Two independent samples of sizes 30 and 40 are randomly selected from two populations to test the difference between the population means, µ 1 - µ 2, where the population variances are unknown but equal. Which of the following best describes the sampling distribution of the sample mean difference $\bar { X } _ { 1 } - \bar { X } _ { 2 }$ ?
A)Normal.
B)t-distributed with 68 degrees of freedom.
C)Approximately normal.
D)None of these choices are correct.

Question

Quizplus · Accepted Answer

Since the sample sizes are relatively large (30 and 40) and the population variances are unknown but assumed to be equal, the sampling distribution of the difference between the two sample means follows a t-distribution. The degrees of freedom for this t-distribution can be calculated using the formula for two independent samples with equal variances, which in this case is the sum of the two sample sizes minus 2 (30 + 40 - 2 = 68).

Two Independent Samples of Sizes 30 and 40 Are Randomly Xˉ1−Xˉ2\bar { X } _ { 1 } - \bar { X } _ { 2 }Xˉ1​−Xˉ2​

Two Independent Samples of Sizes 30 and 40 Are Randomly $\bar { X } _ { 1 } - \bar { X } _ { 2 }$