If the p-value is less than α in a two-tailed test,

Question

BlackTom AI · Accepted Answer

This prompt asks us to evaluate a statement about p-values in a two-tailed test. The given correct answer text is: 'the null hypothesis should be rejected.'

First, recall what a p-value represents: it is the probability, under the null hypothesis, of observing a test statistic as extreme as, or more extreme than, the one observed. In a two-tailed test, 'extreme' refers to deviations in either direction away from the null value.

Now consider the decision rule: if the p-value is less than the pre-specified significance level alpha (α), we reject the null hypothesis. This rule applies regardless of the direction of the observed effect, which is why the test is two-tailed.

Why this leads to rejection when p < α: a small p-value indicates that the observed data are unlikely under H0, suggesting that the null may not adequately explain the data. Setting α before the test controls the probability of a Type I error (falsely rejecting H0) at α, so when the p-value falls below this threshold, the evidence against H0 is considered strong enough to warrant rejection.

Why this statement is correct in the two-tailed context: a two-tailed test looks for unusually large or unusually small test statistics. If either tail yields a p-value portion that accumulates to less than α, the combined p-value is less than α, signaling rejection of H0. The core idea is the same as in one-tailed tests, but the tail(s) considered are both ends of the distribution.

What this reasoning implies about the other potential statements (if any were present): statements claiming that we never reject H0 when p < α would be incorrect, because the conventional decision rule exactly uses p < α to reject. Conversely, statements suggesting we always accept H0 regardless of p-value would run counter to the standard inferential framework and would be misleading. Additionally, it’s important to remember that a p-value does not quantify the probability that H0 is true, nor does it measure practical significance or effect size; it only measures compatibility with H0 under the observed data.

In summary, the logic of rejecting H0 when p < α in a two-tailed test aligns with the defined decision rule, the construction of p-values under the null, and the control of Type I error rate. The statement encapsulates the fundamental test decision criterion, while the surrounding caveats remind us about interpretation and limitations in practice.

If the p-value is less than α in a two-tailed test,单项选择题

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