Chapter 24, Problem 18E

(a)

To determine

To find $95\%$ confidence intervals for the average number of pegs that males and females can each place.

Explanation of Solution

It is given that:

  $\begin{array}{l}{\overline{x}}_1=19.39\\ {\overline{x}}_2=17.91\\ n_1=50\\ n_2=50\\ s_1=2.52\\ s_2=3.39\\ \alpha =0.05\end{array}$

Thus, the degree of freedom is:

  $\begin{array}{l}df=n-1\\ =50-1\\ =49>40\end{array}$

Thus, the t -value is then:

  $t_{\alpha /2}=2.014$

The confidence interval is:

For males:

  $\begin{array}{l}\overline{x}-t_{\alpha /2}\times \frac{s}{\sqrt{n}}\\ =19.39-2.014\times \frac{2.52}{\sqrt{50}}\\ =18.6722\\ \overline{x}-t_{\alpha /2}\times \frac{s}{\sqrt{n}}\\ =19.39+2.014\times \frac{2.52}{\sqrt{50}}\\ =20.2078\end{array}$

For females:

  $\begin{array}{l}\overline{x}-t_{\alpha /2}\times \frac{s}{\sqrt{n}}\\ =17.91-2.014\times \frac{3.39}{\sqrt{50}}\\ =16.9445\\ \overline{x}-t_{\alpha /2}\times \frac{s}{\sqrt{n}}\\ =17.91+2.014\times \frac{3.39}{\sqrt{50}}\\ =18.8755\end{array}$

(b)

To determine

To explain what does this suggest about any sex-based difference in manual agility.

Explanation of Solution

It is given that:

  $\begin{array}{l}{\overline{x}}_1=19.39\\ {\overline{x}}_2=17.91\\ n_1=50\\ n_2=50\\ s_1=2.52\\ s_2=3.39\\ \alpha =0.05\end{array}$

Thus, this suggests about any sex-based difference in manual agility that there is no significant difference in the number of pegs placed between males and females.

(c)

To determine

To find a $95\%$ confidence interval for the difference in the mean number of pegs that could be placed by men and women.

Answer to Problem 18E

The confidence interval is: $(0.2900,2.6700)$ .

Explanation of Solution

It is given that:

  $\begin{array}{l}{\overline{x}}_1=19.39\\ {\overline{x}}_2=17.91\\ n_1=50\\ n_2=50\\ s_1=2.52\\ s_2=3.39\\ \alpha =0.05\end{array}$

The degree of freedom is:

  $df=90>75$

Thus, the t -value is:

  $t=1.992$

The confidence interval is then:

  $\begin{array}{l}({\overline{x}}_1-{\overline{x}}_2)-t_{\alpha /2}\times \sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}\\ =(19.39-17.91)-1.992\times \sqrt{\frac{{2.52}^2}{50}+\frac{{3.39}^2}{50}}\\ =0.2900\\ ({\overline{x}}_1-{\overline{x}}_2)+t_{\alpha /2}\times \sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}\\ =(19.39-17.91)+1.992\times \sqrt{\frac{{2.52}^2}{50}+\frac{{3.39}^2}{50}}\\ =2.6700\end{array}$

Thus, the confidence interval is: $(0.2900,2.6700)$ .

(d)

To determine

To explain what does this interval suggest about any difference in manual agility between men and women.

Explanation of Solution

It is given that:

  $\begin{array}{l}{\overline{x}}_1=19.39\\ {\overline{x}}_2=17.91\\ n_1=50\\ n_2=50\\ s_1=2.52\\ s_2=3.39\\ \alpha =0.05\end{array}$

Thus, the confidence interval is: $(0.2900,2.6700)$ . Therefore, this interval suggests about any difference in manual agility between men and women that there is a significant difference in manual agility between men and women, because the confidence interval does not contain zero.

(e)

To determine

To explain which method is correct: doing two-sample inference or doing one-sample inference twice.

Answer to Problem 18E

Thus, doing the two-sample inference is the correct method.

Explanation of Solution

It is given that:

  $\begin{array}{l}{\overline{x}}_1=19.39\\ {\overline{x}}_2=17.91\\ n_1=50\\ n_2=50\\ s_1=2.52\\ s_2=3.39\\ \alpha =0.05\end{array}$

Thus, doing the two-sample inference is the correct method because it takes into account both samples at the same time and the degrees of freedom are adjusted to the sample size and variability of both samples.

(f)

To determine

To explain why do not the result agree.

Explanation of Solution

It is given that:

  $\begin{array}{l}{\overline{x}}_1=19.39\\ {\overline{x}}_2=17.91\\ n_1=50\\ n_2=50\\ s_1=2.52\\ s_2=3.39\\ \alpha =0.05\end{array}$

Thus, the result do not agree because the degrees of freedom of the two-sample inference are adjusted to the sample size and variability of both samples, which causes the difference in results between the two-sample inference and one-sample inferences.