Question

suppose jack and diane are each attempting to use a simulation to describe the sampling distribution from a population that is skewed right with mean 60 and standard deviation 5. jack obtains 1000 random samples of size n = 3 from the population, finds the mean of the means, and determines the standard deviation of the means. diane does the same simulation, but obtains 1000 random samples of size n = 30 from the population. complete parts (a) through (c)
(a) describe the shape you expect for jacks distribution of sample means. describe the shape you expect for dianes distribution of sample means. choose the correct answer below
oa. jacks distribution is expected to be skewed right, but less skewed than the original distribution. dianes distribution is expected to be approximately normal
ob. dianes distribution is expected to be skewed right, but less skewed than the original distribution. jacks distribution is expected to be approximately normal.
oc. jacks distribution and dianes distribution are expected to be approximately normal. however, jacks will have the smaller standard deviation of the sample mean
od. jacks distribution and dianes distribution are expected to be approximately normal. however, jacks will have the greater standard deviation of the sample mean

Explanation:

The Central Limit Theorem (CLT) states that for a sample size \(n\), the sampling distribution of the sample mean \(\bar{X}\) has mean \(\mu_{\bar{X}}=\mu\) and standard deviation \(\sigma_{\bar{X}}=\frac{\sigma}{\sqrt{n}}\). When the sample size \(n\) is small (\(n = 3\) in Jack's case), if the population is skewed, the sampling distribution of the sample mean will still be skewed (but less so than the population). When \(n\) is large (\(n=30\) in Diane's case, \(n\geq30\) is a common rule - of - thumb for the CLT), the sampling distribution of the sample mean is approximately normal.

Answer:

A. Jack's distribution is expected to be skewed right, but less skewed than the original distribution. Diane's distribution is expected to be approximately normal.