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

• A. The $p$-value is unaffected by changes in sample size.
• B. The $p$-value increases as the sample size increases, making it harder to detect statistically significant results.
• C. The $p$-value tends to decrease as the sample size increases, making it more likely to detect statistically significant results.
• D. The $p$-value always equals the significance level regardless of sample size.

Answer: (C)

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 the square root of the sample size, for example, \(Z = \frac{\bar{X} - \mu_0}{\sigma / \sqrt{n}}\) where \(\bar{X}\) is the sample mean, \(\mu_0\) is the hypothesized mean, and \(\sigma\) is the population standard deviation.
3. Note that as the sample size \(n\) increases, the denominator \(\sigma / \sqrt{n}\) decreases, which tends to increase the absolute value of the test statistic if the effect size (difference between \(\bar{X}\) and \(\mu_0\)) remains constant.
4. Since the p-value is calculated based on the test statistic, a larger test statistic generally leads to a smaller p-value, indicating stronger evidence against the null hypothesis.
5. Therefore, increasing the sample size while keeping the effect size constant tends to decrease the p-value, making it easier to detect statistically significant results.