Chapter 9, Problem 24E

(a)

To determine

To explain why is the $R^2$ higher than the previous model in previous exercise.

Explanation of Solution

From the previous exercise, we have the plot of the age difference of men and women at their first marriage over the years. Now, in this question it has been extended to further years till $2003$ . The dependent variable is age difference. Also, it is given:

  $\begin{array}{l}{R}^2=65.6\%\\ s=0.1869\\ \text{Intercept}=4.88424\\ \text{Year}-1950=-0.029959\end{array}$

From the data provided by the question we can say that, the model in the previous exercise could be approximated by a linear model while this model would need to be approximated by a parabolic model. Thus, the $R^2$ is higher for this model than the previous model in previous exercise.

(b)

To determine

To explain is this linear model appropriate for the post $1975$ data or not.

Answer to Problem 24E

No, it cannot.

Explanation of Solution

From the previous exercise, we have the plot of the age difference of men and women at their first marriage over the years. Now, in this question it has been extended to further years till $2003$ . The dependent variable is age difference. Also, it is given:

  $\begin{array}{l}{R}^2=65.6\%\\ s=0.1869\\ \text{Intercept}=4.88424\\ \text{Year}-1950=-0.029959\end{array}$

This linear model will not be accurate for the post $1975$ data because this model predicts lower than the previous model and also the $R^2$ values are changed for the both models thus they shows different variations and different slopes. So, it is not possible.

(c)

To determine

To explain what does the slope says about the marriages ages.

Explanation of Solution

From the previous exercise, we have the plot of the age difference of men and women at their first marriage over the years. Now, in this question it has been extended to further years till $2003$ . The dependent variable is age difference. Also, it is given:

  $\begin{array}{l}{R}^2=65.6\%\\ s=0.1869\\ \text{Intercept}=4.88424\\ \text{Year}-1950=-0.029959\end{array}$

The slope say about the model is that this new model is more accurate because it is based on data since $1975$ which means that we are not extrapolating and that this model actually fits the data we are trying to predict.

(d)

To determine

To explain why it is not reasonable to interpret the y -intercept.

Explanation of Solution

From the previous exercise, we have the plot of the age difference of men and women at their first marriage over the years. Now, in this question it has been extended to further years till $2003$ . The dependent variable is age difference. Also, it is given:

  $\begin{array}{l}{R}^2=65.6\%\\ s=0.1869\\ \text{Intercept}=4.88424\\ \text{Year}-1950=-0.029959\end{array}$

Now, it is not reasonable to interpret the y -intercept because the predictions made by the second model does not make sense. And the slope of this model has negative relationship. Thus, the plot will be more easier to predict the results than the y -intercept.