To determine the shape of the sampling distribution for the difference in mean times [tex]\(\bar{x}_A - \bar{x}_C\)[/tex] between Alex and Chris, let's break down the given information step by step.
1. Population Distribution:
- Alex's times are Normally distributed with a mean time of 5.28 minutes and a standard deviation of 0.38 seconds.
- Chris's times are Normally distributed with a mean time of 5.45 seconds and a standard deviation of 0.2 seconds.
2. Sample Sizes:
- A sample of 10 times is taken for Alex.
- A sample of 15 times is taken for Chris.
3. Sampling Distribution:
- The sampling distribution of a sample mean (such as [tex]\(\bar{x}_A\)[/tex] or [tex]\(\bar{x}_C\)[/tex]) will be Normally distributed if the population from which the sample is drawn is Normally distributed.
4. Difference in Sample Means:
- When considering the difference between two sample means [tex]\(\bar{x}_A - \bar{x}_C\)[/tex], the sampling distribution of this difference will also be Normally distributed if each individual sample mean has a Normal distribution.
Given that:
- Both Alex's and Chris's times are Normally distributed,
- The sampling distributions of [tex]\(\bar{x}_A\)[/tex] and [tex]\(\bar{x}_C\)[/tex] are Normal because they come from Normally distributed populations,
we can conclude that the distribution of the difference between these two sample means, [tex]\(\bar{x}_A - \bar{x}_C\)[/tex], will also follow a Normal distribution. This holds true regardless of the sample sizes, as long as both original populations are Normally distributed.
Therefore, the correct shape of the sampling distribution for [tex]\(\bar{x}_A - \bar{x}_C\)[/tex] is Normal, because both population distributions are Normal.
