Chapter 10.2, Problem 61E

(a)

To determine

To construct and interpret $95\%$ confidence interval for the difference between the true means.

Answer to Problem 61E

We are $95\%$ confident that the mean number of words spoken in a day by all male students at this university is between $3562.90\text{ words lower and 2778}\text{.90 words higher}$ than the mean number of words spoken in a day by all female students at this university.

Explanation of Solution

It is given that:

  $\begin{array}{l}{\overline{x}}_1=16569\\ {\overline{x}}_2=16177\\ n_1=56\\ n_2=56\\ s_1=9108\\ s_2=7520\end{array}$

  $\alpha =0.05$

There are three conditions to be satisfied:

Random: It is satisfied because the samples are randomly selected from separate populations.

Independent: It is satisfied because the sample of $56$ female students is less than $10\%$ of the population of all female students and the sample of $56$ male students is less than $10\%$ of the population of all male students.

Normal: It is satisfied because both the samples are large as they both have a sample size of at least $30$ .

Thus, all the conditions are satisfied.

Now, the degree of freedom will be:

  $\begin{array}{l}df=\min (n_1-1,n_2-1)\\ =\min (56-1,56-1)\\ =55\end{array}$

Thus the t value will be:

  $t=2.009$

The confidence interval will be as:

  $\begin{array}{l}({\overline{x}}_1-{\overline{x}}_2)-t_{\alpha /2}\times \sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}\\ =(16177-16569)-2.009\times \sqrt{\frac{{9108}^2}{56}+\frac{{7520}^2}{56}}\\ =-392-3170.90\\ =-3562.90\\ ({\overline{x}}_1-{\overline{x}}_2)+t_{\alpha /2}\times \sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}\\ =(16177-16569)+2.009\times \sqrt{\frac{{9108}^2}{56}+\frac{{7520}^2}{56}}\\ =-392+3170.90\\ =2778.90\end{array}$

Thus we conclude that we are $95\%$ confident that the mean number of words spoken in a day by all male students at this university is between $3562.90\text{ words lower and 2778}\text{.90 words higher}$ than the mean number of words spoken in a day by all female students at this university.

(b)

To determine

To explain how the confidence interval provide more information than the test.

Explanation of Solution

The confidence interval gives us more information than a significance test because the confidence interval gives us a range of possible values for the difference between the true means and the significance test only checks one possible value for the difference between the means, that is if the means are equal.