Chapter 11.2, Problem 26E

a.

To determine

Find the least-squares regression line to predict the 15-year rates from 30-year rates.

Answer to Problem 26E

The least-squares regression line to predict the 15-year rates from 30-year rates is $\underset{\_}{\widehat{y}=-0.3351+0.8940x}$.

Explanation of Solution

Calculation:

The interest rates for the 30-year fixed-rate mortgages and the 15-year fixed-rate mortgages are given.

Denote the 30-year rates as x and the 15-year rates as y.

Least-squares regression:

For an ordered pairs of values of variables, (x, y) with respective means $\overline{x}$ and $\overline{y}$, respective standard deviations $s_x$ and $s_y$, correlation coefficient r, the least-squares regression line for predicting the response variable, y, by using the predictor variable, x, is given as:

$\widehat{y}=b_0+b_{1}x$, where the slope is $b_1=r\frac{s_y}{s_x}$  and the y-intercept is $b_0=\overline{y}-b_1\overline{x}$.

Regression:

Software procedure:

Step by step procedure to obtain regression using Minitab software is given as,

• Choose Stat > Regression > Regression > Fit Regression Model.
• In Responses, enter the numeric column containing the response data y.
• In Continuous Predictors, enter the numeric column containing the predictor variable x.
• Choose Results, select Regression equation and click OK.
• Click OK.

Output using MINITAB software is given below:

From the output, the least-squares regression line for the data set is found to be $\underset{\_}{\widehat{y}=-0.3351+0.8940x}$.

b.

To determine

Create a scatterplot of the 15-year rate (y) with the 30-year rate (x) and graph the least squares regression line on the same axes.

Answer to Problem 26E

The scatterplot with the least squares regression line on the same axes is given below:

Explanation of Solution

Calculation:

Scatterplot:

Software procedure:

Step-by-step procedure to obtain the scatterplot using the MINITAB software:

• Choose Graph > Scatterplot > With Regression, click OK.
• In Y variables, enter the column of y and in X variables, enter the column of x.
• Click OK.

c.

To determine

Explain whether it is possible to give an interpretation of the y-intercept of the regression line.

Answer to Problem 26E

It is not possible to give an interpretation of the y-intercept of the regression line.

Explanation of Solution

Calculation:

Comparing the least-squares regression equation, $\widehat{y}=-0.3351+0.8940x$ with the general form of the regression equation, the y-intercept is $b_0=-0.3351$.

The y-intercept would mean that when the 30-year rate is $x=0$, the 15-year rate would be –0.3351.

However, it is practically impossible for the rate of a fixed-rate mortgage to be 0 percent. The 30-year rate would always take some non-zero positive value.

Hence, it is not possible to give an interpretation of the y-intercept of the regression line.

d.

To determine

Find the predicted difference in the 15-year rates, if the 30-year rates differ by 0.3 percent.

Answer to Problem 26E

The predicted difference in the 15-year rates, if the 30-year rates differ by 0.3 percent is 0.2682 percent.

Explanation of Solution

Interpretation:

It is known that when the predictor variable values differ by the amount d, the corresponding response variable values differ by amount $b_1\cdot d$, where $b_1$ is the slope of the least-squares regression line.

Here, $d=0.3$.

From part a, the least-squares regression equation is:

$\widehat{y}=-0.3351+0.8940x$.

Comparing this equation with the general form of the regression equation, $b_1=0.8940$.

Thus,

$\begin{array}{c}{b}_1\cdot d=0.3\times 0.8940\\ =\mathrm{0.2682.}\end{array}$

Hence, the predicted difference in the 15-year rates, if the 30-year rates differ by 0.3 percent is 0.2682 percent.

e.

To determine

Find the 15-year rate for a month which has 30-year rate is 3.5 percent.

Answer to Problem 26E

The 15-year rate for a month which has 30-year rate is 3.5 percent is 2.7939 percent.

Explanation of Solution

Calculation:

From part a, the least-squares regression equation is:

$\widehat{y}=-0.3351+0.8940x$.

For a quarterback whose 30-year rate is 3.5 percent, $x=3.5$. Replace this value of x in the equation:

$\begin{array}{c}\widehat{y}=-0.3351+\left(0.8940\times 3.5\right)\\ =-0.3351+3.129\\ =\mathrm{2.7939.}\end{array}$

Hence, the 15-year rate for a month which has 30-year rate is 3.5 percent is 2.7939 percent.