The null hypothesis and alternative hypothesis are cornerstones in statistics and hypothesis testing. They represent a methodology used to validate or refute claims about a population based on sample data. The process involves assuming the null hypothesis is true and using statistical analysis to test if the observed data fit this assumption. This article delves into the key roles, calculations, and formulation of the null and alternative hypotheses.
Definition: Null and alternative hypotheses
These two hypotheses are used in statistical testing to prove or disprove a theory.
- The null hypothesis always states that there’s no statistically significant relationship between variables.
- The alternative hypothesis states the opposite.
The null hypothesis (H_{0}) assumes there is no significant difference between specified populations or no association among groups. This is often considered the default or status quo hypothesis, indicating no change or effect.
The alternative hypothesis (H_{a} or H_{1}) is the counterpart to the null hypothesis and claims that there is a significant difference or association among groups. It represents a statement of what a statistical hypothesis test is set up to establish.
Null/Alternative hypotheses: Research questions
These hypotheses function as tentative answers to research questions. Therefore, you can’t have an answer to your research questions without confirming or rejecting either hypothesis.
Both hypotheses are tested using statistical tests that compare two population samples/groups. Testing confirms or rejects the hypotheses, by showing whether there’s a relationship between an independent variable and a dependent variable.
The null hypothesis (H_{0})
The null hypothesis states that there’s no statistically significant relationship or effect between variables.
Based on test results, the null hypothesis can be rejected, which means there is a significant relationship between the variables – one affects the other.
When writing about null hypotheses, you can only reject them or fail to reject them. Don’t use expressions like accepting, proving, or disproving.
Depending on sample size and testing method, you can incur errors when determining the validity of null hypotheses.
- A Type I error happens if you reject H_{0} and claim there’s a significant relationship between the variables, even though test results don’t support this claim.
- A Type II error happens if you fail to reject H_{0} and claim there’s no relationship between variables, even though testing proves otherwise.
Examples of null hypotheses
The table below illustrates null hypotheses for their respective research question:
Common statistical tests used to reject H_{0} include:
- t test
- ANOVA
- Pearson correlation
- Linear regression
In the paper’s Methods section, you must indicate which test you used.
The alternative hypothesis (H_{a})
The alternative hypothesis (H_{a} or H_{1}) claims there’s a statistically significant relationship between variables.
Because H_{a} is the opposite of what the null hypothesis claims, accepting H_{a} means rejecting H_{0} and vice versa.
When reporting alternative hypotheses, you can only say that H_{a} is supported or not supported by test data. Don’t use expressions like accept, reject, disprove, confirm, etc.
Examples of alternative hypotheses
The table below shows alternative hypotheses for their respective research question:
Common statistical tests used to support the alternative hypothesis include:
- t test
- ANOVA
- Pearson correlation
- Linear regression
In the paper’s Methods section, you must indicate which test you used.
Null hypothesis vs. alternative hypothesis
Both hypotheses provide possible but mutually exclusive answers to a research question. They can only be rejected or supported through statistical testing.
The differences between them are:
Null hypothesis (H_{0}) | Alternative hypothesis (H_{a}) |
Claims there’s no relationship between variables. | Claims there isa relationship between variables. |
Common expressions used to write it include no relationship, no effect, no difference, no increase, no decrease, and no change. | Common expressions used to write it include a relationship, an effect, a difference, an increase, a decrease, and a change. |
If testing shows there’s a relationship, this is reported as p ≤ α, therefore H_{0} is rejected. | If testing shows there’s a relationship, this is reported as p ≤ α, therefore H_{a} is supported. |
If testing shows there’s no relationship, this is reported as p > α, therefore we fail to reject H_{0}. | If testing shows no relationship, this is reported as p > α, therefore H_{a} is not supported. |
Writing null and alternative hypotheses correctly
To write these hypotheses correctly in your essay, make sure you:
- Write your research question, mentioning both independent and dependent variables
- State the null hypothesis
- State the alternative hypothesis.
For specific tests, use the following wording:
Test name | H_{0} | H_{a} |
Two-sample t test | The mean dependent variable has no effect on sample 1 (µ1) and sample 2 (µ2); µ1 = µ2 | The mean dependent variable has an effect on sample 1 (µ1) and sample 2 (µ2); µ1 ≠ µ2. |
One-way ANOVA | The mean dependent variable has no effect on sample/group 1 (µ1) and sample/group 2 (µ2); µ1 = µ2 | The mean dependent variable (µ1) and sample/group 2 (µ2) are not all equal; µ1 ≠ µ2 ≠ µ3. |
Pearson correlation | There’s no correlation between the independent variable and the dependent variable: ρ = 0. | There's a correlation between the independent variable and ρ = 0 the dependent variable; ρ ≠ 0. |
Simple linear regression | There’s no relationship between the independent variable and the dependent variable; β1 = 0. | There’s a relationship between the independent variable and the dependent variable; β1 ≠ 0. |
Two proportions | The dependent variable expressed as a proportion doesn’t differentiate between sample/group 1 (ρ1) and sample/group 2 (ρ2); ρ1 = ρ2. | The dependent variable expressed as proportion differentiates between sample/group 1 (ρ1) and sample/group 2 (ρ2); p1 ≠ p2. |
FAQs
They’re unproven statements about a research question.
The null hypothesis says there’s a relationship between variables, and the alternative hypothesis claims there isn’t one.
No, these hypotheses are competing statements, so when you test one hypothesis, you automatically test the other.
The mathematical symbol used to write H_{0} is =
For H_{a}, the symbol is ≠
In most cases, one-tailed tests are best.