Suppose you are carrying out a paired [math: t]-test for a sample of data with sample size [math: n=25], and you wish to check for normality. Further suppose that, using the paired differences, you have carried out the Shapiro-Wilk test with the following results: Shapiro-Wilk normality test data: x
W = 0.93963, p-value = 0.1451 Using the results of the Shapiro-Wilk test alone, can normality be assumed?

Question

BlackTom AI · Accepted Answer

A Shapiro–Wilk test result with p = 0.1451 indicates there is not enough evidence to reject the null hypothesis that the paired differences come from a normal distribution. The relevant concepts are: (1) Shapiro–Wilk tests H0: data are normally distributed versus H1: not normal; (2) we compare the p-value to a chosen significance level (commonly α = 0.05); (3) failing to reject H0 means no statistical evidence against normality, not proof of normality.

Step-by-step reasoning:
1. Test statistic and p-value reported: W = 0.93963, p = 0.1451.
2. Choose a significance level. Standard choice is α = 0.05.
3. Compare p-value to α: 0.1451 > 0.05.
4. Decision: fail to reject H0 because the p-value exceeds α.
5. Interpretation: the Shapiro–Wilk test does not provide evidence that the paired differences deviate from normality. Therefore it is reasonable, based on this test alone, to assume normality for the paired differences when proceeding with the paired t-test.

Intermediate calculation shown: p = 0.1451 > α (0.05) ⇒ do not reject H0.

Why the correct answer is correct: The Shapiro–Wilk p-value is sufficiently large so the formal test does not flag non-normality. For n = 25 the Shapiro–Wilk is appropriate and so the standard decision rule supports assuming normality.

Why the other option is incorrect: Option b (“No, normality cannot be assumed”) would be chosen only if p ≤ α (e.g., p ≤ 0.05), which is not the case here. Note the caveat: statistical non-rejection is not proof of normality; one should still inspect plots (histogram, Q–Q plot) and consider sample size and test power. Concluding: based solely on the Shapiro–Wilk result, yes, normality can be assumed (answer a).

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