In statistics, "statistical significance" means that the null hypothesis is ...

Question

BlackTom AI · Accepted Answer

Statistical significance indicates that the observed data are unlikely under the null hypothesis, so the null hypothesis is rejected.

1. Relevant knowledge points and concepts
- Null hypothesis (H0): a baseline statement (e.g., no effect, no difference).
- Alternative hypothesis (H1): the claim we consider if H0 seems implausible.
- Test statistic and p-value: measures how extreme the observed data are under H0.
- Significance level (α): a prechosen threshold (commonly 0.05) for deciding when to reject H0.
- Decision rule: if p-value ≤ α, reject H0; if p-value > α, fail to reject H0.
- Important nuance: statistical significance ≠ practical importance and does not “prove” H1 true; it only provides evidence against H0.

2. Complete reasoning step by step
- Formulate H0 and H1.
- Compute the test statistic from the sample.
- Determine the p-value: the probability of observing data at least as extreme as the sample, assuming H0 is true.
- Compare p-value to α. If p-value is less than or equal to α, the result is called statistically significant; by convention this leads to rejecting H0.

3. Example with intermediate steps
- Suppose α = 0.05. Calculate test statistic and find p-value = 0.03.
- Compare: 0.03 ≤ 0.05 → decision = reject H0.
- If instead p-value = 0.12, 0.12 > 0.05 → fail to reject H0 (not statistically significant).

4. Why the correct answer is correct
- The formal definition: “statistical significance” refers to data that are sufficiently unlikely under H0 to cross the chosen α threshold; this triggers the decision rule to reject H0. Thus “rejected” is the correct characterization.

5. Why other possible phrasings are incorrect
- “Accepted” or “proven true” is incorrect because statistical tests do not prove H0 or H1; they only fail to reject or reject H0.
- “Not rejected” is what you say when results are not statistically significant, the opposite of the correct answer.
- Saying significance means the effect is important (practical significance) is incorrect; significance is purely a probabilistic threshold.

6. Conclusion
The correct interpretation of “statistical significance” is that the null hypothesis is rejected. This matches the provided answer: rejected.

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