a.
Find the least-squares regression line to predict the 15-year rates from 30-year rates.
a.
Answer to Problem 26E
The least-squares regression line to predict the 15-year rates from 30-year rates is
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
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
b.
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.
b.
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.
Explain whether it is possible to give an interpretation of the y-intercept of the regression line.
c.
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,
The y-intercept would mean that when the 30-year rate is
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.
Find the predicted difference in the 15-year rates, if the 30-year rates differ by 0.3 percent.
d.
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
Here,
From part a, the least-squares regression equation is:
Comparing this equation with the general form of the regression equation,
Thus,
Hence, the predicted difference in the 15-year rates, if the 30-year rates differ by 0.3 percent is 0.2682 percent.
e.
Find the 15-year rate for a month which has 30-year rate is 3.5 percent.
e.
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:
For a quarterback whose 30-year rate is 3.5 percent,
Hence, the 15-year rate for a month which has 30-year rate is 3.5 percent is 2.7939 percent.
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Chapter 11 Solutions
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