Chapter 10.2, Problem 62E

(a)

To determine

To construct and interpret $99\%$ confidence interval for the difference between the true means.

Answer to Problem 62E

We are $99\%$ confident that the mean weight of all gray Eastern gray squirrels is between $\text{0}\text{.1471 ounces lower and 2}\text{.3471 ounces higher}$ than the mean weight of all black Eastern gray squirrels in the forest.

Explanation of Solution

It is given that:

  $\begin{array}{l}{\overline{x}}_1=20.3\\ {\overline{x}}_2=19.2\\ n_1=40\\ n_2=40\\ s_1=2.1\\ s_2=1.9\end{array}$

  $\alpha =0.01$

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 $40$ gray Eastern gray squirrels is less than $10\%$ of the population of all gray Eastern gray squirrels and the sample of $40$ black Eastern gray squirrels is less than $10\%$ of the population of all black Eastern gray squirrels.

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:

  $t_{\alpha /2}=2.750$

The confidence interval will be as:

  $\begin{array}{l}({\overline{x}}_1-{\overline{x}}_2)-t_{\alpha /2}\times \sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}\\ =(20.3-19.2)-2.750\times \sqrt{\frac{{2.1}^2}{40}+\frac{{1.9}^2}{40}}\\ =1.1-1.2471\\ =-0.1471\\ ({\overline{x}}_1-{\overline{x}}_2)+t_{\alpha /2}\times \sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}\\ =(20.3-19.2)+2.750\times \sqrt{\frac{{2.1}^2}{40}+\frac{{1.9}^2}{40}}\\ =1.1+1.2471\\ =2.3471\end{array}$

Thus we conclude that we are $99\%$ confident that the mean weight of all gray Eastern gray squirrels is between $\text{0}\text{.1471 ounces lower and 2}\text{.3471 ounces higher}$ than the mean weight of all black Eastern gray squirrels in the forest.

(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.