Null and Alternative Hypotheses Explained
What the null and alternative hypotheses are, how to write them, one- vs two-tailed, and Type I and Type II errors — with a worked example.
6 min read · Reviewed August 2026
What the null and alternative hypotheses are
Every hypothesis test starts with two competing statements. The null hypothesis (H₀) is the default position of “no effect” or “no difference”. The alternative hypothesis (H₁ or Hₐ) is the claim you are actually trying to find evidence for.
A test never “proves” the alternative. Instead it asks whether the data gives enough evidence to reject the null. If it does, you accept the alternative; if not, you fail to reject the null.
How to write them
The null always contains equality; the alternative captures what you want to detect. For a test about a population mean μ against a target value μ₀:
| Test | Null (H₀) | Alternative (H₁) |
|---|---|---|
| Two-tailed | μ = μ₀ | μ ≠ μ₀ |
| One-tailed (greater) | μ ≤ μ₀ | μ > μ₀ |
| One-tailed (less) | μ ≥ μ₀ | μ < μ₀ |
Type I and Type II errors
Because a test decides from limited data, it can be wrong in two ways:
| Error | What happens | Probability |
|---|---|---|
| Type I | Reject a true null (false positive) | α (significance level) |
| Type II | Fail to reject a false null (false negative) | β |
The testing process
- State H₀ and H₁ and choose a significance level α (often 0.05).
- Collect data and compute a test statistic (e.g. a t- or z-score).
- Find the p-value for that statistic.
- Reject H₀ if p ≤ α; otherwise fail to reject it.
- State the conclusion in the context of the problem.
A worked example
A factory claims its bags weigh 500 g. You suspect they are underfilled. You would write H₀: μ ≥ 500 and H₁: μ < 500 (a one-tailed test). You weigh a sample, compute a t-statistic and its p-value, and if that p-value is at or below 0.05 you reject H₀ and conclude the bags are underfilled.
Frequently asked questions
What is a null hypothesis in simple terms?
It is the default assumption that there is no effect, no difference, or no relationship. A hypothesis test looks for evidence strong enough to reject this default in favour of the alternative.
What is the difference between the null and alternative hypothesis?
The null hypothesis (H₀) states there is no effect and always contains an equality. The alternative hypothesis (H₁) is the claim you want to support — that there is a difference, an increase, or a decrease.
Can you prove the null hypothesis?
No. You can only fail to reject it. Failing to reject the null means the data did not provide enough evidence against it — not that it is definitely true.