Chapter 18, Problem 32E

(a)

To determine

To explain what should the theoretical mean and the standard deviation be for each of these sample sizes, according to the Central Limit Theorem.

Answer to Problem 32E

n	Theoretical mean	Theoretical S.D.
30	10307.31	3279.88
50	10307.31	2540.58
100	10307.31	1796.46
200	10307.31	1270.29

Explanation of Solution

In the question it is given that the average compensation of the $800$ largest companies is $10307.31$ and the standard deviation is $17964.62$ . Now, a research organization simulated sample means by drawing samples of $30,50,100\text{ and 200}$ and the histogram is given for them to show the distribution of means for many samples of each sizes. And the summary statistics of the simulations are also given. Thus, according to the central limit theorem we have,

  

(b)

To determine

To explain how close are the theoretical values to what was observed in the simulation.

Answer to Problem 32E

They seems to be very close.

Explanation of Solution

In the question it is given that the average compensation of the $800$ largest companies is $10307.31$ and the standard deviation is $17964.62$ . Now, a research organization simulated sample means by drawing samples of $30,50,100\text{ and 200}$ and the histogram is given for them to show the distribution of means for many samples of each sizes. And the summary statistics of the simulations are also given. Thus, as we see in the table in part (a), we can say from that table that the observed values are all very close to the theoretical values.

(c)

To determine

To find out at what sample size would you be comfortable using the Normal model as an approximation for the sampling distribution.

Answer to Problem 32E

None of them will be appropriate.

Explanation of Solution

In the question it is given that the average compensation of the $800$ largest companies is $10307.31$ and the standard deviation is $17964.62$ . Now, a research organization simulated sample means by drawing samples of $30,50,100\text{ and 200}$ and the histogram is given for them to show the distribution of means for many samples of each sizes. And the summary statistics of the simulations are also given. Thus, as all the sampling distributions are still quite skewed. The normal model would not be appropriate.

(d)

To determine

To explain what about the shape of the distribution of total compensation explains your choice of sample size in part (c).

Answer to Problem 32E

The shape of the distribution of total compensation will take very large sample size to be approximately Normal.

Explanation of Solution

In the question it is given that the average compensation of the $800$ largest companies is $10307.31$ and the standard deviation is $17964.62$ . Now, a research organization simulated sample means by drawing samples of $30,50,100\text{ and 200}$ and the histogram is given for them to show the distribution of means for many samples of each sizes. And the summary statistics of the simulations are also given. Thus, as all the sampling distributions are still quite skewed. The normal model would not be appropriate. So, the distribution of the original data is strongly skewed, so it will take a very large sample size before the distribution of sample means is approximately normal.