1. The shoe size of five randomly selected men in the class is 7, 7.5, 6, 6.5 the shoe size of 4 randomly selected women is 6, 5.

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1 Economics 3 Introduction to Econometrics Winter 2004 Professor Dobkin Name Final Exam (Sample) You must answer all the questions. The exam is closed book and closed notes you may use calculators. You must show your work to receive full credit. The shoe size of five randomly selected men in the class is 7, 7.5, 6, 6.5 the shoe size of 4 randomly selected women is 6, 5.5, 5, 7 a. What is the mean shoe size of men and women? b. What is the Standard Deviation of men and women s shoe size? c. On Average do men or women in these two samples have bigger feet? Are you sure? 2. If the weight of pumpkins is normally distributed with mean 34 pounds and standard deviation 5 pounds. a. What is the probability that a randomly selected pumpkin will weigh between 33 and 40 pounds? b. What is the probability that 4 randomly selected pumpkins will weigh between 33 and 38 pounds? /

2 3. We are interested in the amount of time an athlete spends each week lifting weights and how much this increases the amount they can bench press. We gather the following data at the west field house. Hours lifting Bench a. Compute B 0 and B for the model y = B 0 + B x +u b. Plot the data points and include the fitted regression line c. What are two other variables we may want to include in the regression and why should we include them? d. How much would we expect someone who lifts 40 hours per week to lift? Do we trust the prediction? Why or why not.? e. Is the linear model a good one to fit here? What sort of model might we prefer and why? (Hint Plot the data) 2/2

3 4. How would you reparameterize the following model so that you can run a regression in Stata that will automatically test the hypothesis that one hour running per week running and one hour of weightlifting per week contribute equally to an athletes performance on the mile. Mile time = B 0 + B hr_running + B 2 hr_lifting + B 3 Age+ B 4 alititude + B 5 drugs a. Write out the null and alternative hypothesis b. Rewrite the regression so that you can run it in Stata to estimate your hypothesis directly and get the standard errors. Show your work. c. What new variable will you create for the regression? 5. You are presented with the following regression trying to predict the amount of sleep college students get based on how much they study, how much TV they watch and how many friends they have. Log(sleep) = log(tv) -.9 Log(study) -.6friends (.2) (.0) (.02) N = 330, R 2 =.32 a. Write out the null and alternative hypothesis for the hypothesis that a % increase in TV watching results in a % decrease in sleeping time 3/3

4 b. Did you do a one or two sided t-test? Why? c. Test the hypothesis that a % increase in TV watching results in a % decrease in sleeping time. Is it significant at the 4% level? d. Interpret the coefficient on friends. What is the exact percentage change in the amount of sleep people are getting? 6. You are presented with the following model of worker effort on productivity Productivity = effort -.3 effort 2 a. What is the increase in productivity of a worker increasing their effort by unit if the worker is at an effort level of 7. b. What is the additional productivity of a worker increasing their effort by unit if the worker is at an effort level of 9 c. What level of effort maximizes productivity? 4/4

5 d. If the worker productivity is Productivity = effort -.3 effort effort 3 What is the increase in productivity for an additional unit of effort for a worker that is at an effort level of 7 7. What does it mean that an estimator is consistent? Can a consistent estimator be biased. Please use pictures. 8. You are working as a research assistant and studying the impact of IQ on wage. Your boss gives you the following regression output and tells you to go conduct a onesided test of the impact of IQ on wage at the 5% level. Log(wage) = age +.5 education +.9log(IQ) (.0) (.2) (.03) (.2) N = 24, R 2 =.34 a. State the null and the alternative hypothesis. b. Which side do you conduct the t-test on and why? 5/5

6 c. Does IQ have a statistically significant impact on wage d. Interpret the coefficient on IQ is it practically significant? e. How much will a year increase in education increase a persons wage? 9. We are interested in the amount of bacteria in a flask. We start with 00 bacteria and measure the number of bacteria every hour. We fit the following two models. () Bacteria = B 0 + B log(time) + u (2) Bacteria = B 0 + B time + B 2 time 2 + u The first regression has an SSR of 4504 and a SST The second regression has an SSR of 4554 and a SST We have 2 observations for each regression a. Which model is better and why (hint compare adjusted R-Squareds)? b. Why can t we use the F test to compare these two models? c. Why did we try these two transformations rather than running Bacteria = B 0 + B time + u? 6/6

7 0. B = 2.3 and SE( B ) =.2 they are estimated from a sample of 32 observations. What is the probability that you will get a B bigger than 2.3 in absolute value if the null hypothesis that B =0 is true. (Please draw a picture). Test the hypothesis that conditional on a house being a colonial and on the number of bedrooms the size of the house and the size of the lot it sits on are not important predictors of price.. reg lprice bdrms lotsize colonial sqrft Source SS df MS Number of obs = F( 4, 83) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = lprice Coef. Std. Err. t P> t [95% Conf. Interval] bdrms lotsize 5.65e e e e-06 colonial sqrft _cons reg lprice colonial bdrms Source SS df MS Number of obs = F( 2, 85) =.75 Model Prob > F = Residual R-squared = Adj R-squared = 0.98 Total Root MSE = lprice Coef. Std. Err. t P> t [95% Conf. Interval] colonial bdrms _cons /7

8 n i= B = ( x x)( y y) n i i= i ( x x) ( SSRr SSRur)/ q F = SSR /( n k ) ur i 2 Useful Formulas B = y B x o Sample Variance = n ( X i X ) n i= 2 2 SSR /( n k ) R = SST /( n ) 2 % y = 00[exp( B ) ] 8/8