What Is Accept or Reject Null Hypothesis?
Every hypothesis test ends with a decision about the null hypothesis. You reject it when the evidence against it is strong enough, which means the p-value is at most alpha or the test statistic falls beyond the critical value.
Choose whether you have a p-value or a test statistic. For a statistic, select the distribution, enter the degrees of freedom and pick the tail. The calculator returns the p-value, the critical value, the decision and how far the p-value sits from alpha.
Strictly speaking you never accept the null hypothesis. You fail to reject it, which means the data did not give enough evidence against it. That is not proof that the null hypothesis is true.
How to Use the Accept or Reject Null Hypothesis Calculator
- Choose whether you have a p-value or a test statistic.
- For a statistic, pick z, t, chi-square or F and enter the degrees of freedom.
- Enter alpha and read the decision.
Accept or Reject Null Hypothesis Formula
p-value rule
Critical value rule (two-tailed z)
Accept or Reject Null Hypothesis Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| H₀ | The null hypothesis | |
| α | Significance level, chosen before the test | Commonly 0.05 |
| p-value | Probability of a result at least this extreme if H₀ is true | |
| Critical value | The cutoff beyond which H₀ is rejected | |
| df | Degrees of freedom of the test |
Accept or Reject Null Hypothesis Worked Example
With the example values loaded in the calculator above, the steps are:
- Test statistic = 2.31; p-value = 0.032284; critical value = ± 2.093.
- Reject H₀ when the p-value is at most α (equivalently, when the statistic falls in the rejection region).
- p ≤ α, so reject the null hypothesis.
Result: Decision = Reject H₀; p-value = 0.032284.
Worked example
A t statistic of 2.31 with 19 degrees of freedom, two-tailed, gives p ≈ 0.032. At α = 0.05 the critical value is ±2.093 and 2.31 lies beyond it, so you reject the null hypothesis.
At α = 0.01 the critical value is ±2.861, so the same data would fail to reject. The decision depends on alpha, which is why it must be chosen first.
Why “fail to reject” and not “accept”
A test can only find evidence against the null hypothesis. A large p-value may reflect a small sample or noisy data rather than a true null, so the careful wording is fail to reject.
Where Accept or Reject Null Hypothesis Is Used
Finishing a hypothesis test
Turn a computed statistic into a clear decision.
Exam practice
Check the critical value and p-value for z, t, chi-square and F tests.
Comparing alpha levels
See whether a result survives at 0.05 but not at 0.01.
Common Accept or Reject Null Hypothesis Mistakes to Avoid
- A confidence interval that excludes the null value matches a two-tailed rejection.
- Report the exact p-value, not only the decision.
Accept or Reject Null Hypothesis Terms and Definitions
- Null hypothesis H₀
- The default claim, usually of no effect or no difference.
- Critical value
- The cutoff beyond which the statistic leads to rejection.
Accept or Reject Null Hypothesis FAQs
When do you reject the null hypothesis?
When the p-value is less than or equal to alpha, or equivalently when the test statistic falls beyond the critical value.
Do we ever accept the null hypothesis?
Strictly, no. We fail to reject it, meaning the data did not provide enough evidence against it.
What alpha should I use?
Alpha = 0.05 is common, but fields differ. Set it before collecting or looking at the data.
What if p is exactly equal to alpha?
By the usual rule p ≤ α rejects, but a result that close deserves caution.
Does the tail matter?
Yes. A two-tailed test splits alpha between both tails, so its critical values are larger than a one-tailed test.
What does rejecting the null hypothesis prove?
It shows the data are unlikely if the null were true. It does not prove the alternative or show the effect is large.