Finding a small value of the p-value (e.g. less than 5%)

Question

BlackTom AI · Accepted Answer

A p-value below 5% indicates evidence against the null hypothesis. 1. Relevant knowledge points and concepts - The p-value is the probability, assuming the null hypothesis H0 is true, of obtaining a test statistic as extreme or more extreme than the observed one. - A small p-value means the observed data are unlikely under H0, so it provides evidence against H0. - Common decision rule: if p < α (commonly α = 0.05), reject H0. 2. Step-by-step reasoning - Under H0, compute the test statistic from data (e.g., z or t) and the corresponding p-value. - Compare p to the chosen significance level α. If p < 0.05, the result is considered statistically significant at the 5% level. - "Statistically significant" means the observed outcome is unlikely under H0, so we conclude there is evidence against H0. 3. Example calculation (intermediate steps) - Suppose a two-sided z-test yields p = 0.03. For a two-sided test, p = 2·(1 − Φ(|z|)). Solve for |z|: p/2 = 0.015, so 1 − Φ(|z|) = 0.015 → Φ(|z|) = 0.985 → |z| ≈ 2.17. - Because |z| ≈ 2.17 > 1.96 and p = 0.03 < 0.05, we reject H0. This numeric example shows small p corresponds to a test statistic in the rejection region and thus evidence against H0. 4. Why the correct answer is correct - By definition, a small p-value indicates the data are unlikely under the null model; therefore it counts as evidence against H0. The common threshold 5% is just a convention for deciding when "unlikely" is sufficient to reject H0. 5. Why the other options are incorrect - "Indicates evidence in favor of the null hypothesis." Wrong: a small p-value contradicts H0 rather than supporting it. - "Implies that the t-statistic is less than 1.96." Wrong sign/direction: for a two-sided 5% test, a small p-value corresponds to |t| > 1.96, not < 1.96. - "Will only happen roughly one in twenty samples." Misleading: if H0 is true, p < 0.05 occurs about 5% of the time, but that statement omits the crucial "assuming H0 is true" condition and oversimplifies interpretation; it is not the primary meaning of a small p-value. 6. Conclusion Therefore the correct interpretation is that a small p-value (e.g., < 5%) indicates evidence against the null hypothesis, which matches the provided answer.

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