Need help with f and g!!!

Question 2 (65 points) Suppose that you decided to buy a car. You're considering four used cars, all of the same make, model, and color. The only differences between the four cars are their age, mileage, and price, which are as follows: Table 1: Price, mileage, and age of four used cars. Suppose you wish to estimate the following regression model so that to better understand the impact of mileage and age on car price: Yi=1+2X2,i+3X3,i+ui, for i=1,,4, where X2i is mileage for car i, and X3i is the age. (a) (10 points) Use OLS to obtain an estimate of the coefficients vector ^. Show your work. You may use the fact that (XX)1=1.85410.00420.53490.00420.00150.01550.53490.01550.3719. (b) (10 points) Find the sampling variance-covariance matrix of ^. Circle the sampling variances of ^1,^2 and ^3. You may use the fact that here, ^2=0.0193. (c) (5 points) Using your estimates, at =5% level of significance, test the hypothesis that mileage has a negative effect on the car price. Use the statistical table at the end of this exam booklet to find the critical values. (d) (7 points) Using your estimates, at =5% level of significance, test whether these cars tend to depreciate by more than $2,000 per year, all else equal? (e) (6 points) With 95% confidence, by at most how much would you expect these cars to depreciate in one year, all else equal? (f) (10 points) Suppose that you unexpectedly found another listing for a car of the same make, model, and color as the four cars in your sample. This fifth car is 3 years old, has 15,000 miles on it and costs $8,700. Use your regression estimates to determine whether this represents a good deal. Clearly explain your reasoning. (g) (10 points) Suppose that you find yet another listing for a car of the same make, model, and color as the four cars in your sample. That sixth car is 5 years old, and costs $6,200. There is no mileage information available for that car. Based on your estimates, how many miles would you expect that car to have on it? That is, which mileage would you expect to be associated with such a car? Question 2 (65 points) Suppose that you decided to buy a car. You're considering four used cars, all of the same make, model, and color. The only differences between the four cars are their age, mileage, and price, which are as follows: Table 1: Price, mileage, and age of four used cars. Suppose you wish to estimate the following regression model so that to better understand the impact of mileage and age on car price: Yi=1+2X2,i+3X3,i+ui, for i=1,,4, where X2i is mileage for car i, and X3i is the age. (a) (10 points) Use OLS to obtain an estimate of the coefficients vector ^. Show your work. You may use the fact that (XX)1=1.85410.00420.53490.00420.00150.01550.53490.01550.3719. (b) (10 points) Find the sampling variance-covariance matrix of ^. Circle the sampling variances of ^1,^2 and ^3. You may use the fact that here, ^2=0.0193. (c) (5 points) Using your estimates, at =5% level of significance, test the hypothesis that mileage has a negative effect on the car price. Use the statistical table at the end of this exam booklet to find the critical values. (d) (7 points) Using your estimates, at =5% level of significance, test whether these cars tend to depreciate by more than $2,000 per year, all else equal? (e) (6 points) With 95% confidence, by at most how much would you expect these cars to depreciate in one year, all else equal? (f) (10 points) Suppose that you unexpectedly found another listing for a car of the same make, model, and color as the four cars in your sample. This fifth car is 3 years old, has 15,000 miles on it and costs $8,700. Use your regression estimates to determine whether this represents a good deal. Clearly explain your reasoning. (g) (10 points) Suppose that you find yet another listing for a car of the same make, model, and color as the four cars in your sample. That sixth car is 5 years old, and costs $6,200. There is no mileage information available for that car. Based on your estimates, how many miles would you expect that car to have on it? That is, which mileage would you expect to be associated with such a car