Two-tailed p-value for a z-test.

Two-tailed p-value for a z-test.

• A. P-value equals Φ(z)
• B. P-value equals 1 − Φ(z)
• C. Two-tailed p-value equals 2Φ(z) − 1
• D. P-value equals Φ(−|z|)

Answer: (C)

In a two-tailed z-test, you’re looking at how extreme the observed z-statistic is in either direction under the null hypothesis. The p-value is the probability of getting a value as extreme or more extreme in either tail. For a standard normal Z, that two-tailed p-value is the tail area beyond the absolute value of the observed z: p = P(|Z| ≥ |z|) = 2Φ(-|z|) = 2(1 − Φ(|z|)). If you only look at the probability that Z falls between −|z| and |z|, that central probability is Φ(|z|) − Φ(−|z|) = 2Φ(|z|) − 1, which is not the p-value in the standard definition; it represents the area inside the interval, not the combined tail area beyond it. So the conventional two-tailed p-value uses the tail regions and is 2Φ(−|z|). The expression 2Φ(z) − 1 corresponds to the central probability within ±z, not the p-value itself.

In a two-tailed z-test, you’re looking at how extreme the observed z-statistic is in either direction under the null hypothesis. The p-value is the probability of getting a value as extreme or more extreme in either tail. For a standard normal Z, that two-tailed p-value is the tail area beyond the absolute value of the observed z: p = P(|Z| ≥ |z|) = 2Φ(-|z|) = 2(1 − Φ(|z|)).

If you only look at the probability that Z falls between −|z| and |z|, that central probability is Φ(|z|) − Φ(−|z|) = 2Φ(|z|) − 1, which is not the p-value in the standard definition; it represents the area inside the interval, not the combined tail area beyond it.

So the conventional two-tailed p-value uses the tail regions and is 2Φ(−|z|). The expression 2Φ(z) − 1 corresponds to the central probability within ±z, not the p-value itself.