Question

To test whether body size differs between two populations of a field mouse species, a researcher measured 100 individuals in each population and calculated the statistic 𝑋̅1-𝑋̅2/Sp√1/n1+1/n2 where 𝑋̅1and 𝑋̅2 are the mean body sizes of the two populations, respectively,Sp is the pooled standard deviation, and n1 and n2 are the sample sizes for the two populations, respectively.This statistic is used in the

• A. Chi–square test
• B. Kruskal–Wallis test
• C. Student’s t-test
• D. Mann–Whitney U test

Hint:

If you see \(\overline{X}_1-\overline{X}_2\) divided by \(S_p\sqrt{1/n_1+1/n_2}\) with \(df=n_1+n_2-2\), it’s the \textbf{pooled} two-sample t-test. No pooling (Welch’s t) uses group-wise \(S_1,S_2\) and Satterthwaite \(df\).

Answer 1

The Correct Option is C

Step 1: Recognize the test statistic form.

The statistic has the form: \[ t = \frac{\overline{X}_1 - \overline{X}_2}{S_p \sqrt{\tfrac{1}{n_1}+\tfrac{1}{n_2}}}, \quad S_p = \sqrt{\frac{(n_1-1)S_1^2 + (n_2-1)S_2^2}{\,n_1+n_2-2\,}}. \] This is the two-sample pooled-variance Student’s t-test with \(df = n_1+n_2-2\).

Step 2: Assumptions/setting.

• Two independent random samples
• Populations approximately normal (or large \(n\))
• Equal population variances (\(\sigma_1^2 = \sigma_2^2\)) → use pooled variance

Step 3: Eliminate other options.

• Chi–square test: for categorical data or variance tests → ❌
• Kruskal–Wallis: nonparametric, >2 groups → ❌
• Mann–Whitney U: nonparametric 2-sample test, does not use \(S_p\) → ❌

Final Answer: \[ \boxed{\text{(C) Student’s two-sample (pooled) t-test}} \]