Test-specific
Does tooth flossing affect the number of cavities? Tooth flossing has an effect on the number of cavities. t test:

The mean number of cavities per person differs between the flossing group (µ1) and the non-flossing group (µ2) in the population; µ1 ≠ µ2.

Does the amount of text highlighted in a textbook affect exam scores? The amount of text highlighted in the textbook has an effect on exam scores. Linear regression:

There is a relationship between the amount of text highlighted and exam scores in the population; β1 ≠ 0.

Does daily meditation decrease the incidence of depression? Daily meditation decreases the incidence of depression. Two-proportions z test:

The proportion of people with depression in the daily-meditation group (p1) is less than the no-meditation group (p2) in the population; p1 < p2.

Similarities and differences between null and alternative hypotheses

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.

Null hypotheses (H0) Alternative hypotheses (Ha)
Definition A claim that there is no effect in the population. A claim that there is an effect in the population.
Also known as H0 Ha

H1

Typical phrases used
  • No effect
  • No difference
  • No relationship
  • No change
  • Does not increase
  • Does not decrease
  • An effect
  • A difference
  • A relationship
  • A change
  • Increases
  • Decreases
Symbols used Equality symbol (=, ≥, or ≤) Inequality symbol (≠, <, or >)
p ≤ α Rejected Supported
p > α Failed to reject Not supported

How to write null and alternative hypotheses

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 (H0): Independent variable does not affect dependent variable.
  • Alternative hypothesis (Ha): 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.

Statistical test Null hypothesis (H0) Alternative hypothesis (Ha)
Two-sample t test 

or

One-way ANOVA with two groups

The mean dependent variable does not differ between group 11) and group 22) in the population; µ1 = µ2. The mean dependent variable differs between group 11), group 22) in the population; µ1 ≠ µ2.
One-way ANOVA with three groups The mean dependent variable does not differ between group 11), group 22), and group 33) in the population; µ1 = µ2 = µ3. The mean dependent variable of group 11), group 22), and group 33) are not all equal in the population.
Pearson correlation There is no correlation between independent variable and dependent variable in the population; ρ = 0. There is a correlation between independent variable and dependent variable in the population; ρ ≠ 0.
Simple linear regression There is no relationship between independent variable and dependent variable in the population; β1 = 0. There is a relationship between independent variable and dependent variable in the population; β1 ≠ 0.
Two-proportions z test The dependent variable expressed as a proportion does not differ between group 1 (p1) and group 2 (p2) in the population; p1 = p2. The dependent variable expressed as a proportion differs between group 1 (p1) and group 2 (p2) in the population; p1p2.

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

Frequently asked questions

What is hypothesis testing?

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.

What are null and alternative hypotheses?

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.

What symbols are used to represent null hypotheses?

The null hypothesis is often abbreviated as H0. When the null hypothesis is written using mathematical symbols, it always includes an equality symbol (usually =, but sometimes ≥ or ≤).

What symbols are used to represent alternative hypotheses?

The alternative hypothesis is often abbreviated as Ha or H1. When the alternative hypothesis is written using mathematical symbols, it always includes an inequality symbol (usually ≠, but sometimes < or >).

What’s the difference between a research hypothesis and a statistical hypothesis?

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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Shaun Turney

During his MSc and PhD, Shaun learned how to apply scientific and statistical methods to his research in ecology. Now he loves to teach students how to collect and analyze data for their own theses and research projects.