Question

We saw a plot of total mortgages in the United States (in millions of 2005 dollars) versus the interest rate at various times over the past 26 years. The correlation is \(r=-0.84\). The mean mortgage amount is \(\$ 151.9\) million and the mean interest rate is \(8.88 \%\). The standard deviations are \(\$ 23.86\) million for mortgage amounts and \(2.58 \%\) for the interest rates. a) Is a regression model appropriate for predicting mortgage amount from interest rates? Explain. b) What is the equation that predicts mortgage amount from interest rates? c) What would you predict the mortgage amount would be if the interest rates climbed to \(20 \%\) ? d) Do you have any reservations about your prediction in part c? e) If we standardized both variables, what would be the regression equation that predicts standardized mortgage amount from standardized interest rates? f) If we standardized both variables, what would be the regression equation that predicts standardized interest rates from standardized mortgage amount?

Short answer

a) Yes, due to the strong correlation. b) \(y = 221.83 - 7.77x\). c) \(66.43\) million. d) Yes, it's an extrapolation. e) \(Z_{mortgage} = -0.84 \cdot Z_{interest rate}\). f) \(Z_{interest rate} = -0.84 \cdot Z_{mortgage}\).

Step-by-step solution

Determine Appropriateness of Regression Model

A regression model is appropriate if there is a linear relationship indicated by the data. The correlation coefficient \(r = -0.84\) is relatively strong and negative, suggesting a strong linear relationship between interest rates and mortgage amounts. Thus, using a regression model seems appropriate in predicting mortgage amounts from interest rates.

Calculate the Regression Equation

The regression equation predicts the mortgage amount from interest rates. The formula for the regression line is \(y = a + bx\), where \(b = r \cdot \frac{s_y}{s_x}\) is the slope, and \(a = \bar{y} - b\cdot \bar{x}\) is the intercept.Given:- \(r = -0.84\)- \(s_y = 23.86\)- \(s_x = 2.58\)- \(\bar{y} = 151.9\)- \(\bar{x} = 8.88\)First, calculate the slope:\[b = -0.84 \cdot \frac{23.86}{2.58} = -7.77\]Then, calculate the intercept:\[a = 151.9 - (-7.77 \cdot 8.88) = 221.83\]So, the regression equation is:\[y = 221.83 - 7.77x\]

Predict Mortgage Amount at 20% Interest Rate

Substitute \(x = 20\) into the regression equation to find the predicted mortgage amount.\[y = 221.83 - 7.77 \times 20 = 221.83 - 155.4 = 66.43\]Therefore, the predicted mortgage amount when the interest rate is 20% is \(66.43\) million dollars.

Evaluate Prediction Concerns

The prediction made is based on the assumption of a linear relationship. Predicting at \(20\%\) interest rate is an extrapolation beyond the average range given (mean of \(8.88\%\)), which can introduce uncertainties as the model might not account for behaviors outside the observed data range.

Standardized Regression Equation for Mortgage Amount

When both variables are standardized, the regression equation changes. The standardized slope is equal to the correlation coefficient, \(b = r = -0.84\). Therefore, the standardized regression equation for predicting standardized mortgage amount from standardized interest rates is:\[Z_{mortgage} = -0.84 \cdot Z_{interest rate}\]

Standardized Regression Equation for Interest Rates

Similarly, when predicting the standardized interest rate from standardized mortgage amounts, the slope is the inverse of the correlation coefficient, \(b = \frac{1}{r} = -1.19\). Thus, the standardized regression equation is:\[Z_{interest rate} = -0.84 \cdot Z_{mortgage}\]

Key concepts

Correlation Coefficient

The correlation coefficient, often represented as \(r\), is a statistical measure that describes the strength and direction of a linear relationship between two variables. Here, we have \(r = -0.84\), which suggests a strong, negative linear relationship between interest rates and mortgage amounts. This means that as interest rates increase, mortgage amounts tend to decrease, and vice versa. The range of values for the correlation coefficient is between -1 and 1.

• An \(r\) value of 1 indicates a perfect positive linear relationship.
• An \(r\) value of -1 indicates a perfect negative linear relationship.
• An \(r\) value of 0 suggests no linear relationship.

In this scenario, the strong negative correlation justifies using a regression model to predict mortgage amounts from interest rates.

Standard Deviation

Standard deviation is a crucial concept in statistics that represents the amount of variation or dispersion in a dataset. In the context of mortgage amounts and interest rates:

• The standard deviation of mortgage amounts is \(23.86\) million, indicating how much mortgage amounts typically deviate from their average value.
• The standard deviation of interest rates is \(2.58\)%.

A smaller standard deviation indicates that the data points tend to be close to the mean, while a larger standard deviation implies more spread out data points. In regression analysis, standard deviations are used to calculate the slope of the regression line, helping to understand the variability in the data and how different variables are related.

Linear Relationship

A linear relationship suggests a straight-line relationship between two variables. In regression analysis, understanding this relationship is key to predicting one variable from another. The negative correlation coefficient \(-0.84\) implies a strong linear relationship that is negative, meaning:

• As one variable increases, the other decreases.
• In this exercise, as interest rates increase, mortgage amounts decrease.

This linear relationship allows us to construct a regression equation of the form \(y = a + bx\), where \(a\) is the intercept and \(b\) is the slope. The equation \(y = 221.83 - 7.77x\) illustrates this relationship and enables us to make predictions based on interest rate changes.

Standardized Variables

Standardizing variables involves converting them into a common scale with a mean of 0 and a standard deviation of 1. This process simplifies comparison between different datasets or variables. When variables are standardized, the regression equation between them reflects the strength of their relationship as indicated by the correlation coefficient.

• The standardized regression equation for predicting mortgage amounts based on interest rates uses the correlation coefficient as the slope: \(Z_{mortgage} = -0.84 \cdot Z_{interest rate}\).
• Similarly, the standardized regression for predicting interest rates from mortgage amounts is \(Z_{interest rate} = -0.84 \cdot Z_{mortgage}\).

These equations are valuable as they show that even when variables are measured on different scales, their relationship remains consistent when standardized.