If a test statistic follows a standard normal distribution under H0, which distribution is used to compute a two-sided p-value?

If a test statistic follows a standard normal distribution under H0, which distribution is used to compute a two-sided p-value?

• A. t distribution.
• B. Chi-square distribution.
• C. F distribution.
• D. Standard normal distribution (Z distribution).

Answer: (D)

When the null distribution of the test statistic is standard normal, you use the standard normal distribution to compute the two-sided p-value because the p-value is the probability, under H0, of observing a result as extreme or more extreme in either direction. If you observe a value z0, the two-sided p-value is P(|Z| ≥ |z0|) with Z ~ N(0,1). Because the standard normal is symmetric, this equals 2 * P(Z ≥ |z0|) = 2 * [1 − Φ(|z0|)], where Φ is the standard normal CDF. This captures extreme values in both tails, which is what a two-sided test looks for. Other distributions (like the t, chi-square, or F) are used in different testing scenarios, but when the statistic under H0 follows a standard normal, the two-sided p-value is computed from that standard normal distribution.

When the null distribution of the test statistic is standard normal, you use the standard normal distribution to compute the two-sided p-value because the p-value is the probability, under H0, of observing a result as extreme or more extreme in either direction.

If you observe a value z0, the two-sided p-value is P(|Z| ≥ |z0|) with Z ~ N(0,1). Because the standard normal is symmetric, this equals 2 * P(Z ≥ |z0|) = 2 * [1 − Φ(|z0|)], where Φ is the standard normal CDF. This captures extreme values in both tails, which is what a two-sided test looks for.

Other distributions (like the t, chi-square, or F) are used in different testing scenarios, but when the statistic under H0 follows a standard normal, the two-sided p-value is computed from that standard normal distribution.