Question

Students M and N sampled body lengths of two populations of a fish species using the same study design. The table below summarizes their data as mean \(\pm\) standard error.

Body length (cm)	Population I	Population II
Student M	\(8.0 \pm 1.2\)	\(6.2 \pm 0.7\)
Student N	\(10.0 \pm 0.6\)	\(5.1 \pm 0.3\)

Given that the data are normally distributed, when comparing the means of the populations using a t-test, which student will find a lower p-value?

• A. Student M
• B. Student N
• C. Insufficient information to comment
• D. p-values will be zero in both cases

Hint:

A smaller p-value comes from a bigger mean difference relative to a smaller combined standard error; compare that ratio for each student.

Answer 1

The Correct Option is B

Step 1: Think about what makes a p-value small.
A t-test gives a small p-value when the gap between the two means is large compared to how uncertain each mean is. Here, uncertainty is measured by the standard error (SE), so we just need to compare each student's mean gap against their SE spread.

Step 2: Look at Student M's numbers.
Student M got $8.0\pm1.2$ for Population I and $6.2\pm0.7$ for Population II. The gap between the means is $8.0-6.2=1.8$. Adding the two SEs gives roughly $1.2+0.7=1.9$, which is even bigger than the gap itself. The two estimates almost overlap, so this difference looks weak.

Step 3: Look at Student N's numbers.
Student N got $10.0\pm0.6$ for Population I and $5.1\pm0.3$ for Population II. The gap between the means is $10.0-5.1=4.9$, while the SEs are tiny, only $0.6$ and $0.3$. The gap is more than five times either SE, so the two means sit far apart with very little uncertainty around each.

Step 4: Compare the two students.
Student N has both a bigger mean gap and smaller SEs than Student M. Both of these push the t-statistic up and the p-value down. Student M's numbers barely separate once you allow for the SE, while Student N's numbers separate cleanly.

Step 5: Conclude.
Since a cleaner, more precise separation between means always gives a smaller p-value in a t-test, Student N will find the lower p-value. Option (C) fails because there is enough data here to compare directly, and option (D) fails because p-values are never truly zero.
\[ \boxed{\text{Student N}} \]