In the context of hypothesis testing, what is the impact of increasing the sample size $n$ on the $p$-value, assuming the effect size remains constant?

• A. The $p$-value remains unchanged regardless of the sample size.
• B. The $p$-value increases as the sample size $n$ increases, making it harder to detect statistically significant results.
• C. The $p$-value becomes exactly $0$ when the sample size is large.
• D. The $p$-value tends to decrease as the sample size $n$ increases, making it more likely to detect statistically significant results.

Answer: (D)

Verified step by step guidance

1. Understand that the p-value in hypothesis testing measures the probability of observing data as extreme as, or more extreme than, the sample data assuming the null hypothesis is true.
2. Recognize that the test statistic often depends on the sample size \( n \). For many tests, the test statistic is proportional to \( \sqrt{n} \) times the effect size.
3. Since the effect size is constant, increasing the sample size \( n \) increases the test statistic's magnitude, making the observed data less likely under the null hypothesis.
4. A larger test statistic corresponds to a smaller p-value, because the p-value is the tail probability beyond the test statistic value.
5. Therefore, as the sample size increases, the p-value tends to decrease, which increases the likelihood of rejecting the null hypothesis and detecting statistically significant results.