Chapter 18, Problem 3E

(a)

To determine

To explain what should the theoretical mean and standard deviations be for these sample sizes, according to the central limit theorem.

Answer to Problem 3E

The theoretical mean for all the sample sizes be $0.05$ . And the theoretical standard deviation is:

  $0.0486,0.0313,0.0217,0.0154$ respectively for the sample sizes.

Explanation of Solution

It is given in the question, the responses of the mails are recorded by the organization for different sample sizes. Thus, as given in the question the theoretical mean for all the sample sizes be $0.05$ . And the theoretical standard deviation can be calculated as:

  $S.D.=\sqrt{\frac{pq}{n}}$

Thus, for all the sample sizes it can be calculated as:

  $\begin{array}{l}S.D.=\sqrt{\frac{pq}{n}}\\ =\sqrt{\frac{0.0497(1-0.0497)}{20}}\\ =0.0486\\ S.D.=\sqrt{\frac{pq}{n}}\\ =\sqrt{\frac{0.0516(1-0.0516)}{50}}\\ =0.0313\\ S.D.=\sqrt{\frac{pq}{n}}\\ =\sqrt{\frac{0.0497(1-0.0497)}{100}}\\ =0.0217\\ S.D.=\sqrt{\frac{pq}{n}}\\ =\sqrt{\frac{0.0501(1-0.0501)}{200}}\\ =0.0154\end{array}$

(b)

To determine

To explain how close are those theoretical values to what was observed in these simulations.

Answer to Problem 3E

They look very similar in values.

Explanation of Solution

It is given in the question, the responses of the mails are recorded by the organization for different sample sizes. As we see at the table for the observed means and the standard deviation and the theoretical means and the standard deviation as follows:

n	Observed Mean	Theoretical Mean	Observed deviation	Theoretical deviation
20	0.0497	0.05	0.0479	0.0486
50	0.0516	0.05	0.0309	0.0313
100	0.0497	0.05	0.0215	0.0217
200	0.0501	0.05	0.0152	0.0154

Thus, by comparing the theoretical vales with the observed values we can see that they seems very similar in 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, by looking at the histogram in previous exercise.

Answer to Problem 3E

The sample size would you be comfortable using the Normal model as an approximation for the sampling distribution is the sample size of $200$ .

Explanation of Solution

It is given in the question, the responses of the mails are recorded by the organization for different sample sizes. As we see at the table for the observed means and the standard deviation and the theoretical means and the standard deviation as follows:

n	Observed Mean	Theoretical Mean	Observed deviation	Theoretical deviation
20	0.0497	0.05	0.0479	0.0486
50	0.0516	0.05	0.0309	0.0313
100	0.0497	0.05	0.0215	0.0217
200	0.0501	0.05	0.0152	0.0154

Thus, the sample size would you be comfortable using the Normal model as an approximation for the sampling distribution, by looking at the histogram in previous exercise is the sample size of $200$ as the histogram of this sample size is uni-modal and symmetric in nature and it is very similar to normal graph than compared to other graphs of the other sample sizes.

(d)

To determine

To explain what does the success or failure condition say about the choice you made in part (c).

Explanation of Solution

It is given in the question, the responses of the mails are recorded by the organization for different sample sizes. As we see at the table for the observed means and the standard deviation and the theoretical means and the standard deviation as follows:

n	Observed Mean	Theoretical Mean	Observed deviation	Theoretical deviation
20	0.0497	0.05	0.0479	0.0486
50	0.0516	0.05	0.0309	0.0313
100	0.0497	0.05	0.0215	0.0217
200	0.0501	0.05	0.0152	0.0154

Thus, the sample size would you be comfortable using the Normal model as an approximation for the sampling distribution, by looking at the histogram in previous exercise is the sample size of $200$ . Thus, the success and the failure can be calculated for this sample size as:

  $\begin{array}{l}200(0.05)=10\\ 200(1-0.05)\\ =200(0.95)\\ =190\end{array}$

Thus, it is viable because there are at least ten people for both the successes and failures.