Chapter 18, Problem 30E

(a)

To determine

To describe the histogram of total compensation.

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 $17964.62$ . And the histogram of their annual compensation is also given. Thus, we can say about the distribution that the distribution of total compensation for the CEOs of the $800$ largest US companies is uni-modal but strongly skewed to the right by looking at the histogram with several large outliers.

(b)

To determine

To explain how these histograms demonstrate what the central limit theorem says about the sampling distribution model for sample means and also talk about the shape, centre and spread.

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. Thus, we can say that all histograms are centered near $\$10,000,000$ . As n gets larger the standard deviation decreases and the histograms approach the shape of a normal distribution. However, they are still visibly skewed to the right.

(c)

To determine

To comment on the “rule of thumb” that “with a sample size of at least $30$ , the sampling distribution of the mean is 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. Thus, the “rule of thumb” that “with a sample size of at least $30$ , the sampling distribution of the mean is Normal”, for this rule we can say that this does not seem to apply to highly skewed distributions.