Null and Alternative Hypotheses  Definitions & Examples
The null and alternative hypotheses are two competing claims that researchers weigh evidence for and against using a statistical test:
 Null hypothesis (H_{0}): There’s no effect in the population.
 Alternative hypothesis (H_{A}): There’s an effect in the population.
The effect is usually the effect of the independent variable on the dependent variable.
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 (H_{0}) answers “No, there’s no effect in the population.” On the other hand, the alternative hypothesis (H_{A}) answers “Yes, there is an effect in the population.”
The null and alternative are always claims about the population. That’s because the goal of hypothesis testing is to make inferences about a population based on a sample. Often, we infer whether there’s an effect in the population by looking at differences between groups or relationships between variables in the sample.
You can use a statistical test to decide whether the evidence favors the null or alternative hypothesis. Each type of statistical test comes with a specific way of phrasing the null and alternative hypothesis. However, the hypotheses can also be phrased in a general way that applies to any test.
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.
Null hypotheses often include phrases such as “no effect”, “no difference”, or “no relationship”. When written in mathematical terms, they always include an equality (usually =, but sometimes ≥ or ≤).
Examples of null hypotheses
The table below gives examples of research questions and null hypotheses. There’s always more than one way to answer a research question, but these null hypotheses can help you get started.
Research question  Null hypothesis (H_{0})  
General  Testspecific  
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 (µ_{1}) and the nonflossing group (µ_{2}) in the population; µ_{1} = µ_{2}. 
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; β_{1} = 0. 
Does daily meditation decrease the incidence of depression?  Daily meditation does not decrease the incidence of depression.*  Twoproportions z test:
The proportion of people with depression in the dailymeditation group (p_{1}) is greater than or equal to the nomeditation group (p_{2}) in the population; p_{1} ≥ p_{2}. 
*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}.
What is an alternative hypothesis?
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.
Research question  Alternative hypothesis (H_{a})  
General  Testspecific  
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 (µ_{1}) and the nonflossing group (µ_{2}) in the population; µ_{1} ≠ µ_{2}. 
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; β_{1} ≠ 0. 
Does daily meditation decrease the incidence of depression?  Daily meditation decreases the incidence of depression.  Twoproportions z test:
The proportion of people with depression in the dailymeditation group (p_{1}) is less than the nomeditation group (p_{2}) in the population; p_{1} < p_{2}. 
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 (H_{0})  Alternative hypotheses (H_{a})  
Definition  A claim that there is no effect in the population.  A claim that there is an effect in the population. 
Also known as  H_{0}  H_{a}
H_{1} 
Typical phrases used 


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 testspecific template sentences. Otherwise, you can use the general template sentences.
General
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.
Testspecific
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 (H_{0})  Alternative hypothesis (H_{a}) 
Twosample t test
or Oneway ANOVA with two groups 
The mean dependent variable does not differ between group 1 (µ_{1}) and group 2 (µ_{2}) in the population; µ_{1} = µ_{2}.  The mean dependent variable differs between group 1 (µ_{1}) and group 2 (µ_{2}) in the population; µ_{1} ≠ µ_{2}. 
Oneway ANOVA with three groups  The mean dependent variable does not differ between group 1 (µ_{1}), group 2 (µ_{2}), and group 3 (µ_{3}) in the population; µ_{1} = µ_{2} = µ_{3}.  The mean dependent variable of group 1 (µ_{1}), group 2 (µ_{2}), and group 3 (µ_{3}) 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; β_{1} = 0.  There is a relationship between independent variable and dependent variable in the population; β_{1} ≠ 0. 
Twoproportions z test  The dependent variable expressed as a proportion does not differ between group 1 (p_{1}) and group 2 (p_{2}) in the population; p_{1} = p_{2}.  The dependent variable expressed as a proportion differs between group 1 (p_{1}) and group 2 (p_{2}) in the population; p_{1} ≠ p_{2}. 
Note: The template sentences above assume that you’re performing onetailed tests. Onetailed tests are appropriate for most studies.
Frequently asked questions about null and alternative hypotheses
 What symbols are used to represent null hypotheses?

The null hypothesis is often abbreviated as H_{0}. When the null hypothesis is written using mathematical symbols, it always includes an equality symbol (usually =, but sometimes ≥ or ≤).
 What symbols are used to represent alternative hypotheses?

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 >).
 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 welldesigned study, the statistical hypotheses correspond logically to the research hypothesis.
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