False. Model 2 is not a special case of Model 1, because Model 2 includes X5, which is not part of Model 1. What she ought to do is estimate

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1 Sociology 59 - Research Statistics I Final Exam Answer Key December 6, 00 Where appropriate, show your work - partial credit may be given. (On the other hand, don't waste a lot of time on excess verbiage.) Do not spend too much time on any one problem. You are free to refer to anything that was demonstrated in the homework or handouts.. (5 points each, 0 points total). For each of the following, indicate whether the statement is true or false. If you think the statement is false, indicate how the statement could be corrected. NOTE: These are all pretty easy, but you could waste a great deal of time on some of them or make stupid mistakes if you don't happen to see what the easiest way to approach each problem is. a. N = 7, K = 6, Adjusted R =.37. If the researcher adds a 7 th variable to her model, Adjusted R will either increase or stay the same. False. As more variables are added to an equation, R² will either stay the same or increase. But, Adjusted R² can go down and even become negative. For example, as the following formula shows, if additional variables are added (i.e., K increases) but R² stays the same, the numerator will stay the same while the denominator decreases, causing Adjusted R² to decrease. Adjusted R ( N )( R ) = ( N K ) b. A researcher has estimated the following two models: : yˆ = a + b X : yˆ = a + b X + b X + b3 X 3 + b4 X + b X + b X She can now use the results from these two models to test H 0 : β 3 = β 4 = 0 H A : β 3 and/or β 4 0 False. is not a special case of, because includes X5, which is not part of. What she ought to do is estimate : yˆ = a + b X : yˆ = a + b X + b X + b X + b X b X 4 4 c. Multicollinearity causes least-squares estimates to be biased. False. When multicollinearity occurs, the least-squares estimates are still unbiased and efficient. The problem is that the estimated standard errors of the coefficients tend to be inflated. That is, the standard error tends to be larger than it would be in the absence of multicollinearity because the estimates are very sensitive to changes in the Sociology 59 Final Exam Answer Key December 6, 00 Page

2 sample observations or in the model specification. Hence, including or excluding a particular variable or certain observations may greatly change the estimated coefficients. d. It is March, 003. The Notre Dame Men s basketball team is playing in its first ever National Championship Game. Their opponent is the only school to defeat them this season the # ranked and undefeated Creighton Bluejays from Omaha, Nebraska. ND coach Mike Brey knows that he has to keep Creighton s scoring down to a minimum if ND is to pull off the upset. He must choose between three defensive strategies: play a zone defense most of the game; use man to man defense most of the game; or play zone and man to man about equally in the game. The students in Sociology 59 have collected data from ND games over the past three years to help Brey determine what defensive strategy will be most effective. Three variables have been computed: ZONE (coded when ND plays a zone defense most of the game, 0 otherwise), MANMAN (coded when ND plays man to man defense most of the game, 0 otherwise) and BOTH (coded when ND plays both zone and man to man defense about equally during a game, 0 otherwise). The dependent variable is OSCORE, the number of points scored by ND s opponent in the game. The following results are obtained: Variable B ZONE 8 MANMAN Constant 70 Based on these results, Notre Dame s best strategy is to play a zone defense most of the game. That is, of the three defensive strategies ND can use (mostly zone, mostly man to man, or an even mix of both), Creighton will likely score its fewest points if ND plays mostly zone. False. Brey should use both types of defense about equally. According to the model, yˆ = * Zone + * MANMAN If he plays zone, Creighton can be expected to score about 78 points (70 + 8* + *0); if he goes man to man, Creighton s expected score is 8 (70 + 8*0 + *); but if he uses both types of defense, Creighton s expected score is only 70 (70 + 8*0 + * 0). More simply, we can just note that, because the coefficients for the two dummy variables are both positive, the average score must be lower for the reference category (Both defenses used about equally) than it is for the other two categories.. Short answer problems. (0 points each, 30 points total, up to 5 points extra credit). Answer three of the following. You will get up to five points extra credit if you can solve all four problems. a. SSR = 00, K = 0, F = 4, N =. Construct the ANOVA table and compute R. Note that MSR = SSR/K = 00/0 = 0; also, F = 4 = MSR/MSE = 0/MSE implying MSE = 5. The rest follows easily. Sociology 59 Final Exam Answer Key December 6, 00 Page

3 Source SS d.f. MS F Regression (or explained) SSR = 00 K = 0 SSR / K = 0 MSR/MSE = 4 Error (or residual) SSE = 500 N - K - = 00 SSE / (N-K-) = 5 Total SST = 700 N - = 0 SST / (N - ) = R = SSR/SST = 00/700 =.857 b. r Y =.60, r Y =.6, r =.98. What are the standardized betas when Y is regressed on X and X? What warning might you give to a researcher who got these results? b ' = (r y - r * r y ) / ( r ) = ( *.6) / ( -.98 ) = b ' = (r y - r * r y ) / ( r ) = ( *.60) / ( -.98 ) = Because of the very high correlation between the IVs, multicollinearity may be a problem. These results suggest that X has a much larger effect than X, even though the two variables have almost identical correlations with Y. Parameter estimates can be highly volatile when there is high multicollinearity and hence a slightly different sample might yield very different results. c. Fifty blacks, fifty whites and fifty members of other races have been interviewed. Y = Political liberalism, measured on a scale that ranges from a low of 0 to a high of 00. X = if white, 0 if black, - if other. X = 0 if white, if black, - if other. The following results are obtained: Variable B X -8 X +5 Constant +50 Compute the average Liberalism (Y) scores for whites, blacks and others, and for the sample as a whole. According to the model, y ˆ = 50 (8* X) + (5* X ) For whites, y ˆ = 50 (8*) + (5*0) = 4 For blacks, y ˆ = 50 (8*0) + (5*) = 55 For others, y ˆ = 50 (8* ) + (5* ) = 53 For the sample as a whole, since effect coding is used and group sizes are equal, the overall mean is equal to the constant, 50. Sociology 59 Final Exam Answer Key December 6, 00 Page 3

4 d. When Y is regressed on X, X, X3, an X4, sr =.3, sr =.4, sr 3 =., sr 4 =.. The tolerances for all 4 X s equal. How much would R decline if both X and X were dropped from the model? The fact that the tolerances all equal means that the Xs are uncorrelated with each other. Hence, each X makes a totally unique contribution to R, and that contribution will be completely lost if the variable is dropped. Therefore, if X and X are both dropped, R will decline by =.5. Graphically, the described relationships look something like this: 3. (50 points total; up to 5 points extra credit. This example is adapted from Hamilton, Statistics with Stata, Version 7). A researcher wants to examine how characteristics of states are related to student performance on the Scholastic Aptitude Test (SAT). She has the following data, which were collected for all 50 states during the years : Variable CSAT PERCENT EXPENSE INCOME HIGH COLLEGE NEAST Question Mean composite SAT (Scholastic Aptitude Test) score % High School graduates taking the SAT Per pupil expenditures for primary and secondary schools Median household income % of those over 5 with a High School % of those over 5 with a bachelor's degree or higher Is state in the Northeastern United States? = Northeastern state, 0 = State from another part of the country She gets the following results: Sociology 59 Final Exam Answer Key December 6, 00 Page 4

5 Regression Descriptive Statistics CSAT Mean composite SAT score PERCENT % HS graduates taking SAT EXPENSE Per pupil expenditures prim&sec INCOME Median household income HIGH % over 5 w/hs COLLEGE % over 5 w/bachelor's degree + NEAST Is state in the Mean Std. Deviation N Pearson Correlation CSAT Mean composite SAT score PERCENT % HS graduates taking SAT EXPENSE Per pupil expenditures prim&sec INCOME Median household income HIGH % over 5 w/hs COLLEGE % over 5 w/bachelor's degree + NEAST Is state in the CSAT Mean composite SAT score PERCENT % HS graduates taking SAT Correlations EXPENSE Per pupil expenditures prim&sec INCOME Median household income HIGH % over 5 w/hs COLLEGE % over 5 w/bachelor's degree + NEAST Is state in the [] Summary Adjusted Std. Error of R R Square R Square the Estimate.9377 a a. Predictors: (Constant), NEAST Is state in the, COLLEGE % over 5 w/bachelor's degree +, PERCENT % HS graduates taking SAT Sociology 59 Final Exam Answer Key December 6, 00 Page 5

6 ANOVA b Sum of Squares df Mean Square F Sig. Regression [].000 a Residual Total a. Predictors: (Constant), NEAST Is state in the, COLLEGE % over 5 w/bachelor's degree +, PERCENT % HS graduates taking SAT b. Dependent Variable: CSAT Mean composite SAT score (Constant) PERCENT % HS graduates taking SAT COLLEGE % over 5 w/bachelor's degree + NEAST Is state in the Unstandardized Coefficients a. Dependent Variable: CSAT Mean composite SAT score Standardized Coefficients Coefficients a B Std. Error Beta [3].000 Correlations t Sig. Zero-order Partial Part Collinearity Statistics Tolerance VIF [4] [5] [6] [7].854 Excluded Variables b EXPENSE Per pupil expenditures prim&sec INCOME Median household income HIGH % over 5 w/hs Beta In t Sig. Collinearity Statistics Partial Correlation Tolerance VIF Minimum Tolerance.036 a a a a. Predictors in the : (Constant), NEAST Is state in the, COLLEGE % over 5 w/bachelor's degree +, PERCENT % HS graduates taking SAT b. Dependent Variable: CSAT Mean composite SAT score a. ( points) Fill in the missing items [] [7]. Here are uncensored versions of the three parts of the printout with items missing: Sociology 59 Final Exam Answer Key December 6, 00 Page 6

7 Pearson Correlation CSAT Mean composite SAT score PERCENT % HS graduates taking SAT EXPENSE Per pupil expenditures prim&sec INCOME Median household income HIGH % over 5 w/hs COLLEGE % over 5 w/bachelor's degree + NEAST Is state in the CSAT Mean composite SAT score PERCENT % HS graduates taking SAT Correlations EXPENSE Per pupil expenditures prim&sec INCOME Median household income HIGH % over 5 w/hs COLLEGE % over 5 w/bachelor's degree + NEAST Is state in the ANOVA b Sum of Squares df Mean Square F Sig. Regression a Residual Total a. Predictors: (Constant), NEAST Is state in the, COLLEGE % over 5 w/bachelor's degree +, PERCENT % HS graduates taking SAT b. Dependent Variable: CSAT Mean composite SAT score (Constant) PERCENT % HS graduates taking SAT COLLEGE % over 5 w/bachelor's degree + NEAST Is state in the Unstandardized Coefficients a. Dependent Variable: CSAT Mean composite SAT score Standardized Coefficients Coefficients a B Std. Error Beta Correlations t Sig. Zero-order Partial Part Collinearity Statistics Tolerance VIF To confirm that SPSS got the numbers right: [] = r NEAST,EXPENSE = r EXPENSE,NEAST =.6366 (i.e. the correlation matrix is symmetric, so each correlation appears in two different places in the table) [] = F = MSR/MSE = / =.76. Or, if you prefer to do it the hard way, F = R * ( N K ).87935* (50 3 ) = = =.76 ( R ) * K (.87935) * Sociology 59 Final Exam Answer Key December 6, 00 Page 7

8 [3] = t Constant = b Constant /s Constant = /9.887 = [4] = b Percent = s bpercent * t Percent =.98 * = -3.. Or, alternatively, b Percent = b Percent scsat * =.68* = 3. s Percent [5] = r College,Csat = -.33 (see the correlation matrix) [6] = b Neast = b NEast * s NEast / s CSat = *.38809/65.95 =.39 [7] = Tol NEast = /VIF NEast = /.854 =.539 b. (5 points) If she had used forward stepwise regression, what variable would have been entered first? Why? What would the R have been for that first model? PERCENT has the largest bivariate correlation with CSAT, so it would have been entered first. The R would be r Percent, Csat = =.758 c. (4 points) If the researcher were to use stepwise regression now, would she add additional variables to her model, would she delete variables, or would she stick with the three variables she has? Explain why. All of the variables currently in the model are statistically significant, so none would be dropped. None of the variables not in the model would significantly increase R (see Excluded Variables) so no more variables would get added. Hence, she would stick with the 3 variables she has. d. (5 points) Do an incremental F test of the hypothesis H 0 : β College = β NEast = 0 The Unconstrained R (for the model with all three variables) is given in the above printout and equals The constrained R (for the model which has only Percent) was computed in Part b and equals.758. Further, N = 50, K = 3, J =. Hence, F ( R = Rc ) *( N K ) ( ) *(50 3 ) = = ( R ) * J (.87935) *.43 u J, N K = u 3. e. (5 points) Interpret the results. Be sure to answer the following questions. Explain what information from the printout supports your conclusions.. What proportion of the states are in the In what part of the country (the Northeast or outside the Northeast) are high school graduates most likely to take the SAT? Sociology 59 Final Exam Answer Key December 6, 00 Page 8

9 As the means show, 8% of the states are in the Northeast. The positive correlation between NEAST and PERCENT (.67736) shows that students in the Northeastern states are more likely to take the SAT than students in other parts of the country.. After controlling for other variables, what are the characteristics of those states that tend to have higher SAT scores? For example, are they wealthier, better educated, or what? Again, after controlling for other variables, what variables do not have significant effects on the state s SAT scores? The higher the percentage of students in the state taking the SAT, the lower the state s SAT score tends to be (the b value for Percent is -3.). The positive coefficient for College (5.345) means that the more college graduates a state has, the higher its SAT scores are (i.e. states with better-educated adults tend to have higher-scoring students, once other variables are controlled). The positive coefficient for NEast (54.96) means that students in the Northeast score about 54 points higher, on average, than do students in otherwise comparable states in the rest of the country. None of the other variables (Expense, Income, High) have statistically significant effects. 3. The effect of Percent (% High School graduates taking the SAT) is negative that is, the more students in the state that take the SAT, the lower the mean composite SAT score tends to be. Offer a plausible substantive explanation for this relationship. This may mean that, when relatively more students in a state take the SAT, more relatively weak students tend to be taking the test. That is, outside the Northeast, only the very best students tend to take the test, whereas in the Northeast both strong and weak students tend to take the test. This might mean that both weak and strong students in the Northeast try to go to college, whereas in other parts of the country weaker students are less likely to try for college. (This is what you could reasonably guess from the results; as noted below, there is a better explanation, which uses additional info not contained in the printout.) f. (5 points extra credit) The bivariate correlation of NEAST and CSAT is negative, which implies that students in Northeastern states tend to have lower SAT scores than students in the rest of the country. However, once other variables are controlled, the effect of NEAST on CSAT is positive, which implies that students in the Northeast actually tend to do better on the SATs once you take other variables into account. Offer a plausible substantive explanation for this seeming inconsistency. [HINT: Your answers to part e may give you some insights.] Relatively more students in the Northeast tend to take the SAT, hence the Northeast has relatively more of its weaker students taking the exam than do other parts of the country. Once you take this into account (by controlling for Percent), you see that students in the Northeast tend to outscore comparable students in other parts of the country. That is, the bivariate correlation alone creates a misleading impression, because the best and the weakest students combined in the Northeast are being compared to the best students from other parts of the country. To put it another way suppressor effects are present. Students in the Northeast are more likely to take the exam, and when more students take the exam, overall scores tend to be lower (probably because the additional students taking the exam tend to be the weaker students). But, Northeast students tend to score higher than do otherwise Sociology 59 Final Exam Answer Key December 6, 00 Page 9

10 comparable students from the rest of the country. This might be due to better schools in the Northeast. In case you are wondering why students in the Northeast are more likely to take the SAT you can t tell this from the information given, but the reason this occurs is because there are two major college admission exams given in the US: The SAT and the ACT. Schools in the Northeast primarily rely on the SAT, while schools in other parts of the country are more likely to be using the ACT for admission. Hence, in states like Nebraska and Iowa, most college bound students take the ACT, since that is what is used by colleges in their areas. The Nebraska/Iowa students who do take the SAT tend to be the brighter students who want to go to a strong school in another part of the country. That is, outside the Northeast, a smaller but more elite group of students tends to take the test. Hence, Nebraska and Iowa (along with North Dakota, South Dakota, Kansas and Utah) are among the highest scoring states on the SAT. This isn t so much because they have the smartest students in the country; rather, it is because their smartest students are the ones who tend to take the SAT test. Their less bright students tend to skip the SAT and only take the ACT (or neither test at all). Hamilton s Stata book discusses this problem in additional detail. Sociology 59 Final Exam Answer Key December 6, 00 Page 0

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