Given below are two statements: Statement I: The p-value of a test depends on…

Concept

In a fixed hypothesis test, the p-value is calculated from a standardized test statistic. The denominator of that statistic contains the standard error of the estimated effect, which generally decreases as sample size increases; for the same observed effect and variability, changing sample size can therefore change the statistic and the p-value.

Statistical significance measures evidence against a null hypothesis, whereas practical importance concerns the size and real-world meaning of an effect. A small sample usually has lower statistical power, so even an important effect may remain statistically non-significant.

Application

Statement I follows because sample size changes the estimated effect’s standard error and therefore the standardized test statistic used to calculate the p-value. Statement II follows because a small sample has lower power, making a meaningful effect harder to distinguish from sampling variation.

Contrast

The four combinations differ in how they treat these two sampling mechanisms:

• Both statements true: sample size influences the p-value through standard error, and a small sample can fail to detect an important effect.

• Both statements false: this would require both p-values and detection power to be independent of sample size.

• Statement I true and Statement II false: this accepts the standard-error mechanism but rejects the loss of power in small samples.

• Statement I false and Statement II true: this rejects the standard-error mechanism while accepting the loss of power in small samples.

Cross-check

As sample size grows, the standard error of an estimated effect commonly falls and power commonly rises. Thus, small effects can become statistically significant in large samples, while important effects can be missed in small samples.

Result

Therefore, both Statement I and Statement II are true.