Explanation

When you run a hypothesis test, you start with a null hypothesis (H0) that represents the status quo or no effect. The p‑value is the probability of seeing data at least as extreme as what you observed, assuming H0 is true. In symbols,

$$ p\text{-value} = P(\text{data} \ge \text{observed} \mid H_0). $$

If this probability is very small, it suggests that the observed data would be unlikely under H0, giving you reason to consider rejecting H0. The conventional threshold (α) is often set at 0.05, meaning you would reject H0 if the p‑value is below 5%.

Importantly, a p‑value is not the probability that H0 is true, nor is it the probability of making a mistake. It only quantifies how surprising the data are under the null model. You should interpret a small p‑value as evidence against H0, but you still need to consider effect size, study design, and the broader context before drawing conclusions.

Naive Response

I think a p‑value tells you how likely the null hypothesis is true. If the p‑value is low, you just reject the null and accept the alternative. Usually you compare it to 0.05 and that’s all you need to decide.

Good Response

I start by stating the null hypothesis (H0) and the alternative (H1). The p‑value is the probability of obtaining data at least as extreme as what we observed, assuming H0 holds, which I write as $$ p = P(\text{data} \ge \text{observed} \mid H_0). $$

If the p‑value is below a pre‑chosen significance level α (commonly 0.05), I consider the evidence against H0 strong enough to reject it. However, I remind myself that the p‑value does not give the probability that H0 is true, nor does it measure the size of the effect. I also check the practical significance, confidence intervals, and whether the test assumptions (e.g., independence, normality) are met before making a final decision.

Why the Good Response is Better

The good response correctly defines the p‑value as a conditional probability under H0, using proper notation.
It emphasizes the distinction between statistical significance and practical relevance, and warns against common misinterpretations.
It mentions the role of the significance level, test assumptions, and complementary information (effect size, confidence intervals), providing a more complete decision framework.

135. Interpreting P Value

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