For a given sample, the P-value for upper tail test is x, and the P-value for two-tailed test is y, then

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BlackTom AI · Accepted Answer

The question compares the P-values for an upper tail test (x) and a two-tailed test (y) for the same observed statistic.
Option 1: x = y. This would only occur in very special circumstances where the two-tailed test does not double the one-tailed area, which does not generally hold for symmetric distributions like Z or t when the statistic has a symmetric null distribution. In practice, the two-tailed P-value typically incorporates both tails, so it is not equal to the single upper tail P-value.
Option 2: x < y. For many standard test statistics with a symmetric null distribution (e.g., Z, t) the two-tailed P-value is computed as y = 2 * P(Z ≥ z_obs) when the observed statistic lies in the upper tail, which makes y roughly twice as large as the upper-tail P-value x. Therefore, except in degenerate cases, the two-tailed P-value is larger, so x is smaller than y. This aligns with the usual relationship between one-sided and two-sided P-values for symmetric distributions.
Option 3: The relative size of x and y depends on α. The P-values are functions of the observed statistic, not the significance level α. While α influences decision rules (e.g., comparing P to α to reject), it does not change the mathematical relationship between the computed P-values x and y for a given data result. Thus this statement misstates what α controls in the calculation.
Option 4: None of the other three. Given the standard relationship described in Option 2, this choice would be incorrect because there is a well-known general comparison between one-sided and two-sided P-values for the usual symmetric tests.
Therefore, considering how one-sided and two-sided P-values are derived for symmetric null distributions, the typical ordering is that the upper-tail P-value is smaller than the corresponding two-tailed P-value, with y usually equal to approximately 2x (for a nontrivial observed effect).

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For a given sample, the P-value for upper tail test is x, and the P-value for two-tailed test is y, then

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For a given sample, the P-value for upper tail test is x, and the P-value for two-tailed test is y, then

View Explanation

Log in for full answers

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