Question

Two sets of sample data, \(\mathrm{A}\) and \(\mathrm{B}\), are given. Without doing any calculations, indicate in which set of sample data, \(\mathrm{A}\) or \(\mathrm{B}\), there is likely to be stronger evidence of a difference in the two population means. Give a brief reason, comparing means and variability, for your answer. $$ \begin{array}{cc|cc} \hline {\text { Dataset A }} & {\text { Dataset B }} \\ \hline \text { Group 1 } & \text { Group 2 } & \text { Group 1 } & \text { Group 2 } \\ \hline 13 & 18 & 13 & 48 \\ 14 & 19 & 14 & 49 \\ 15 & 20 & 15 & 50 \\ 16 & 21 & 16 & 51 \\ 17 & 22 & 17 & 52 \\ \bar{x}_{1}=15 & \bar{x}_{2}=20 & \bar{x}_{1}=15 & \bar{x}_{2}=50 \end{array} $$

Short answer

Dataset B demonstrates a stronger evidence of a difference in the population means, mainly because it has a larger difference between group means.

Step-by-step solution

Observe Mean Values

We will start by examining the mean (\(\overline{x}\)) values provided. For Dataset A, the mean value of group 1 is 15 and for group 2 it's 20. For Dataset B, the mean value of group 1 is 15 (same as Dataset A), but for group 2 it's 50. Therefore, the difference between the mean values for Dataset A is 5 (20 - 15 = 5) and for Dataset B is 35 (50 - 15 = 35).

Compare the Mean Differences

As observed in Step 1, the difference in mean values between the two groups is greater in Dataset B than in Dataset A.

Consider Variability

The difference in each individual value from dataset A to the corresponding value in dataset B is constant and equals to 30, so the variability doesn't add any extra information to explain the difference between dataset A and B.

Conclude

From the above comparisons, it's clear that Dataset B provides stronger evidence of a difference in the two population means as its mean difference is significantly larger than the one in Dataset A, despite the variability being constant in both datasets.