Chapter 24, Problem 35E

(a)

To determine

To construct and interpret a $95\%$ confidence interval for the difference between a female and male swimmers to swim the lake.

Answer to Problem 35E

We are $95\%$ confidence that the mean amount of time for females swimmers is between less

  $103.6508\text{ and }253.3308$ seconds more than the mean amount of time for male swimmers.

Explanation of Solution

It is given that:

  $\begin{array}{l}{\overline{x}}_1=1271.59\\ {\overline{x}}_2=1196.75\\ n_1=22\\ n_2=20\\ s_1=261.111\\ s_2=304.369\\ \alpha =0.05\end{array}$

The mean and the standard deviation is calculated by using excel by going to the data tab and then to the analysis tab and then selecting the data analysis option. Then we go to the t -test option in the dialogue box that appeared on screen. Then we select the data and the result will come out.

Thus, the degrees of freedom will be as:

  $\begin{array}{l}df=\frac{(\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2})}{\frac{{(\raisebox{1ex}{$s_1^2$}\!\left/ \!\raisebox{-1ex}{$n_1$}\right.)}^2}{n_1-1}+\frac{{(\raisebox{1ex}{$s_2^2$}\!\left/ \!\raisebox{-1ex}{$n_2$}\right.)}^2}{n_2-1}}\\ =37>35\end{array}$

Thus, the t -value will be as:

  $t=2.030$

Thus, the confidence interval will be:

  $\begin{array}{l}({\overline{x}}_1-{\overline{x}}_2)-t_{\alpha /2}\times \sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}\\ =(1271.59-1196.75)-2.030\times \sqrt{\frac{{261.111}^2}{22}+\frac{{304.369}^2}{20}}\\ =-103.6508\\ ({\overline{x}}_1-{\overline{x}}_2)+t_{\alpha /2}\times \sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}\\ =(1271.59-1196.75)+2.030\times \sqrt{\frac{{261.111}^2}{22}+\frac{{304.369}^2}{20}}\\ =253.3308\end{array}$

Thus, we conclude that we are $95\%$ confidence that the mean amount of time for females

swimmers is between less $103.6508\text{ and }253.3308$ seconds more than the mean amount of time for male swimmers.

(b)

To determine

To comment on the assumptions and conditions.

Explanation of Solution

Now, let us check the conditions and assumptions for inference, that is:

Random condition: It is not satisfied because the swimmers were not selected at random but they might be representative for the population.

Independent condition: It is satisfied because assuming that the individuals and the swims between the two groups are independent.

Normal condition: It is not satisfied because the distribution for men is strongly right-skewed and the sample size is small.

Thus, the random and the normal conditions are not satisfied in this context.