Chapter 4.2, Problem 26E

Mortgage payments: The following table presents interest rates, in percent, for 30-year and

15-year fixed-rate mortgages, for January through December, 2012.

1. Compute the least-squares regression line for predicting the 15-year rate from the 30-year rate.
2. Construct a scatter-plot of the 15-year rate (y) versus the 30-year rate (x). Graph the least-squares regression line on the same axes.
3. Is it possible to interpret die y-intercept? Explain.
4. If the 30-year rate differs by 0.3 percent from one month to the next, by how much would you predict the 15-year rate to differ?
5. Predict the 15-year rate for a month when the 30-year rate is 3.5 percent.

To determine

To compute:The least squares regression line for the $15-$ year rate and the $30-$ year rate.

Answer to Problem 26E

The least square regression line of the given data set is,

  $\widehat{y}=-0.3351+0.8940x$

Explanation of Solution

The mortgage rate for $15-$ year and $30-$ year mortgages in the twelve months of $2012$ are given in the following table.

  $\begin{array}{cccc}30-\text{Year}& 15-\text{Year}& 30-\text{Year}& 15-\text{Year}\\ 3.92& 3.20& 3.55& 2.85\\ 3.89& 3.16& 3.60& 2.86\\ 3.95& 3.20& 3.47& 2.78\\ 3.91& 3.14& 3.38& 2.69\\ 3.80& 3.03& 3.35& 2.66\\ 3.68& 2.95& 3.35& 2.66\end{array}$

Calculation:

The least-square regression is given by the formula,

  $\widehat{y}=b_0+b_{1}x$

Where $b_1=r\frac{s_y}{s_x}$ and $b_0=\overline{y}-b_1\overline{x}$

  $r$ is the correlation coefficient.

  $s_x$ is the standard deviation of $x$ .

  $s_y$ is the standard deviation of $y$ .

The correlation coefficient is given by the formula,

  $r=\frac{1}{n-1}{\displaystyle \sum \left(\frac{x-\overline{x}}{s_x}\right)\left(\frac{y-\overline{y}}{s_y}\right)}$

Let $x$ be the $30-$ year rate and $y$ be the $15-$ year rate. Here MINITAB is used.

  

The correlation coefficient can be obtained by the following table.

  $\begin{array}{l}\begin{array}{ccccc}x& y& \frac{x-\overline{x}}{s_x}& \frac{y-\overline{y}}{s_y}& \left(\frac{x-\overline{x}}{s_x}\right)\left(\frac{y-\overline{y}}{s_y}\right)\\ 3.92& 3.20& 1.1297& 1.2709& 1.4357\\ 3.89& 3.16& 1.0022& 1.0814& 1.0838\\ 3.95& 3.20& 1.2572& 1.2709& 1.5978\\ 3.91& 3.14& 1.0887& 0.9867& 1.0728\\ 3.80& 3.03& 0.6197& 0.4657& 0.2886\\ 3.68& 2.95& 0.1098& 0.0868& 0.0095\\ 3.55& 2.85& -0.4427& -0.3868& 0.1712\\ 3.60& 2.86& -0.2302& -0.3394& 0.0781\\ 3.47& 2.78& -0.7827& -0.7183& 0.5622\\ 3.38& 2.69& -1.1651& -1.1446& 1.3336\\ 3.35& 2.66& -1.2926& -1.2867& 1.6632\\ 3.35& 2.66& -1.2926& -1.2867& 1.6632\end{array}\\ {\displaystyle \sum \left(\frac{x-\overline{x}}{s_x}\right)\left(\frac{y-\overline{y}}{s_y}\right)}=10.9598\end{array}$

Hence, the correlation coefficient is,

  $\begin{array}{c}r=\frac{1}{12-1}\times 10.9598\\ =\frac{10.9598}{11}\\ r=0.9963\end{array}$

Then, the coefficient $b_1$ should be,

  $\begin{array}{c}{b}_1=r\frac{s_y}{s_x}\\ =0.9963\times \frac{0.2111}{0.2353}\\ b_1=0.8940\end{array}$

Therefore,

  $\begin{array}{c}{b}_0=\overline{y}-b_1\overline{x}\\ =2.9317-0.8940\times 3.6542\\ =2.9317-3.2668\\ b_0=-0.3351\end{array}$

Conclusion:

The least square regression line is found to be,

  $\widehat{y}=-0.3351+0.8940x$

To determine

To graph:The scatter plot for the two mortgage rates.

Explanation of Solution

Graph:

Taking the $30-$ year ate as $x$ axis and the $15-$ year rate as $y$ axis, the following plot can be constructed.

  

Interpretation:

We can clearly observe that there is a strong linear relationship between these two parameters in the positive direction.

To determine

To explain:The interpretation of the $y$ intercept.

Answer to Problem 26E

No, the $y$ intercept cannot be interpreted.

Explanation of Solution

The lease-square regression line has been computed in the part (a) s,

  $\widehat{y}=-0.3351+0.8940x$

By the constant $b_0$ , the $y$ intercept is denoted. In this case, the value of $b_0$ is said to be $-0.3351$ . Hence, for this relationship there is a negative intercept.

The $y$ intercept is defined as the value the variable $y$ approaches when $x$ goes to zero. Here, the $x$ variable is defined as the $30-$ year mortgage rate while $y$ is the $15-$ year mortgage rate. Regarding the subjected scenario, when the $30-$ year mortgage rate is zero, the $15-$ year mortgage rate should be $-0.3351$ .

Note that a rate of mortgage cannot be negative.

Conclusion:

Therefore, the $y$ intercept cannot be interpreted.

To determine

To calculate:The difference in the percentage of the $15-$ year rate for $0.3$ percent change in the $30-$ year rate.

Answer to Problem 26E

The $15-$ year mortgage rate differs $1.7880$ percent for a $0.3$ percent difference of $30-$ year rate.

Explanation of Solution

The lease-square regression line has been computed in the part (a) s,

  $\widehat{y}=-0.3351+0.8940x$

Calculation:

Let the initial $30-$ year ratebe $x_0$ . The corresponding $15-$ year rate can be obtained as,

  ${\widehat{y}}_{x_0}=-0.3351+0.8940x_0$

Also, the increased rate should be $x_0+0.3$ and the $15-$ year rate should be,

  ${\widehat{y}}_{x_0+0.3}=-0.3351+0.8940\left(x_0+0.3\right)$

Simplifying the obtained weight,

  $\begin{array}{c}{\widehat{y}}_{x_0+0.3}=-0.3351+0.8940\left(x_0+0.3\right)\\ =\underset{{\widehat{y}}_{x_0}}{\underbrace{-0.3351+0.8940\times x_0}}+0.8940\times 2\\ {\widehat{y}}_{x_0+0.3}={\widehat{y}}_{x_0}+0.2082\end{array}$

Therefore, the difference of two weights should be,

  ${\widehat{y}}_{x_0+0.3}-{\widehat{y}}_{x_0}=0.2082$

Interpretation:

According to the calculation, the $15-$ year mortgage rate differs $0.2082$ percent for a $0.3$ percent difference of $30-$ year rate.

To determine

To find:The predicted $15-$ year mortgage rate when $30-$ year rate is $3.5$ percent.

Answer to Problem 26E

When $30-$ year mortgage rate is $3.56$ percent, the $15-$ year mortgage rate should be $2.7938$ percent.

Explanation of Solution

Calculation:

When the $30-$ year mortgage rate is $3.5$ percent, the variable $x$ should be $3.5$ .

  $x=3.5$ .

By substituting this value into the least-square regression line of the relationship, we can obtain the corresponding $y$ value which is the predicted percentage of $15-$ year rate.

  $\begin{array}{c}\widehat{y}=-0.3351+0.8940\times 3.5\\ =-0.3351+3.1289\\ \widehat{y}=2.7938\end{array}$

Conclusion:

The predicted $15-$ year mortgage rate is said to be $2.7938$