Chapter 10, Problem R10.5RE

(a)

To determine

To construct and interpret a $90\%$ confidence interval for the difference in mean score for male and female young adults.

Answer to Problem R10.5RE

We are $90\%$ confident that the mean score for male young adults is between $6.7972\text{ lower and 2}\text{.1372 higher}$ than the mean score for female young adults.

Explanation of Solution

It is given in the question that:

  $\begin{array}{l}{n}_1=840\\ n_2=1077\\ {\overline{x}}_1=272.40\\ {\overline{x}}_2=274.73\\ s_1=59.2\\ s_2=57.5\\ c=90\%=0.90\end{array}$

Now, to find the t -value by looking in the row starting with degrees of freedom,

  $\begin{array}{l}df=\min (n_1-1,n_2-1)\\ =\min (840-1,1077-1)\\ =839\end{array}$

And in the column with $c=90\%$ , in the students t distribution table:

  $t_{\alpha /2}=1.660$

The margin of error is:

  $\begin{array}{l}E=t_{\alpha /2}\sqrt{\frac{s_1{}^2}{n_1}+\frac{s_2{}^2}{n_2}}\\ =1.660\times \sqrt{\frac{{59.2}^2}{840}+\frac{{57.5}^2}{1077}}\\ =4.4672\end{array}$

The endpoints of the confidence interval for ${\mu }_1-{\mu }_2$ are:

  $\begin{array}{l}({\overline{x}}_1-{\overline{x}}_2)-E=(272.40-274.73)-4.4672\\ =-2.33-4.4672\\ =-6.7972\\ ({\overline{x}}_1-{\overline{x}}_2)+E=(272.40-274.73)+4.4672\\ =-2.33+4.4672\\ =2.1372\end{array}$

Thus, we conclude that we are $90\%$ confident that the mean score for male young adults is between $6.7972\text{ lower and 2}\text{.1372 higher}$ than the mean score for female young adults.

(b)

To determine

To find out is there convincing evidence of a difference in mean score for male and female young adults or not.

Answer to Problem R10.5RE

No.

Explanation of Solution

There is not a convincing evidence of a difference in mean score for male and female young adults since zero is contained in this interval, so that is why we can say that there is not sufficient evidence to say that there is a difference in test scores between men and women.