When should you reject the null hypothesis?

Question

BlackTom AI · Accepted Answer

Reject the null hypothesis when the p-value is less than or equal to the chosen significance level α, because the p-value measures how likely the observed data (or more extreme) are if the null hypothesis is true.

1. Relevant concepts
- Null hypothesis (H0): a statement to be tested.
- Significance level (α): the maximum probability of a Type I error (rejecting a true H0) you are willing to accept (commonly 0.05).
- P-value: probability, under H0, of obtaining data as extreme as (or more extreme than) what was observed.
- Decision rule: compare p-value to α to decide whether evidence against H0 is strong enough.

2. Step-by-step reasoning
- Compute the test statistic from sample data and the corresponding p-value.
- Compare p-value to α.
  - If p-value ≤ α, the observed result is sufficiently unlikely under H0, so reject H0.
  - If p-value > α, the observed result is not unlikely enough, so fail to reject H0.
Example: α = 0.05 and p = 0.03. Since 0.03 ≤ 0.05, reject H0. If p = 0.08, 0.08 > 0.05, fail to reject H0.

3. Intermediate/illustrative inequality
- Decision rule written formally: Reject H0 if p ≤ α; Fail to reject H0 if p > α.

4. Why this is correct
A p-value at or below α indicates the data are inconsistent with H0 at the specified significance threshold; rejecting H0 controls the long-run Type I error at α.

5. Why the other options are incorrect
- (a) "The test statistic falls inside the confidence interval": If the null parameter value lies inside a (1 − α) confidence interval, you do not reject H0; rejecting corresponds to the null value being outside the interval.
- (b) "The sample size is sufficiently large": Sample size affects power and precision but by itself does not determine whether H0 is rejected.
- (d) "The p-value is greater than the significance level": That implies insufficient evidence against H0, so you fail to reject H0, not reject it.

Conclusion: The correct answer is c. The p-value is less than or equal to the significance level. This matches the provided answer.

When should you reject the null hypothesis?单项选择题

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