Chapter 12.1, Problem 2E

(a)

To determine

To find: the average high temperature on days.

Answer to Problem 2E

  ${57.4}^{\circ }F$

Explanation of Solution

Given:

  $\begin{array}{l}{\mu }_y=16.6+1.02x\\ \sigma =6.64\end{array}$

Calculation:

For the average high temperature = ${40}^{\circ }F$

  $\begin{array}{c}{\mu }_y=16.6+1.02x\\ =16.6+1.02\left(40\right)\\ =16.6+40.8\\ =57.4\end{array}$

Therefore the average high temperature as per the population regression line is ${57.4}^{\circ }F$ .

(b)

To determine

To find: the percent of high temperature greater than ${70}^{\circ }F$ .

Answer to Problem 2E

2.87%

Explanation of Solution

Given:

  $\begin{array}{l}{\mu }_y=16.6+1.02x\\ \sigma =6.64\end{array}$

Formula used:

  $z=\frac{x-\mu }{\sigma }$

Calculation:

Average mean

  $\begin{array}{c}{\mu }_y=16.6+1.02x\\ =16.6+1.02\left(40\right)\\ =16.6+40.8\\ =57.4\end{array}$

  $\sigma =6.64$

Z score is

  $\begin{array}{c}z=\frac{x-\mu }{\sigma }\\ =\frac{70-57.4}{6.64}\\ =1.90\end{array}$

Corresponding probability for the greater than ${70}^{\circ }F$

  $\begin{array}{c}P\left(X>70\right)=P\left(Z>1.90\right)\\ =1-P\left(Z<1.90\right)\\ =1-0.9713\\ =2.87\%\end{array}$

Therefore about 2.87 percent of the days with a low temperature of ${40}^{\circ }F$ are expected to being having high temperature that is greater than ${70}^{\circ }F$

(c)

To determine

To find: that the slope of the sample regression line be exactly 1.02, yes or not.

Answer to Problem 2E

No

Explanation of Solution

No, the reason is that the slope of the population regression line is 1.02 and it is predicted that the slope of the regression line of sample is very near to 1.02 but it is not exactly so there would be some sampling variability in a sample.