Chapter 13.1, Problem 3E

a:

To determine

The difference between the two population means.

Explanation of Solution

The unequal variance estimator (v) can be calculated as follows:

$\begin{array}{l}\nu =\frac{{(s_1^2/n_1+s_2^2/n_2)}^2}{\frac{{(s_1^2/n_1)}^2}{n_1-1}+\frac{{(s_2^2/n_2)}^2}{n_2-1}}\\ =64.8\end{array}$

The value of unequal variance is 64.8.

The difference between the two population means (DP) can be calculated as follows:

$\begin{array}{c}\text{DP}=({\overline{x}}_1-{\overline{x}}_2)\pm t_{\alpha /2}\sqrt{\left(\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}\right)}\\ =({\overline{x}}_1-{\overline{x}}_2)\pm t_{\frac{90\%}{2}}\sqrt{\left(\frac{{18}^2}{50}+\frac{7^2}{45}\right)}\\ =(63-60)\pm 1.667\sqrt{\left(\frac{{18}^2}{50}+\frac{7^2}{45}\right)}\\ =-1.59\text{ and 7}\text{.59}\end{array}$

The difference between the two population means lies between -1.59 and 7.59.

b:

To determine

The difference between the two population means with increasing standard deviation.

Explanation of Solution

The unequal variance estimator (v) can be calculated as follows:

$\begin{array}{l}\nu =\frac{{(s_1^2/n_1+s_2^2/n_2)}^2}{\frac{{(s_1^2/n_1)}^2}{n_1-1}+\frac{{(s_2^2/n_2)}^2}{n_2-1}}\\ =63.1\end{array}$

The value of unequal variance is 63.1.

The difference between the two population means (DP) can be calculated as follows:

$\begin{array}{c}\text{DP}=({\overline{x}}_1-{\overline{x}}_2)\pm t_{\alpha /2}\sqrt{\left(\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}\right)}\\ =({\overline{x}}_1-{\overline{x}}_2)\pm t_{\frac{90\%}{2}}\sqrt{\left(\frac{{41}^2}{50}+\frac{{15}^2}{45}\right)}\\ =(63-60)\pm 1.667\sqrt{\left(\frac{{41}^2}{50}+\frac{{15}^2}{45}\right)}\\ =-7.38\text{ and 13}\text{.3}\text{.8}\end{array}$

The difference between the two population means lies between -7.08 and 13.08.

c:

To determine

The impact of increasing standard deviation.

Explanation of Solution

When the standard deviation increases, it increases the difference in the interval.

d:

To determine

The difference between the two population means with increasing sample size.

Explanation of Solution

The unequal variance estimator (v) can be calculated as follows:

$\begin{array}{l}\nu =\frac{{(s_1^2/n_1+s_2^2/n_2)}^2}{\frac{{(s_1^2/n_1)}^2}{n_1-1}+\frac{{(s_2^2/n_2)}^2}{n_2-1}}\\ =131\end{array}$

The value of unequal variance is 131.

The difference between the two population means (DP) can be calculated as follows:

$\begin{array}{c}\text{DP}=({\overline{x}}_1-{\overline{x}}_2)\pm t_{\alpha /2}\sqrt{\left(\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}\right)}\\ =({\overline{x}}_1-{\overline{x}}_2)\pm t_{\frac{90\%}{2}}\sqrt{\left(\frac{{18}^2}{100}+\frac{7^2}{90}\right)}\\ =(63-60)\pm 1.667\sqrt{\left(\frac{{18}^2}{100}+\frac{7^2}{90}\right)}\\ =-0.22\text{ and 6}\text{.22}\end{array}$

The difference between the two population means lies between -0.22 and 6.22.

e:

To determine

The impact of increasing sample size.

Explanation of Solution

When the sample size increases, it reduces the difference in the interval.