Chapter 9, Problem 22E

(a)

To determine

To find out what is the correlation between age difference and year.

Answer to Problem 22E

The correlation between the age difference and the year is $0.867$ .

Explanation of Solution

In the question it is given the graph which shows the ages of both men and women at first marriage and the pattern of men and women seems familiar. Also the time plot of difference in average age at first marriage over the years and the dependent variable is age difference. Also:

  $\begin{array}{l}{R}^2=75.1\%\\ s=0.2333\\ \text{Intercept}=35.0617\\ \text{Year}-1950=-0.016565\end{array}$

As we know that the formula for calculation correlation from the R -squared is:

  $r^2=R^2$

This implies we have,

  $\begin{array}{l}{r}^2=R^2\\ =0.751\\ \Rightarrow r=\sqrt{0.751}\\ \Rightarrow r=0.867\end{array}$

Thus, the correlation between the age difference and the year is $0.867$ .

(b)

To determine

To interpret the slope of this line.

Explanation of Solution

In the question it is given the graph which shows the ages of both men and women at first marriage and the pattern of men and women seems familiar. Also the time plot of difference in average age at first marriage over the years and the dependent variable is age difference. Also:

  $\begin{array}{l}{R}^2=75.1\%\\ s=0.2333\\ \text{Intercept}=35.0617\\ \text{Year}-1950=-0.016565\end{array}$

Thus, we know that the slope is defined as the age difference by year. Therefore, we interpret from the slope of this line that as the year increases by one, the age difference between men and women decrease by $0.016565$ .

(c)

To determine

To predict the average age difference in $2015$ .

Answer to Problem 22E

The average age difference in $2015$ is $1.638225$ years.

Explanation of Solution

In the question it is given the graph which shows the ages of both men and women at first marriage and the pattern of men and women seems familiar. Also the time plot of difference in average age at first marriage over the years and the dependent variable is age difference. Also:

  $\begin{array}{l}{R}^2=75.1\%\\ s=0.2333\\ \text{Intercept}=35.0617\\ \text{Year}-1950=-0.016565\end{array}$

To predict the average age difference in $2015$ can be:

  $\begin{array}{l}\text{Age difference}=35.0617-0.016565x\\ =35.0617-0.016565(2015)\\ =1.638225\end{array}$

Thus, the average age difference in $2015$ is $1.638225$ years.

(d)

To determine

To describe the reason why you would not place much faith in this prediction.

Explanation of Solution

In the question it is given the graph which shows the ages of both men and women at first marriage and the pattern of men and women seems familiar. Also the time plot of difference in average age at first marriage over the years and the dependent variable is age difference. Also:

  $\begin{array}{l}{R}^2=75.1\%\\ s=0.2333\\ \text{Intercept}=35.0617\\ \text{Year}-1950=-0.016565\end{array}$

Thus, the reason why we would not place much faith in this prediction because we are extrapolating $2015$ extends beyond the scope of the linear model because the linear model was only created for data up until the year $2000$ .