To determine the shape of the sampling distribution for [tex]\(\bar{x}_A - \bar{x}_C\)[/tex], we start by examining the properties of the given distributions.
1. Distribution Properties:
- Alex's times are Normally distributed with a mean [tex]\(\mu_A = 5.28\)[/tex] minutes and a standard deviation [tex]\(\sigma_A = 0.38\)[/tex] seconds.
- Chris's times are Normally distributed with a mean [tex]\(\mu_C = 5.45\)[/tex] minutes and a standard deviation [tex]\(\sigma_C = 0.2\)[/tex] seconds.
2. Sample Selection:
- We are taking random samples of 10 times from Alex’s distribution and 15 times from Chris’s distribution.
- This gives us sample means [tex]\(\bar{x}_A\)[/tex] from Alex's sample and [tex]\(\bar{x}_C\)[/tex] from Chris's sample.
3. Shape of Sampling Distributions:
- According to the Central Limit Theorem, if a population has a normal distribution, the sampling distribution of the sample mean will also be normal, no matter the sample size.
- Since Alex's and Chris's times are normally distributed, the means [tex]\(\bar{x}_A\)[/tex] and [tex]\(\bar{x}_C\)[/tex] will both have normal distributions, regardless of their sample sizes.
4. Difference of Two Normally Distributed Variables:
- The difference between two normally distributed variables is also normally distributed.
- Therefore, the distribution of the difference in sample means [tex]\(\bar{x}_A - \bar{x}_C\)[/tex] will follow a normal distribution.
Given these steps:
- The shape of the sampling distribution for [tex]\(\bar{x}_A - \bar{x}_C\)[/tex] is Normal.
- The correct choice is: Normal, because both population distributions are Normal.
