Chapter 18, Problem 31E

(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 31E

n	Theoretical mean	Theoretical S.D.
2	36.33	2.842
5	36.33	1.797
10	36.33	1.271
20	36.33	0.897

Explanation of Solution

In the question it is given that researchers measured the waist sizes of $250$ men in a study of body fat. The true mean and the standard deviation are $\text{36}\text{.33 and 4}\text{.019}$ inches. 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 31E

They seems to be very close.

Explanation of Solution

In the question it is given that researchers measured the waist sizes of $250$ men in a study of body fat. The true mean and the standard deviation are $\text{36}\text{.33 and 4}\text{.019}$ inches. 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 31E

The sample size of $\text{5}$ would be appropriate.

Explanation of Solution

In the question it is given that researchers measured the waist sizes of $250$ men in a study of body fat. The true mean and the standard deviation are $\text{36}\text{.33 and 4}\text{.019}$ inches. And the summary statistics of the simulations are also given. Thus, the sample size would you be comfortable using the Normal model as an approximation for the sampling distribution is of the sample size of $\text{5}$ , as the sampling distribution of sample means is uni-modal and symmetric. And the normal model would be appropriate.

(d)

To determine

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

Answer to Problem 31E

The shape of the distribution of the waist size is approximately Normal.

Explanation of Solution

In the question it is given that researchers measured the waist sizes of $250$ men in a study of body fat. The true mean and the standard deviation are $\text{36}\text{.33 and 4}\text{.019}$ inches. And the summary statistics of the simulations are also given. Thus, the sample size would you be comfortable using the Normal model as an approximation for the sampling distribution is of the sample size of $\text{5}$ . So, we can say that the distribution of the original data is nearly uni-modal and symmetric so it does not take a very large sample size for the distribution of sample means to be approximately normal.