A two-sided independent samples t-test with n1=33 and n2=38 yielded a test statistic of 2.03. Assuming unequal variances, which of the following is the correct p-value for this hypothesis test? Be sure to find the p-value using the t-distribution probability table.

Question

A two-sided independent samples t-test with n1=33 and n2=38 yielded a test statistic of 2.03. Assuming unequal variances, which of the following is the correct p-value for this hypothesis test? Be sure to find the p-value using the t-distribution probability table.

BlackTom AI · Accepted Answer

Context: we’re evaluating a two-sided independent samples t-test with n1 = 33 and n2 = 38, t = 2.03, and the assumption of unequal variances (Welch t-test). The p-value depends on the degrees of freedom used for the t-distribution, which, under unequal variances, is approximated by the Welch–Satterthwaite formula and typically ends up somewhere between the smaller sample size minus one and the combined total minus two.

Option A: 0.01 < p-value < 0.025
This would correspond to a fairly small p-value, implying stronger evidence against the null hypothesis. Given t = 2.03 with moderate to large degrees of freedom, the two-sided p-value is unlikely to be this small unless the df are very low. Since our df are not extremely small (n1-1 = 32, n2-1 = 37), this range is not the most plausible.

Option B: 0.05 < p-value < 0.10
A p-value in this interval would indicate weaker evidence against the null, typically associated with t-values near 1.7 to 2.0 when df are moderate and the p-value just above 0.05. With t = 2.03, especially under unequal variances, the p-value is commonly around or slightly below 0.05 rather than clearly above it, making this range unlikely.

Option C: 0.02 < p-value < 0.05
This interval captures the scenario where the test statistic is around 2.0 and the df are not very small, yielding a two-sided p-value that straddles the 0.05 threshold. Given t = 2.03 and typical Welch df for these sample sizes, p-value in this vicinity is quite plausible.

Option D: p-value = 0.05
A precise equality to 0.05 is very specific and unlikely for a rounded t-value of 2.03 unless the df align exactly to produce a tail area of 0.025 per side. In practice, the p-value from a t-table lookup with these sample sizes and unequal variances would be reported as a range, not an exact 0.05, so this exact equality is improbable.

Summary of reasoning for each option:
- The first option suggests a much smaller p-value than is typical for t ≈ 2.03 with these sample sizes; while possible in very low df, our df under Welch are not extremely small, making this option less likely.
- The second option would imply insufficient evidence against the null, which contradicts a t-value around 2.03 that commonly yields a p-value near or below 0.05 rather than well above it.
- The third option aligns with expectations for t ≈ 2.03 and moderate-to-large df: a two-sided p-value that falls between 0.02 and 0.05 is a reasonable and common outcome in this scenario.
- The fourth option asserts an exact p-value of 0.05, which is unlikely because p-values from t-distributions with noninteger df are continuous quantities and seldom land exactly at a round threshold without precise df alignment.

Note on the role of the unequal-variances assumption: using the Welch approach generally reduces the df relative to the pooled-variance case, which tends to increase the p-value slightly compared to the equal-variances case, but not enough to push it well above 0.05 for a t-statistic as large as 2.03 with these sample sizes. This context supports Option C as the most plausible range among those listed.

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A two-sided independent samples t-test with n1=33 and n2=38 yielded a test statistic of 2.03. Assuming unequal variances, which of the following is the correct p-value for this hypothesis test? Be sure to find the p-value using the t-distribution probability table.

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