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.
a. jacks distribution is expected to be skewed right, but less skewed than the original distribution. dianes distribution is expected to be approximately normal.
b. dianes distribution is expected to be skewed right, but less skewed than the original distribution. jacks distribution is expected to be approximately normal.
c. jacks distribution and dianes distribution are expected to be approximately normal. however, jacks will have the smaller standard deviation of the sample mean.
d. jacks distribution and dianes distribution are expected to be approximately normal. however, jacks will have the greater standard deviation of the sample mean.
(b) what do you expect the mean of jacks distribution to be? what do you expect the mean of dianes distribution to be?
jacks distribution is expected to have a mean of □. dianes distribution is expected to have a mean of □
(type integers or decimals rounded to two decimal places as needed.)

Explanation:

Step1: Apply Central Limit Theorem

The Central Limit Theorem (CLT) states that for a sample size \(n\), if \(n\) is large (\(n\geq30\)), the sampling distribution of the sample mean \(\bar{X}\) is approximately normal, regardless of the shape of the population distribution. The mean of the sampling distribution of the sample mean \(\mu_{\bar{X}}=\mu\), and the standard deviation of the sampling distribution of the sample mean (standard error) \(\sigma_{\bar{X}}=\frac{\sigma}{\sqrt{n}}\).

For Jack, \(n = 3\). Since \(n<30\) and the population is skewed right, the sampling distribution of the sample mean for Jack will still be skewed right, but less skewed than the population (because averaging reduces the skewness to some extent).

For Diane, \(n = 30\). By the CLT, since \(n = 30\geq30\), the sampling distribution of the sample mean for Diane is approximately normal.

Step2: Calculate the mean of the sampling distribution

The mean of the sampling distribution of the sample mean \(\mu_{\bar{X}}\) is equal to the population mean \(\mu\). Given that the population mean \(\mu = 60\).

For Jack, \(\mu_{\bar{X}_1}=\mu=60\)

For Diane, \(\mu_{\bar{X}_2}=\mu = 60\)

Answer:

(a) 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.

(b) Jack’s distribution is expected to have a mean of \(60\). Diane’s distribution is expected to have a mean of \(60\).