B01.1305
FINAL EXAM
This is the question sheet. There are 10 questions, each worth 10 points. Please write all answers in the answer book, and justify your answers. Good Luck!
In questions 1-6, we consider data on home prices in the U.S. The data are annual, from 1975 to 2005 (n=31). All Dollar amounts are in 2005 U.S. Dollars.
The y-variable is Home Price, themedian sales price for existing single-family homes.
The first explanatory variable the Mortgage Rate (in percent) based on a Federal Housing Finance Board survey.
The second explanatory variable is Monthly Income, the median income for all owners and renters.
1) Figure 1 plots Home Price vs. Mortgage Rate. The corresponding Minitab simple regression output is given below.
Regression Analysis: Home Price versus Mortgage Rate (%)
The regression equation is
Home Price = 199705 - 5597 Mortgage Rate (%)
Predictor Coef SE Coef T P
Constant 199705 11472 17.41 0.000
Mortgage Rate (%) -5597 1240 -4.51 0.000
S = 16127.1 R-Sq = 41.3% R-Sq(adj) = 39.2%
Analysis of Variance
Source DF SS MS F P
Regression 1 5298869444 5298869444 20.37 0.000
Residual Error 29 7542452012 260084552
Total 30 12841321456
A)Does Figure 1 indicate a linear relationship between Home Price and Mortgage rate? (2 points).
B)Why would we expect that if there is a linear relationship between the variables, it would be a negative one? (1 point).
C)Is there indeed statistical evidence of a negative linear relationship between Home Price and Mortgage Rate? (3 points).
D)According to the model, if the Mortgage Rate goes up by 1%, what happens to the Home Price? (4 points).
2) Figure 2 gives the Fitted Line Plot for Home Price vs. Monthly Income. The corresponding Minitab simple regression output is given below.
Regression Analysis: Home Price versus Monthly Income
The regression equation is
Home Price = - 124066 + 60.8 Monthly Income
Predictor Coef SE Coef T P
Constant -124066 63730 -1.95 0.061
Monthly Income 60.78 14.14 4.30 0.000
S = 16445.7 R-Sq = 38.9% R-Sq(adj) = 36.8%
Analysis of Variance
Source DF SS MS F P
Regression 1 4997965967 4997965967 18.48 0.000
Residual Error 29 7843355489 270460534
Total 30 12841321456
A)Does the fitted line plot suggest any problems with this model? (2 Points).
B)In this model, is there evidence at the 5% level of significance that the true intercept is greater than zero? (3 Points).
C)Construct a 95% confidence interval for the true coefficient of Monthly Income in this model. (3 Points).
D)In 2005, the Monthly Income was $4672, and the Home Price was $219,000. Calculate the residual for this data point.(2 Points).
3)Based on the output for Problems 1 and 2, which single predictor variable (Mortgage Rate or Monthly Income) gives the better predictor of Home Prices? (10 Points).
For Problem 4-6 below, we consider a multiple regression of Home Price on Mortgage Rate and Monthly Income. The Minitab Regression output is given here.
Regression Analysis: Home Price versus Mortgage Rate (%), Monthly Income
The regression equation is
Home Price = 49214 - 3474 Mortgage Rate (%) + 29.2 Monthly Income
Predictor Coef SE Coef T P
Constant 49214 122131 0.40 0.690
Mortgage Rate (%) -3474 2110 -1.65 0.111
Monthly Income 29.20 23.60 1.24 0.226
S = 15981.3 R-Sq = 44.3% R-Sq(adj) = 40.3%
Analysis of Variance
Source DF SS MS F P
Regression 2 5690047870 2845023935 11.14 0.000
Residual Error 28 7151273586 255402628
Total 30 12841321456
4)
A) Test the null hypothesis that the true coefficient of Mortgage Rate in this model is negative, at the 5% level of significance. (3 Points).
B)Does the Minitab output for this multiple regression provide any evidence that either of the two variables is useful for predicting Home Prices? Explain. (5 points).
C) Figure 3 below gives a scatterplot of Mortgage Rate vs. Monthly Income. Does this plot shed any additional light on your discussion in B) above? (2 Points).
5)
A) What does this model predict for the Home Price if the Monthly Income is $5000 and the Mortgage Rate is 8 percent? (5 Points)
B) Use the AICC to decide which model is preferred: the multiple regression, or one of the two simple regressions. For your calculations, you can use log(7542452012)=22.74381, log(7843355489)=22.78293, log(7151273586)=22.69056.
6) Do the plots of Residuals vs. Fitted Values and Residuals vs. Year given in Figs 4 and 5 below suggest any problems with the multiple regression model? (10 Points).
7)Suppose that you have a null hypothesis , an alternative hypothesis , where μ is the population mean. Assume that the population is normally distributed.
A) Given a random sample of size n=10 from this population, suppose you get a sample mean of 5.2 and a sample standard deviation of 2.1. Can the null hypothesis be rejected at the 5% level of significance? (5 Points).
B) Given a random sample of size n=50 from this population, if the sample mean is 5.1 and the sample standard deviation is .3, compute the p-value and interpret the results. (5 Points).
8) In simple linear regression, suppose we have s=0.
Must all the points lie on a straight line? If so, explain why. If not, provide an example. (10 Points).
9) In a simple linear regression with a sample size of n=10, assuming that all of the model assumptions hold, what is the probability that 8 of the errorswill be positive and two will be negative? Give a numerical answer. (10 Points).
10) In a multiple regression, with 3 explanatory variables, suppose that
we obtain a 95% confidence interval for of (1.13274, 6.86077).
Suppose also that the standard error for (that is, SE Coef from Minitab) is 1.211. What is the sample size? (10 Points).