Chapter 24, Problem 12E

(a)

To determine

To explain what do the boxplot suggest about the differences between male and female pulse rate.

Answer to Problem 12E

There is no difference between male and female pulse rates, because the boxplot of females completely overlaps the boxplot of males.

Explanation of Solution

In this question, a researcher wanted to see whether there is a significant difference in resting pulse rates for men and women. The boxplot is shown in the question according to the data collected. The data is as follows:

  $\begin{array}{l}{\overline{x}}_1=72.75\\ {\overline{x}}_2=72.625\\ n_1=28\\ n_2=24\\ s_1=5.37225\\ s_2=7.69987\\ \alpha =0.10\end{array}$

	Male	Female
Median	73	73
Range	20	29
IQR	9	12.5

Thus, the boxplot suggest about the differences between male and female pulse rate is that there is no difference between male and female pulse rates, because the boxplot of females completely overlaps the boxplot of males.

(b)

To determine

To explain is it appropriate to analyse these data using the methods of inference discussed in this chapter.

Answer to Problem 12E

Yes, it is appropriate to analyse these data.

Explanation of Solution

In this question, a researcher wanted to see whether there is a significant difference in resting pulse rates for men and women. The boxplot is shown in the question according to the data collected. The data is as follows:

  $\begin{array}{l}{\overline{x}}_1=72.75\\ {\overline{x}}_2=72.625\\ n_1=28\\ n_2=24\\ s_1=5.37225\\ s_2=7.69987\\ \alpha =0.10\end{array}$

	Male	Female
Median	73	73
Range	20	29
IQR	9	12.5

Thus, the conditions are as follows:

Random condition: It is satisfied because we assume that the men and women were randomly selected.

Independent condition: It is satisfied because the groups of men and women are independent.

Normal condition: It is satisfied because we assume that the pulse rates are approximately normally distributed.

Therefore, all the conditions are met. And yes, it is appropriate to analyse these data using the methods of inference discussed in this chapter.

(c)

To determine

To create a $90\%$ confidence interval for the difference in mean pulse rates.

Answer to Problem 12E

The confidence interval is $(-3.0260,3.2760)$ .

Explanation of Solution

In this question, a researcher wanted to see whether there is a significant difference in resting pulse rates for men and women. The boxplot is shown in the question according to the data collected. The data is as follows:

  $\begin{array}{l}{\overline{x}}_1=72.75\\ {\overline{x}}_2=72.625\\ n_1=28\\ n_2=24\\ s_1=5.37225\\ s_2=7.69987\\ \alpha =0.10\end{array}$

	Male	Female
Median	73	73
Range	20	29
IQR	9	12.5

Now let us determine the degrees of freedom:

  $\begin{array}{l}df=\frac{(\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2})}{\frac{{(\raisebox{1ex}{$s_1^2$}\!\left/ \!\raisebox{-1ex}{$n_1$}\right.)}^2}{n_1-1}+\frac{{(\raisebox{1ex}{$s_2^2$}\!\left/ \!\raisebox{-1ex}{$n_2$}\right.)}^2}{n_2-1}}\\ =\frac{(\frac{{5.37225}^2}{28}+\frac{{7.69987}^2}{24})}{\frac{{(\raisebox{1ex}{${5.37225}^2$}\!\left/ \!\raisebox{-1ex}{$28$}\right.)}^2}{28-1}+\frac{{(\raisebox{1ex}{${7.69987}^2$}\!\left/ \!\raisebox{-1ex}{$24$}\right.)}^2}{24-1}}\\ =40\end{array}$

Therefore, the t -value will be:

  $t=1.684$

Thus, the confidence interval will be:

  $\begin{array}{l}({\overline{x}}_1-{\overline{x}}_2)-t_{\alpha /2}\times \sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}\\ =(72.75-72.625)-1.684\times \sqrt{\frac{{5.37225}^2}{28}+\frac{{7.69987}^2}{24}}\\ =-3.0260\\ ({\overline{x}}_1-{\overline{x}}_2)+t_{\alpha /2}\times \sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}\\ =(72.75-72.625)+1.684\times \sqrt{\frac{{5.37225}^2}{28}+\frac{{7.69987}^2}{24}}\\ =3.2760\end{array}$

The confidence interval is $(-3.0260,3.2760)$ .

(d)

To determine

To explain does the confidence interval confirm your answer to part (a).

Answer to Problem 12E

Yes, it confirms.

Explanation of Solution

In this question, a researcher wanted to see whether there is a significant difference in resting pulse rates for men and women. The boxplot is shown in the question according to the data collected. The data is as follows:

  $\begin{array}{l}{\overline{x}}_1=72.75\\ {\overline{x}}_2=72.625\\ n_1=28\\ n_2=24\\ s_1=5.37225\\ s_2=7.69987\\ \alpha =0.10\end{array}$

	Male	Female
Median	73	73
Range	20	29
IQR	9	12.5

And the confidence interval is $(-3.0260,3.2760)$ . Thus, we can say that the confidence interval confirm your answer to part (a) because the confidence interval contains zero, which indicates that there is no difference in the mean pulse rate of men and women and this thus agrees with the conclusion in part (a).