STA 4163 Lecture 10: Practice Problems

Transcription

1 STA 463 Lecture 0: Practice Problems Problem.0: A study was conducted to determine whether a student's final grade in STA406 is linearly related to his or her performance on the MATH ability test before entering college. OBS MATH FINAL Given that SSxx = 2474, SSxy = 894, SSyy = 2056, x = 46, and y = 76 (a) Fit a least squares line to relate the math score and the final grade.

2 (b) Based on Figure.8, do you think the straight-line model fits the data well? 2

3 3 (c) Calculate SSE, s 2, and s. (d) Do the data provide sufficient evidence to indicate that a positive relationship exists between math score and final grade at α = 0.05? (e) Find the observed significance level for the test in part (d) using the following SAS printout. Model: MODEL Dependent Variable: FINAL Analysis of Variance Sum of Mean Source DF Squares Square F Value Prob>F Model Error 8 C Total 9 Root MSE R-square Dep Mean Adj R-sq C.V Parameter Estimates Parameter Standard T for H0: Variable DF Estimate Error Parameter=0 Prob > T INTERCEP MATH

4 4 (f) Calculate the coefficient of determination and the sample correlation coefficient. (g) Find a 95% confidence interval for the mean final grade for students scoring 35 on the math exam. (h) Find a 90% prediction interval when math score is 50. (i) Find a 95% confidence interval for the slope β. Problem.. Each year U.S. News & World Report conducts its Survey of America s Best Graduate and Professional Schools. The top 25 business schools in 99, as determined by reputation, student selectivity, placement success, and graduation rate, are listed in the table. For each school, three variables were measured: () GMAT score for the typical incoming student; (2) student acceptance rate (percentage accepted of all students who applied); and (3) starting salary of the typical graduating student. School GMAT Acc. Rate Salary. Harvard % $63, Stanford , Penn , Northwestern , MIT , Chicago , Duke , Dartmouth , Virginia , Michigan ,300. Columbia , Cornell , CMU , UNC ,800

5 5 5. Cal-Berkeley , UCLA , Texas , Indiana ,9 9. NYU ,6 20. Purdue , USC , Pittsburgh , Georgetown , Maryland , Rochester ,499 The academic advisor wants to predict the typical starting salary of a graduate at a top business school using GMAT score of the school as a predictor variable. A simple linear regression of SALARY versus GMAT using the 25 data points in the table are shown below. 2 β = β = 228 s= 323 R =. 66 r =. 8 df = 23 t = Write the equation of the probabilistic model of interest. a. Salary = β 0 + β (GMAT) b. Salary = β 0 + β ( GMAT) + c. GMAT = β 0 + β (SALARY) d. GMAT = β + β (SALARY) Write the equation of the least squares line. a. SALARY = (GMAT) b. SALARY = (GMAT)

6 6 c. GMAT = (SALARY) d. GMAT = (SALARY) 3. Give a practical interpretation of β 0 = a. We expect to predict SALARY to within 2(92040) = $84,080 of its true value using GMAT in a straight-line model. b. We estimate SALARY to decrease $92,040 for every -point increase in GMAT. c. We estimate the base SALARY of graduates of a top business school to be $ 92,040. d. The value has no practical interpretation since a GMAT of 0 is nonsensical and outside the range of the sample data. 4. Give a practical interpretation of β = 228. a. We expect to predict SALARY to within 2(228) = $456 of its true value using GMAT in a straight-line model. b. We estimate SALARY to increase $228 for every -point increase in GMAT. c. We estimate GMAT to increase 228 points for every $ increase in SALARY. d. The value has no practical interpretation since a GMAT of 0 is nonsensical and outside the range of the sample data. 5. Give a practical interpretation of R 2 =.66. a. We expect to predict SALARY to within 2d. 66i= $, 620 of its true value using GMAT in a straight-line model. b. We estimate SALARY to increase $.66 for every -point increase in GMAT. c. 66 % of the sample variation in SALARY can be explained by using GMAT in a straight-line model. d. We can predict SALARY correctly 66 % of the time using GMAT in a straight-line model.

7 7 6. Give a practical interpretation of r =.8. a. We estimate SALARY to increase 8 % for every -point increase in GMAT. b. There appears to be a positive correlation between SALARY and GMAT. c. 8 % of the sample variation in SALARY can be explained by using GMAT in a straight-line model. d. We can predict SALARY correctly 8 % of the time using GMAT in a straight-line model. and GMAT. 7. Set up the null and alternative hypotheses for testing whether a positive linear relationship exists between SALARY a. H0: β = 0 vs. Ha: β > 0 b. H0: β = 0 vs. Ha: β 0 c. H0: β = 228 vs. Ha: β > 228 d. H0: β > 0 vs. Ha : β < 0 8. A 95 % prediction interval for SALARY when GMAT = 600 is ($37,95, $5,984). Interpret this interval. a. We are 95 % confident that the SALARY of a top business school graduate with a GMAT of 600 will fall between $37,95 and $5,984. b. We are 95 % confident that the mean SALARY of all top business school graduates with GMATs of 600 will fall between $37,95 and $5,984. c. We are 95 % confident that the increase in SALARY for a 600-point increase in GMAT will fall between $37,95 and $5,984. d. We are 95 % confident that the SALARY of a top business school graduate will fall between $37,95 and $5, Which of the following is not an assumption required for the simple linear regression analysis to be valid? a. The errors of predicting SALARY are normally distributed.

8 8 b. The errors of predicting SALARY have a mean of 0. c. The errors of predicting SALARY have a variance that is constant for any given value of GMAT. d. SALARY is independent of GMAT. 0. The Real Estate Appraiser (992) reported the results of a regression analysis designed to predict the price (y) measured in dollars of residential properties recently sold in a northern Virginia subdivision. One independent variable used to predict sale price is GLA gross living area (x) measured in square feet. Data for 57 properties were used to fit the model The results of the simple linear regression are provided below. 2 y = 96, x s= 6500 R =. 77 t 6. (for testing β ) E( y) = β + β x Which of the following four assumptions concerning the probability distribution of the random error term is stated incorrectly? a. The distribution is normal. b. The mean of the distribution is 0. c. The variance of the distribution increases as x increases. d. The errors are independent. 0 Please complete these problems on your own. You may then check your answers when you are finished. Course Pages maintained by: Dr. Morgan C. Wang University of Central Florida, Copyright 2008.