If two random samples of sizes $n _ { 1 }$ and $n _ { 2 }$ are selected independently from two non-normally distributed populations, then the sampling distribution of the sample mean difference, $\bar { X } _ { 1 } - \bar { X } _ { 2 }$ , is:
A) always non-normal.
B) always normal.
C) approximately normal only if $n _ { 1 }$ and $n _ { 2 }$ are both larger than 30.
D) approximately normal regardless of $n _ { 1 }$ and $n _ { 2 }$ .

Question

Quizplus · Accepted Answer

The sampling distribution of the sample mean difference, $\bar { X } _ { 1 } - \bar { X } _ { 2 }$, can be approximated to a normal distribution through the central limit theorem only if both the sample sizes ($n _ { 1 }$ and $n _ { 2 }$) are larger than 30. If the sample sizes are smaller than 30, we cannot assume approximate normality, and if the populations are highly skewed or have heavy tails, the sample mean difference may not be normal even for larger sample sizes. Therefore, option A is incorrect as the sampling distribution is not always non-normal, and option B and D are incorrect as the normality of the sampling distribution depends on the sample sizes.

If Two Random Samples of Sizes n1n _ { 1 }n1​

If Two Random Samples of Sizes $n _ { 1 }$