Solutions Sample Final Exam 8AM (1)

Problem 1. (42 points) A dataset contains 93 observations (93 cars) of City Fuel Efficiency (y) vs Hi xbar = 29.183 r_xy 0.94045 SST 2897.172 SSE 334.7799 R^2 0.884446 SSR 2562.392 MSE 3.6789 alpha =.05 H0: Beta1 =0 HA: Beta1 neq 0 RR: |Z| > 1.96 since we have 91 df test stat: t = b1/sb1 Since |26.39148| > 1.96, at alpha =.05, we have evidence that Beta1 neq 0. .9867 +/- 1.645*(.037387) We are 90% confident Beta1 is in the above calculated interval. This is because if we w intervals would include Beta1. Beta1 is the slope of the true regression City Fuel Eff = we expect coty fuel eff to change by Beta1 units. xbar 29.183 29.183 +/- 1.645*(5.3486/sqrt(93)) s_x 5.348645 93 S 2 x = 28.608 S 2 y = 31.491 b 0 = -6.32 b 1 = .9867 A.     Find r x,y , R 2 , s b1 , s e , SSR, SSE, SST, MSE. (16 points) B.     Is there evidence that Beta1 is not equal to 0? Use alpha =.05. (8 points) C.     Construct a 90% CI for Beta1 and interpret it within the context of this problem. (9 D.     Construct a 90% CI for mu x , the true population mean of the x values (Highway M

ighway Fuel Efficiency (x). s_e 1.918046 sb1 0.037387 SS_xx 2631.936 tc 26.39148 were to repeat the process many times, approx 90% of the resulting Beta0 + Beta1*Hwy eff + epsilon. As Hwy eff increases by 1 unit 9 points) MPG). (9 points)

hows by household) versus Show Type (Comedy, Drama, or Reality/Participation). Type is an important predictor of Household Rating. rocess many times, approx 90% of the resulting intervals a =.025. Explain your answer. (8 points) rpret the CI? (8 points) y shows. (4 points)

Problem 3. (24 points) Consider the following regression of Household rating (a rating that measures the popularity of tv shows Coefficients Term Coef SE Coef T-Value P-Value VIF Constant 4.008 0.409 9.8 0 1.229 0.441 2.79 0.006 1.38 1.283 0.464 2.76 0.007 1.41 1.893 0.461 4.11 0 1.31 -1.464 0.433 -3.38 0.001 1.31 Model Summary S R-sq R-sq(adj) 1.83362 39.15% 36.64% 32.68% Analysis of Variance Source DF Adj SS Adj MS F-Value P-Value Regression 4 209.840085 52.46002125 15.60209532 0 Error 97 326.149915 3.362370258 Total 101 535.99 alpha =.01 H0: Beta1=Beta2=Beta3=Beta4=0 HA: At least one such Beta neq 0 test stat: F = MSR/MSE p-value is 0. Since 0 < .01, at alpha = .01, we have evidence that at least one such Be for describing Household Rating. -1.464 +/- 1.96*.433. We are 95% confident that Beta4 is in the above calculated interval. This is because if Beta4 is the expected difference in Household rating for NBC network and ABC Networ The above calculated interval has both endpoints negative, thus, there is evidence tha This model is the complete one and Prob 2 model is the reduced one. Type_Dra ma Type_Reali ty/ Participatio n Network_C BS Network_N BC R- sq(pred) A.       Is there evidence of model utility for describing/explaining Household Rating? Use alp B.       Construct a 95% CI for the coefficient in front of NBC and interpret it in the context of C.       Compare this regression model to the one from problem 2. Use alpha = .05. (7 points

s by household) versus Show Type (Comedy, Drama, or Reality/Participation) and Network (ABC, C eta neq 0 and hence evidence that the model is useful we were to repeat this process many times, approx 95% of the intervals would include Beta4. rk in a multiple regression with categorical predictors TYPE and NETWORK, holding TYPE fixed. at NBC has an expected lower rating than ABC holding TYPE fixed. pha =.01. (7 points) f this model. (10 points) s)

at alpha =.05, we have evidence that the complete model is better than the reduced model. 0

Problem 4. (14 points) Consider the following regression of Household rating (a rating that measures the popularity of tv shows Coefficients Term Coef SE Coef T-Value P-Value VIF Constant 4.44 0.545 8.14 0 0.541 0.729 0.74 0.46 3.87 0.745 0.771 0.97 0.337 3.99 1.249 0.79 1.58 0.117 3.96 -2.135 0.771 -2.77 0.007 4.29 DramaCBS 1.49 1.04 1.42 0.158 4.03 DramaNBC 0.46 1.07 0.43 0.668 3.19 RealityCBS -0.47 1.26 -0.38 0.708 2.29 1.23 1.05 1.18 0.242 4.52 Model Summary S R-sq R-sq(adj) 1.80848 43.25% 38.37% 32.41% Analysis of Variance Source DF Adj SS Adj MS F-Value P-Value Regression 8 231.815675 28.97695938 8.859581498 0 Error 93 304.174325 3.270691667 Total 101 535.99 alpha =.05 H0: Beta5 = Beta6 = Beta7 = Beta8 =0 HA: At least one such Beta neq 0 SSR_c 231.8157 F num 5.493898 Since 1.68 is not greater than 2.4 SSR_r 209.8401 F den 3.270692 At level of significance alpha =.05 SSE_c 304.1743 F 1.679736 is better than model from Proble k 8 the Main Effects model. g 4 n-k-1 93 Type_Dra ma Type_Reali ty/ Participatio n Network_C BS Network_N BC RealityNB C R- sq(pred) A.       Fill in the blanks (6 points) B.       Compare this regression model to the regression model from problem 3 using a Neste

Compare our F test stat value to F.05 with 4 num and 93 den df 2.46

, and NBC) and their interactions. 3 or from Problem 4. (8 points)