ECO 578 EXAM SPRING 2016 Page 1 of 20 Name: _______________ ID: _______________ Final Exam Final Exam There are 4 parts: Part A: Select the correct answer for the following questions (1-11) Part B: True/ False (12-20) Part C: Fill in the blank and circle your decision or answer the following questions (21-25). Part D:Work Problem (26-38) **Excel is not allowed. &All work must be shown step by step** Two different ways to submit your answer sheet 1. Scan your answer sheet and place it in ONE FILE at drop-box. (preferable) 2. Use MS-Word and place it in a drop-box. **Excel is not acceptable for this test **Deadline: Monday, 9th of May 2016 by noon **Part D: All work must be shown step by step in order to receive Part A: Multiple Choice (1-11) ______1. Using the \"eyeball\" method, the regression line ^y = 2+2x has been fitted to the data points (x = 2, y = 1), (x = 3, y = 8), and (x = 4, y = 7). The sum of the squared residuals will be a. 7 b. 19 c. 34 d. 8 ______2. A computer statistical package has included the following quantities in its output: SST = 50, SSR = 35, and SSE = 15. How much of the variation in y is explained by the regression equation? a. 49% b. 70% c. 35% d. 15% ______3. a. = In testing the significance of b1 b. 0 b1 , the null hypothesis is generally that c. = 0 d. = r ______4. Testing whether the slope of the population regression line could be zero is equivalent to testing whether the population _____________ could be zero. a. standard error of estimate c. y-intercept b. prediction interval d. coefficient of correlation ECO 578 EXAM SPRING 2016 Page 2 of 20 ______5. A multiple regression equation includes 4 independent variables, and the coefficient of multiple determination is 0.64. How much of the variation in y is explained by the regression equation? a. 80% b. 16% c. 32% d. 64% ______6. A multiple regression analysis results in the following values for the sum-ofsquares terms: SST = 50.0, SSR = 35.0, and SSE = 15.0. The coefficient of multiple determination will be 2 2 2 R = 0.35 a. b. R = 0.30 c. R = 0.70 d. R 2 = 0.50 ______7. In testing the overall significance of a multiple regression equation in which there are three independent variables, the null hypothesis is H0 1= 2= 3=0 a. : b. H0 : 1 2 2 3 c. H0 : 1= 2= 3=1 d. H0 : 1= 2= 3= ______8. In a multiple regression analysis involving 25 data points and 4 independent variables, the sum-of-squares terms are calculated as SSR = 120, SSE = 80, and SST = 200. In testing the overall significance of the regression equation, the calculated value of the test statistic will be a. F = 1.5 c. F = 5.5 b. F = 2.5 d. F = 7.5 ______9. For a set of 15 data points, a computer statistical package has found the multiple x x x regression equation to be ^y = -23 + 20 1 + 5 2 + 25 3 and has listed the tratio for testing the significance of each partial regression coefficient. Using the 0.05 b1 level in testing whether = 20 differs significantly from zero, the critical t values will be a. b. c. d. t = -1.960 and t= +1.960 t = -2.132 and t = +2.132 t = -2.201 and t = +2.201 t = -1.796 and t = +1.796 ECO 578 EXAM SPRING 2016 Page 3 of 20 ______10. Computer analyses typically provide a p-Value for each partial regression b coefficient. In the case of 1 , this is the probability that a. 1 =0 b. b1 = 1 c. the absolute value of b1 could be this large if 1 d. the absolute value of b1 could be this large if 1 ______11. In the multiple regression equation, ^y is the estimated household income, head of the household, and x2 x1 ^y 1 = 20,000 + 0.05 x1 + 4500 x2 , is the amount of life insurance held by the is a dummy variable ( mutual funds, 0 if it doesn't). The interpretation of a. b. c. d. =0 b2 x2 = 1 if the family owns = 4500 is that owing mutual funds increases the estimated income by $4500 the average value of a mutual funds portfolio is $4500 45% of the persons in the sample own mutual funds the sample size must have been at least n = 4500 Part B: True or False (12-20) _______ 12. The usual objective of regression analysis is to predict estimate the value of one variable when the value of another variable is known. _______ 13. Correlation analysis is concerned with measuring the strength of the relationship between two variables. _______ 14. In the least squares model, the explained sum of squares is always smaller than the regression sum of squares. _______ 15. The sample correlation coefficient and the sample slope will always have the same sign. ^ (^ ) _______ 16. An important relationship in regression analysis is ( Y iY )= Y Y +(Y iY ) . _______ 17. If in a regression analysis the explained sum of squares is 75 and the unexplained sum of square is 25, r2 = 0.33. ECO 578 EXAM SPRING 2016 Page 4 of 20 _______ 18. When small values of Y tend to be paired with small values of X, the relationship between X and Y is said to be inverse. _______ 19. The probability that the test statistic will fall in the critical region, given that H0 is true, represents the probability of making a type II error. _______ 20. When we reject a true null hypothesis, we commit a Type I error. Part C: Please fill in the blank and circle your decision or answer the following questions (21-25). 21. State whether you would reject or fail to reject the null hypothesis in each of the following cases (two-tailed): Make a decision. a) P = 0.12 ; =0.05 Decision Reject / Fail to reject b) P = 0.03 ; =0.05 Decision Reject / Fail to reject c) P = 0.001 ; =0.01 22. State whether F =computed F H0 Decision Reject / Fail to reject should be accepted or rejected for =0.05 , given the following; (fill in the blank and circle your decision) a) F = 2.34; df = 2 and 11 Computed F Critical F Decision ECO 578 EXAM SPRING 2016 ________ Page 5 of 20 ________ Reject / Fail to reject Computed F Critical F Decision ________ ________ Reject / Fail to reject Computed F Critical F Decision ________ ________ Reject / Fail to reject b) F = 2.52; df = 4 and 20 c) F = 4.29; df = 3 and 24 23. Given the following, complete the ANOVA table and make the correct inference. Using F-value to make a decision. Source SS df MS F Treatments _____ 2 3.24 _____ Error _____ 17 _____ Total 40.98 _____ ANSWER a) What is the hypothesis being tested in this problem? b) In the above ANOVA table, is the factor significant at the 5% level? c) What is observations? the number of 24. Given the following, complete the ANOVA table and make the correct inference. Using F-value to make a decision. Source SS df MS F Treatments ____ 5 205.5 ____ ECO 578 EXAM SPRING 2016 Page 6 of 20 Error 637 ____ Total ____ 25 ____ ANSWER a) What is the hypothesis being tested in this problem? b) In the above ANOVA table, is the factor significant at the 5% level? c) What is observations? the number of 25. Given the following, complete the ANOVA table and make the correct inference. Using F-value to make a decision. Source SS df MS F Treatments ____ 3 ____ ____ Error 88.8 ____ ____ Total 435 19 ANSWER a) What is the hypothesis being tested in this problem? b) In the above ANOVA table, is the factor significant at the 5% level? c) What is observations? the number of Part D: Must show all your work step by step in order to receive the full credit; Excel is not allowed. (26-38) 26. Consider the following hypothesis test. Ho: = 17 Ha: 17 ECO 578 EXAM SPRING 2016 Page 7 of 20 A sample of 25 gives a sample mean of 14.2 and sample variance of 25. a ) At 5% should the null be rejected? b ) Compute the value of the test statistic c ) What is the p-value? d ) What is your conclusion? a ) 27. Consider the following hypothesis test Ho: 10 Ha: < 10 A sample of 50 provides a sample mean of 9.46 and sample variance of 4. b At 5% should the null be rejected? ) Compute the value of the test statistic ECO 578 EXAM SPRING 2016 c ) What is the p-value? Page 8 of 20 d ) What is your conclusion? 28. Use problem 13 on page 9-20 to answer the following questions. a) What is the sample mean b) What is the sample standard deviation c) Use Z or T test? And why? d) What is your hypothesis test ECO 578 EXAM SPRING 2016 Page 9 of 20 e) At = 0.05, what is the rejection rule? f) Compute the value of the test statistic. g) What is the p-value? h) What is your conclusion? 29. Use problem 7 on page 9-36 to answer to following questions. a ) At 5% should the null be rejected? b ) Compute the value of the test statistic ECO 578 EXAM SPRING 2016 c ) What is the p-value? Page 10 of 20 d ) What is your conclusion? ECO 578 EXAM SPRING 2016 Page 11 of 20 30. For n=6 data point, the following quantities have been calculated. a)Find b1 xy = 400 x = 40 y = 76 x2 = 346 b)Find b0 y2 = 1,160 ECO 578 EXAM SPRING 2016 c)Write the equation Page 12 of 20 d)Find 2 R Se e)Find SST f)Find g) Construct the 95% confidence interval for the mean of y when x = 7 h) Construct the 95% confidence interval for the individual value of y when x = 7 31. Please fill in the computer printout and answer the following questions. Given that =0.05 ECO 578 EXAM SPRING 2016 Page 13 of 20 SUMMARY OUTPUT Regression Statistics Multiple R 0.9037 R Square ______ Adjusted R Square ______ Standard Error ______ Observations 5 ANOVA df SS MS F Significance F Regression 1 4.9 ______ ______ 0.03535 Residual 3 1.1 ______ ______ ______ Total Coefficients Standard Error t Stat Intercept -0.1 ______ -0.15746 X1 0.7 ______ 3.65563 P-value 0.8848 8 0.0353 5 Lower95 % Upper 95% Lower 95.0% Upper 95.0% -2.12112 1.92112 -2.12112 1.92112 0.09061 1.30939 ______ ______ a) What percent of the variation is explained by the regression equation? b) What is the standard error of regression? c) What is the critical value of the F-statistic? d) What sample size is used in the print out? ECO 578 EXAM SPRING 2016 Page 14 of 20 32. The following regression equation was obtained using the five independent variables. Given that =0.05 a) What percent of the variation is explained by the regression equation b) What is the standard error of regression? c) Write the estimated equation. d) What is the critical value of the F-statistic? ECO 578 EXAM SPRING 2016 e) What sample size is used in the print out? Page 15 of 20 f) What is the variance of the slope coefficient of income? g) Assuming that you are using a two-tailed test make a decision using the computed P-value. 33. Use problem 5 on page 15-9 to answer the following questions. (a-d) a) What is you hypothesis testing b) What is your 2 ? ECO 578 EXAM SPRING 2016 c) Is the 2 Page 16 of 20 value significant at 5% level of significance? d) Write the conclusion for this question. 34. Eighty Frat Rats were blindfolded and given a taste test on different brands of beer. The table below shows the beer brands preferred. Test to see if the data fits a uniform distribution (i.e., what would happen if the Frat Rats can't tell one beer from another). Use 5% level of significance. ECO 578 EXAM SPRING 2016 Beer Brand Page 17 of 20 Number of Preferring that Brand 32 18 14 16 Bud Light Coors Light Molton Golden Miller Light a) What is you hypothesis testing b) What is your c) Is the 2 ? 2 value significant at 5% level of significance? d) Write the conclusion for this question. ECO 578 EXAM SPRING 2016 Page 18 of 20 35. Use the computer printout below to answer the following questions. Coefficients 729.8665 -10.887 0.0465 Intercept Price Advertising Std. Error 169.25751 3.4952397 0.0176228 t-Stat 4.3121659 -3.1148078 2.6386297 P-value 0.0010099 0.0089406 0.0216284 F 37.56127994 Significance F 0.00000683 ANOVA Regression Residual Total Se =12.86986 df 2 12 14 SS 12442.8 1987.6 14430.4 R-sq = 0.862263 MS 6221.4 165.63333 R-sq(adj) = 0.8393068 =5 a) Write and interpret the multiple regression equation b) Does the model with Price and Advertising contribute to the prediction of Y? c) Which independent variable appears to be the best predictor of sales? Explain. d) What is the number of observations used in this study? e)Find SST Assuming that the coefficient on Advertising has Ha: B1 > 0, what's statistical decision ? f) What is the standard error of estimate? Can you use this statistic to assess the model's fit? If so, how? g) What is the coefficient of determination, and what does it tell you about the regression model? h) What is the coefficient of determination, adjusted for degrees of freedom? What do this statistic and the statistic referr to in part (g) tell you about how well this model fits that data. ECO 578 EXAM SPRING 2016 Page 19 of 20 i) Test the overall utility of the model. What does the p-value of the test statistic tell you? 36. Explain why a 95% confidence interval estimate for the mean value of y at a particular x is narrower than a 95% confidence interval for an individual y value at the same value of x. 37. If 2 R =0.95 , n=11, and (Y Y )2=100, 2 what is S e ? 38. You are given the following information from fitting a multiple regression with three variables to 30 sample data points: b1=1.3 b2=10.0 b3 =25.2 S b =0.3 1 S b =2.0 Test the following hypotheses, Use =0.05 2 S b =15.0 3 ECO 578 EXAM SPRING 2016 a) 1=0 b) 2=0 a) 3=0 Page 20 of 20 Use problem 13 on page 9-20 Use problem 7 on page 9-36 - Use problem 5 on page 15-9 Name: _______________ ID: _______________ Final Exam Final Exam There are 4 parts: Part A: Select the correct answer for the following questions (1-11) Part B: True/ False (12-20) Part C: Fill in the blank and circle your decision or answer the following questions (21-25). Part D:Work Problem (26-38) **Excel is not allowed. &All work must be shown step by step** Two different ways to submit your answer sheet 1. Scan your answer sheet and place it in ONE FILE at drop-box. (preferable) 2. Use MS-Word and place it in a drop-box. **Excel is not acceptable for this test **Deadline: Monday, 9th of May 2016 by noon **Part D: All work must be shown step by step in order to receive Part A: Multiple Choice (1-11) ______1. Using the \"eyeball\" method, the regression line ^y = 2+2x has been fitted to the data points (x = 2, y = 1), (x = 3, y = 8), and (x = 4, y = 7). The sum of the squared residuals will be a. 7 b. 19 c. 34 d. 8 ______2. A computer statistical package has included the following quantities in its output: SST = 50, SSR = 35, and SSE = 15. How much of the variation in y is explained by the regression equation? a. 49% b. 70% c. 35% d. 15% ______3. a. = In testing the significance of b1 b. 0 b1 , the null hypothesis is generally that c. = 0 d. = r ______4. Testing whether the slope of the population regression line could be zero is equivalent to testing whether the population _____________ could be zero. a. standard error of estimate c. y-intercept b. prediction interval d. coefficient of correlation ______5. A multiple regression equation includes 4 independent variables, and the coefficient of multiple determination is 0.64. How much of the variation in y is explained by the regression equation? a. 80% b. 16% c. 32% d. 64% ______6. A multiple regression analysis results in the following values for the sum-ofsquares terms: SST = 50.0, SSR = 35.0, and SSE = 15.0. The coefficient of multiple determination will be 2 2 2 R = 0.35 a. b. R = 0.30 c. R = 0.70 d. R 2 = 0.50 ______7. In testing the overall significance of a multiple regression equation in which there are three independent variables, the null hypothesis is H0 1= 2= 3=0 a. : b. H0 : 1 2 2 3 c. H0 : 1= 2= 3=1 d. H0 : 1= 2= 3= ______8. In a multiple regression analysis involving 25 data points and 4 independent variables, the sum-of-squares terms are calculated as SSR = 120, SSE = 80, and SST = 200. In testing the overall significance of the regression equation, the calculated value of the test statistic will be a. F = 1.5 c. F = 5.5 b. F = 2.5 d. F = 7.5 ______9. For a set of 15 data points, a computer statistical package has found the multiple x x x regression equation to be ^y = -23 + 20 1 + 5 2 + 25 3 and has listed the tratio for testing the significance of each partial regression coefficient. Using the 0.05 b1 level in testing whether = 20 differs significantly from zero, the critical t values will be a. b. c. d. t = -1.960 and t= +1.960 t = -2.132 and t = +2.132 t = -2.201 and t = +2.201 t = -1.796 and t = +1.796 ______10. Computer analyses typically provide a p-Value for each partial regression b coefficient. In the case of 1 , this is the probability that a. 1 =0 b. b1 = 1 c. the absolute value of b1 could be this large if 1 d. the absolute value of b1 could be this large if 1 ______11. In the multiple regression equation, ^y is the estimated household income, head of the household, and x2 x1 ^y 1 = 20,000 + 0.05 x1 + 4500 x2 , is the amount of life insurance held by the is a dummy variable ( mutual funds, 0 if it doesn't). The interpretation of a. b. c. d. =0 b2 x2 = 1 if the family owns = 4500 is that owing mutual funds increases the estimated income by $4500 the average value of a mutual funds portfolio is $4500 45% of the persons in the sample own mutual funds the sample size must have been at least n = 4500 Part B: True or False (12-20) True ____ 12. The usual objective of regression analysis is to predict estimate the value of one variable when the value of another variable is known. True _______ 13. Correlation analysis is concerned with measuring the strength of the relationship between two variables. False_ 14. In the least squares model, the explained sum of squares is always smaller than the regression sum of squares. True____ 15. The sample correlation coefficient and the sample slope will always have the same sign. ^ (^ ) True ___ 16. An important relationship in regression analysis is ( Y iY )= Y Y +( Y iY ) . True ___ 17. If in a regression analysis the explained sum of squares is 75 and the unexplained sum of square is 25, r2 = 0.33. False_____ 18. When small values of Y tend to be paired with small values of X, the relationship between X and Y is said to be inverse. False ___ 19. The probability that the test statistic will fall in the critical region, given that H0 is true, represents the probability of making a type II error. True_____ 20. When we reject a true null hypothesis, we commit a Type I error. Part C: Please fill in the blank and circle your decision or answer the following questions (21-25). 21. State whether you would reject or fail to reject the null hypothesis in each of the following cases (two-tailed): Make a decision. a) P = 0.12 ; =0.05 Decision Reject / Fail to reject b) P = 0.03 ; =0.05 Decision Reject / Fail to reject c) P = 0.001 ; =0.01 22. State whether F =computed F H0 Decision Reject / Fail to reject should be accepted or rejected for =0.05 , given the following; (fill in the blank and circle your decision) a) F = 2.34; df = 2 and 11 Computed F Critical F Decision __2.34______ ___ Reject / Fail to reject 3.982_____ b) F = 2.52; df = 4 and 20 Computed F Critical F Decision _____2.52___ Reject / Fail to reject c) F = 4.29; df = 3 and 24 Computed F Critical F Decision ___4.29_____ _3.87_ Reject / Fail to reject 23. Given the following, complete the ANOVA table and make the correct inference. Using F-value to make a decision. Source SS Treatment ___6.48_ s _ ___34.5_ Error _ Total 40.98 Df MS F __1.6__ _ 2 3.24 17 ___2.029 __ ____19 _ ANSWER a) What is the hypothesis being tested in this problem? Ho: b1=0, No factor significant b) In the above ANOVA table, is the factor significant at the 5% level? c) What is observations? the number of No 20 24. Given the following, complete the ANOVA table and make the correct inference. Using F-value to make a decision. Source Treatmen ts SS ___1027. 5_ df MS F 5 205.5 ___6_ Error 637 _20___ __31.85 __ Total ___1664.5 25 ANSWER a) What is the hypothesis being tested in this problem? B1=B2=B3=B4=0 b) In the above ANOVA table, is the factor significant at the 5% level? c) What is observations? the number Yes of 26 25. Given the following, complete the ANOVA table and make the correct inference. Using F-value to make a decision. Source Treatmen ts Error Total SS __346.2 __ 88.8 435 df MS F __115.4 ____20.7 3 __ 9 __16__ __5.55__ 19 ANSWER a) What is the hypothesis being tested in this problem? B1=B2=B3=0 b) In the above ANOVA table, is the factor significant at the 5% level? YES c) What is observations? 20 the number of Part D: Must show all your work step by step in order to receive the full credit; Excel is not allowed. (26-38) 26. Consider the following hypothesis test. Ho: = 17 Ha: 17 A sample of 25 gives a sample mean of 14.2 and sample variance of 25. a ) At 5% should the null be rejected? b ) Compute the value of the test statistic Yes T= c ) What is the p-value? d.f=25-1=24 d ) 14.217 5/ 25 = -2.8 What is your conclusion? 0.0001 a ) 17 27. Consider the following hypothesis test Ho: 10 Ha: < 10 A sample of 50 provides a sample mean of 9.46 and sample variance of 4. b At 5% should the null be rejected? ) Compute the value of the test statistic No Z= c ) What is the p-value? d ) 9.4610 2/ 50 = -1.909 What is your conclusion? 0.028 Ho: 10 28. Use problem 13 on page 9-20 to answer the following questions. a) What is the sample mean b) What is the sample standard deviation S=sqrt(1/9(219)= Mean=1189/10= 118.9 c) Use Z or T test? And why? t- as sample size is n<30 and population 4.931757 d) what is your hypothesis test h0: manufacturing process standard deviation unknown not functioning properly ha: 118.91 2 0 t =4.93/ 1 -0.7055 e) at the rejection rule? reject ho p<0.05 f) p-value? 9 degrees of freedom 0.2490 g) conclusion? rejected we conclude that 29. use problem 7 on page 9-36 to answer following questions. a ) 5% should null be rejected? b compute value statistic h0 5.66.9 z =c 0.4219 d 482 17 + 64 38 =0.197 mean are different 30. for n =6 data point, quantities have been calculated. a)find b1 xy =400 x =40 y =76 x2 =346 b)find b0 =b0 7634640400 6346402 y2 =1,160 >7.8147 Observed is greater Rejectt IID U(0, 1) 35. Use the computer printout below to answer the following questions. Intercept Coefficients 729.8665 Std. Error 169.25751 t-Stat 4.3121659 P-value 0.0010099 Price Advertising -10.887 0.0465 3.4952397 0.0176228 -3.1148078 2.6386297 0.0089406 0.0216284 F 37.56127994 Significance F 0.00000683 ANOVA Regression Residual Total df 2 12 14 Se =12.86986 SS 12442.8 1987.6 14430.4 R-sq = 0.862263 a) Write and interpret the multiple regression equation Y=729.8665-10887X+0.0465 Where y= 729.8665 is the value of y when determinants MS 6221.4 165.63333 R-sq(adj) = 0.8393068 =5 b) Does the model with Price and Advertising contribute to the prediction of Y? Yes, both has p-value less than 0.05 are equal to zero -10.887 is the change in y as a result of unit change in price holding advertising as a constant. 0.0465 is the change in y as a result of unit change in advertising holding price as a constant. c) Which independent variable appears to be the best predictor of sales? Explain. Price as it has the smallest p value e)Find SST Assuming that the coefficient on Advertising has Ha: B1 > 0, what's statistical decision ? SST=14430.4 Reject H0 d) What is the number of observations used in this study? n=14+1=15 f) What is the standard error of estimate? Can you use this statistic to assess the model's fit? If so, how? Se =12.86986 No g) What is the coefficient of determination, and what does it tell you about the regression model? h) What is the coefficient of determination, adjusted for degrees of freedom? What do this statistic and the statistic referr to in part (g) tell you about how well this model fits that data. R-sq(adj) = 0.8393068 R2=0.862263 R2=0.862263 means that 86.2263% of y is explained by the model i) Test the overall utility of the model. What does the p-value of the test statistic tell you? P value=0.00000683 The model is significant 36. Explain why a 95% confidence interval estimate for the mean value of y at a particular x is narrower than a 95% confidence interval for an individual y value at the same value of x. This is caused by the standard error for mean value of y at a particular x is largr than standard error individual y value at the same value of x. 37. If R= R2=0.95 , n=11, and 1 (Y Y )2=100, sse sst 0.95=1-sse/100 Sse=5 S 2e =sse/n-k-1=5/(11-1-1)=5/9 2 what is S e ? 38. You are given the following information from fitting a multiple regression with three variables to 30 sample data points: b1=1.3 b2=10.0 b3 =25.2 S b =0.3 1 S b =2.0 Test the following hypotheses, Use =0.05 a) 1=0 H0: 1=0 Ha: 1 0 t =1.3/0.3=4.2 t critical at 27(n-k)d.f=2.0518 4.2>2.0518, Reject Ho, Conclude b) 1 0 2=0 H0: 2=0 Ha: 2 0 T=10/2=5 t critical at 27(n-k)d.f=2.0518 5>2.05, reject Ho a) H0: 3=0 3=0 2 S b =15.0 3 Ha: 3 0 T=25.2/15=1.68 t critical at 27(n-k)d.f=2.0518 1.68<2.0518 Hence we fail to reject Ho B3 is actually no significant Name: _______________ ID: _______________ Final Exam Final Exam There are 4 parts: Part A: Select the correct answer for the following questions (1-11) Part B: True/ False (12-20) Part C: Fill in the blank and circle your decision or answer the following questions (21-25). Part D:Work Problem (26-38) **Excel is not allowed. &All work must be shown step by step** Two different ways to submit your answer sheet 1. Scan your answer sheet and place it in ONE FILE at drop-box. (preferable) 2. Use MS-Word and place it in a drop-box. **Excel is not acceptable for this test **Deadline: Monday, 9th of May 2016 by noon **Part D: All work must be shown step by step in order to receive Part A: Multiple Choice (1-11) ______1. Using the \"eyeball\" method, the regression line ^y = 2+2x has been fitted to the data points (x = 2, y = 1), (x = 3, y = 8), and (x = 4, y = 7). The sum of the squared residuals will be a. 7 b. 19 c. 34 d. 8 ______2. A computer statistical package has included the following quantities in its output: SST = 50, SSR = 35, and SSE = 15. How much of the variation in y is explained by the regression equation? a. 49% b. 70% c. 35% d. 15% ______3. a. = In testing the significance of b1 b. 0 b1 , the null hypothesis is generally that c. = 0 d. = r ______4. Testing whether the slope of the population regression line could be zero is equivalent to testing whether the population _____________ could be zero. a. standard error of estimate c. y-intercept b. prediction interval d. coefficient of correlation ______5. A multiple regression equation includes 4 independent variables, and the coefficient of multiple determination is 0.64. How much of the variation in y is explained by the regression equation? a. 80% b. 16% c. 32% d. 64% ______6. A multiple regression analysis results in the following values for the sum-ofsquares terms: SST = 50.0, SSR = 35.0, and SSE = 15.0. The coefficient of multiple determination will be 2 2 2 R = 0.35 a. b. R = 0.30 c. R = 0.70 d. R 2 = 0.50 ______7. In testing the overall significance of a multiple regression equation in which there are three independent variables, the null hypothesis is H0 1= 2= 3=0 a. : b. H0 : 1 2 2 3 c. H0 : 1= 2= 3=1 d. H0 : 1= 2= 3= ______8. In a multiple regression analysis involving 25 data points and 4 independent variables, the sum-of-squares terms are calculated as SSR = 120, SSE = 80, and SST = 200. In testing the overall significance of the regression equation, the calculated value of the test statistic will be a. F = 1.5 c. F = 5.5 b. F = 2.5 d. F = 7.5 ______9. For a set of 15 data points, a computer statistical package has found the multiple x x x regression equation to be ^y = -23 + 20 1 + 5 2 + 25 3 and has listed the tratio for testing the significance of each partial regression coefficient. Using the 0.05 b1 level in testing whether = 20 differs significantly from zero, the critical t values will be a. b. c. d. t = -1.960 and t= +1.960 t = -2.132 and t = +2.132 t = -2.201 and t = +2.201 t = -1.796 and t = +1.796 ______10. Computer analyses typically provide a p-Value for each partial regression b coefficient. In the case of 1 , this is the probability that a. 1 =0 b. b1 = 1 c. the absolute value of b1 could be this large if 1 d. the absolute value of b1 could be this large if 1 ______11. In the multiple regression equation, ^y is the estimated household income, head of the household, and x2 x1 ^y 1 = 20,000 + 0.05 x1 + 4500 x2 , is the amount of life insurance held by the is a dummy variable ( mutual funds, 0 if it doesn't). The interpretation of a. b. c. d. =0 b2 x2 = 1 if the family owns = 4500 is that owing mutual funds increases the estimated income by $4500 the average value of a mutual funds portfolio is $4500 45% of the persons in the sample own mutual funds the sample size must have been at least n = 4500 Part B: True or False (12-20) True ____ 12. The usual objective of regression analysis is to predict estimate the value of one variable when the value of another variable is known. True _______ 13. Correlation analysis is concerned with measuring the strength of the relationship between two variables. False_ 14. In the least squares model, the explained sum of squares is always smaller than the regression sum of squares. True____ 15. The sample correlation coefficient and the sample slope will always have the same sign. ^ (^ ) True ___ 16. An important relationship in regression analysis is ( Y iY )= Y Y +( Y iY ) . True ___ 17. If in a regression analysis the explained sum of squares is 75 and the unexplained sum of square is 25, r2 = 0.33. False_____ 18. When small values of Y tend to be paired with small values of X, the relationship between X and Y is said to be inverse. False ___ 19. The probability that the test statistic will fall in the critical region, given that H0 is true, represents the probability of making a type II error. True_____ 20. When we reject a true null hypothesis, we commit a Type I error. Part C: Please fill in the blank and circle your decision or answer the following questions (21-25). 21. State whether you would reject or fail to reject the null hypothesis in each of the following cases (two-tailed): Make a decision. a) P = 0.12 ; =0.05 Decision Reject / Fail to reject b) P = 0.03 ; =0.05 Decision Reject / Fail to reject c) P = 0.001 ; =0.01 22. State whether F =computed F H0 Decision Reject / Fail to reject should be accepted or rejected for =0.05 , given the following; (fill in the blank and circle your decision) a) F = 2.34; df = 2 and 11 Computed F Critical F Decision __2.34______ ___ Reject / Fail to reject 3.982_____ b) F = 2.52; df = 4 and 20 Computed F Critical F Decision _____2.52___ Reject / Fail to reject c) F = 4.29; df = 3 and 24 Computed F Critical F Decision ___4.29_____ _3.87_ Reject / Fail to reject 23. Given the following, complete the ANOVA table and make the correct inference. Using F-value to make a decision. Source SS Treatment ___6.48_ s _ ___34.5_ Error _ Total 40.98 Df MS F __1.6__ _ 2 3.24 17 ___2.029 __ ____19 _ ANSWER a) What is the hypothesis being tested in this problem? Ho: b1=0, No factor significant b) In the above ANOVA table, is the factor significant at the 5% level? c) What is observations? the number of No 20 24. Given the following, complete the ANOVA table and make the correct inference. Using F-value to make a decision. Source Treatmen ts SS ___1027. 5_ df MS F 5 205.5 ___6_ Error 637 _20___ __31.85 __ Total ___1664.5 25 ANSWER a) What is the hypothesis being tested in this problem? B1=B2=B3=B4=0 b) In the above ANOVA table, is the factor significant at the 5% level? c) What is observations? the number Yes of 26 25. Given the following, complete the ANOVA table and make the correct inference. Using F-value to make a decision. Source Treatmen ts Error Total SS __346.2 __ 88.8 435 df MS F __115.4 ____20.7 3 __ 9 __16__ __5.55__ 19 ANSWER a) What is the hypothesis being tested in this problem? B1=B2=B3=0 b) In the above ANOVA table, is the factor significant at the 5% level? YES c) What is observations? 20 the number of Part D: Must show all your work step by step in order to receive the full credit; Excel is not allowed. (26-38) 26. Consider the following hypothesis test. Ho: = 17 Ha: 17 A sample of 25 gives a sample mean of 14.2 and sample variance of 25. a ) At 5% should the null be rejected? b ) Compute the value of the test statistic Yes T= c ) What is the p-value? d.f=25-1=24 d ) 14.217 5/ 25 = -2.8 What is your conclusion? 0.0001 a ) 17 27. Consider the following hypothesis test Ho: 10 Ha: < 10 A sample of 50 provides a sample mean of 9.46 and sample variance of 4. b At 5% should the null be rejected? ) Compute the value of the test statistic No Z= c ) What is the p-value? d ) 9.4610 2/ 50 = -1.909 What is your conclusion? 0.028 Ho: 10 28. Use problem 13 on page 9-20 to answer the following questions. a) What is the sample mean b) What is the sample standard deviation S=sqrt(1/9(219)= Mean=1189/10= 118.9 c) Use Z or T test? And why? t- as sample size is n<30 and population 4.931757 d) what is your hypothesis test h0: manufacturing process standard deviation unknown not functioning properly ha: 118.91 2 0 t =4.93/ 1 -0.7055 e) at the rejection rule? reject ho p<0.05 f) p-value? 9 degrees of freedom 0.2490 g) conclusion? rejected we conclude that 29. use problem 7 on page 9-36 to answer following questions. a ) 5% should null be rejected? b compute value statistic h0 5.66.9 z =c 0.4219 d 482 17 + 64 38 =0.197 mean are different 30. for n =6 data point, quantities have been calculated. a)find b1 xy =400 x =40 y =76 x2 =346 b)find b0 =b0 7634640400 6346402 y2 =1,160 >7.8147 Observed is greater Rejectt IID U(0, 1) 35. Use the computer printout below to answer the following questions. Intercept Coefficients 729.8665 Std. Error 169.25751 t-Stat 4.3121659 P-value 0.0010099 Price Advertising -10.887 0.0465 3.4952397 0.0176228 -3.1148078 2.6386297 0.0089406 0.0216284 F 37.56127994 Significance F 0.00000683 ANOVA Regression Residual Total df 2 12 14 Se =12.86986 SS 12442.8 1987.6 14430.4 R-sq = 0.862263 a) Write and interpret the multiple regression equation Y=729.8665-10887X+0.0465 Where y= 729.8665 is the value of y when determinants MS 6221.4 165.63333 R-sq(adj) = 0.8393068 =5 b) Does the model with Price and Advertising contribute to the prediction of Y? Yes, both has p-value less than 0.05 are equal to zero -10.887 is the change in y as a result of unit change in price holding advertising as a constant. 0.0465 is the change in y as a result of unit change in advertising holding price as a constant. c) Which independent variable appears to be the best predictor of sales? Explain. Price as it has the smallest p value e)Find SST Assuming that the coefficient on Advertising has Ha: B1 > 0, what's statistical decision ? SST=14430.4 Reject H0 d) What is the number of observations used in this study? n=14+1=15 f) What is the standard error of estimate? Can you use this statistic to assess the model's fit? If so, how? Se =12.86986 No g) What is the coefficient of determination, and what does it tell you about the regression model? h) What is the coefficient of determination, adjusted for degrees of freedom? What do this statistic and the statistic referr to in part (g) tell you about how well this model fits that data. R-sq(adj) = 0.8393068 R2=0.862263 R2=0.862263 means that 86.2263% of y is explained by the model i) Test the overall utility of the model. What does the p-value of the test statistic tell you? P value=0.00000683 The model is significant 36. Explain why a 95% confidence interval estimate for the mean value of y at a particular x is narrower than a 95% confidence interval for an individual y value at the same value of x. This is caused by the standard error for mean value of y at a particular x is largr than standard error individual y value at the same value of x. 37. If R= R2=0.95 , n=11, and 1 (Y Y )2=100, sse sst 0.95=1-sse/100 Sse=5 S 2e =sse/n-k-1=5/(11-1-1)=5/9 2 what is S e ? 38. You are given the following information from fitting a multiple regression with three variables to 30 sample data points: b1=1.3 b2=10.0 b3 =25.2 S b =0.3 1 S b =2.0 Test the following hypotheses, Use =0.05 a) 1=0 H0: 1=0 Ha: 1 0 t =1.3/0.3=4.2 t critical at 27(n-k)d.f=2.0518 4.2>2.0518, Reject Ho, Conclude b) 1 0 2=0 H0: 2=0 Ha: 2 0 T=10/2=5 t critical at 27(n-k)d.f=2.0518 5>2.05, reject Ho a) H0: 3=0 3=0 2 S b =15.0 3 Ha: 3 0 T=25.2/15=1.68 t critical at 27(n-k)d.f=2.0518 1.68<2.0518 Hence we fail to reject Ho B3 is actually no significant