A team of researchers runs a controlled experiment to test whether a new cognitive‑training program improves participants’ memory performance compared to a standard program. After randomly assigning participants to the two programs and measuring their post‑training memory scores, they conduct a statistical test and obtain a p‑value of 0.03. Which of the following interpretations of this p‑value is most appropriate?

Question

BlackTom AI · Accepted Answer

A p-value of 0.03 indicates that, assuming the null hypothesis is true, the observed data are fairly unlikely: about a 3% chance of getting results as extreme (or more extreme) as those seen.

1. Relevant concepts
- Null hypothesis (H0): no effect of the new program on memory.
- P-value: P(observed data or more extreme | H0 true). It is a conditional probability given H0.
- Significance level (α): a pre-chosen threshold (commonly 0.05) for deciding whether to reject H0; p-values are compared to α but are not themselves α.

2. Step-by-step reasoning
- Define H0: the new program has no effect on memory performance; alternative H1: it does.
- Collect data and compute a test statistic summarizing the difference in post-training scores.
- Under H0, determine the sampling distribution of that statistic and compute the probability of observing a statistic at least as extreme as the one obtained.
- That computed probability is the p-value = 0.03.

3. Calculations
- No numeric derivation is required here beyond the reported p-value. Conceptually: p = P(T ≥ t_obs | H0) (or two-sided: P(|T| ≥ |t_obs| | H0)) = 0.03.

4. Why the correct answer is correct
- The provided statement exactly matches the formal definition of a p-value: it conditions on the null being true and gives the probability of seeing data at least as extreme as observed. Thus it correctly interprets p = 0.03.

5. Why common incorrect interpretations are wrong
- “There is a 3% chance the null hypothesis is true” is incorrect: p-values do not give P(H0 | data).
- “There is a 97% chance the new program works” is incorrect for the same reason.
- “This measures effect size or practical importance” is wrong: p-values reflect evidence against H0, not magnitude of effect.
- “It equals the probability the result is due to random chance” is imprecise; more correct is “probability of such data under H0.”

6. Conclusion
The correct interpretation is: If the new program truly had no effect (H0 true), the probability of observing differences at least as large as those seen is about 3% — which matches the provided answer.

类似问题

Quackers conducted his first study! His p-value is .063. What can he conclude?

Quackers conducted his first study! His p-value is .027. What can he conclude?

Statistical tests such as a t-test generate a p-value. The p-value is best described as the probability of which of the following?

What does a p-value of 0.85 mean?

这句话是真还是假？ “所有假设检验中，符号感值必须达到0.05，以确保实验间的一致性。”

True or False: The p-value represents the probability that you will correctly reject the null hypothesis when it is truly incorrect. (Assume no other sources of the following: bias, confounding, effect modification or measurement error)

What does a p-value represent in hypothesis testing?

Finding a small value of the p-value (e.g. less than 5%)

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