Consider the following statistical references. Which one is impossible?

Question

BlackTom AI · Accepted Answer

A t statistic of 0.04 with df = 21 cannot produce a p-value of .001. A t value so close to zero indicates essentially no difference/effect, so the p-value must be very large, not extremely small. 1) Relevant concepts - The p-value is the probability of observing a test statistic at least as extreme as the observed one under the null. For t-tests, extremely small |t| gives large p; extremely large |t| gives small p. - Reported t-tests are usually two-tailed unless stated otherwise. - For χ² with df = 2, the distribution is exponential: P(χ² ≥ x) = e^{-x/2}. - For F with numerator df1 and denominator df2, values < 1 commonly give p-values > 0.5. 2) Step-by-step reasoning and calculations - For t(21) = 0.04, approximate the two-tailed p using the normal (z) approximation because t with df = 21 is very close to normal for such a tiny t: z ≈ 0.04 → one-tailed p ≈ 1 - Φ(0.04) ≈ 1 - 0.51598 = 0.48402. two-tailed p ≈ 2 × 0.48402 = 0.96804. Thus the two-tailed p is about 0.97 (and the one-tailed p would be ≈0.484). Neither is anywhere near .001. - For χ²(2) = 2.36: because χ²(df=2) has CDF = 1 − e^{-x/2}, the upper-tail p = e^{-2.36/2} = e^{-1.18} ≈ 0.307. The reported p = .301 is essentially the same (rounding differences), so this reference is plausible. - For F(2,105) = 0.66: an F value less than 1 generally yields a p-value > 0.5. Reporting p = .518 is consistent with that behavior and therefore plausible. 3) Why the correct answer is correct - A t statistic of 0.04 represents virtually no evidence against the null, so the p-value cannot be as small as .001. The computed two-tailed p ≈ 0.968 (or one-tailed ≈ 0.484) directly contradicts p = .001, making this reference impossible. 4) Why the other options are not incorrect - χ²(2) = 2.36, p ≈ 0.307 (reported .301): mathematically consistent using the exponential form for df = 2. - F(2,105) = 0.66, p = .518: since F < 1, a p-value slightly above .5 is reasonable. Conclusion: The impossible reference is "t(21) = 0.04, p = .001," which matches the provided correct answer.

Consider the following statistical references. Which one is impossible?Single choice

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