7 EXERCISES (SHOW ALL WORK and EXPLAIN ALL ANSWERS) A large company recently laid off several workers. Most layoffs were temporary. For workers who were ultimately hired back, the company recorded their Weeks of layoff, their Age when laid off, their years of Education, and their years of Tenure with the company. They then fit the three regression models shown below. Questions 1 & 2 concern these models. A. Regression Analysis: Weeks versus Age Model Summary S 19.5342 R-sq ? _ R-sq(adj) ? ________ Coefficients Term Constant Age Coef -8.9 1.509 SE Coef 11.0 0.308 T-Value -0.80 4.90 P-Value 0.425 0.000 B. Regression Analysis: Weeks versus Age, Tenure Model Summary S 19.4083 R-sq ? R-sq(adj) ? Coefficients Term Constant Age Tenure Coef -7.6 1.308 0.681 SE Coef 11.0 0.344 0.534 T-Value -0.69 3.80 1.27 P-Value 0.493 0.000 0.209 C. Regression Analysis: Weeks versus Age, Educ, Tenure Model Summary S 19.5360 R-sq ? R-sq(adj) ? Coefficients Term Constant Age Educ Tenure Coef 0.7 1.313 -0.66 0.729 SE Coef 17.3 0.347 1.05 0.543 T-Value 0.04 3.79 -0.62 1.34 P-Value 0.968 0.000 0.537 0.186 1. Which of the three models has the highest multiple R-squared? Which has the highest adjusted R-squared? (Remember to explain your answers.) 2. Write the regression equation of Model C and interpret the slope of the Tenure predictor. Regression Analysis: Hwy MPG versus Displacement, Fuel, Drive Analysis of Variance Source Regression DF 4 Adj SS 7308.5 Adj MS 1827.14 F-Value 207.31 P-Value 0.000 Model Summary S 2.96876 R-sq 73.17% R-sq(adj) 72.82% R-sq(pred) 72.40% Coefficients Term Constant Displacement Fuel Regular Drive F R Coef 35.827 -3.242 SE Coef 0.770 0.194 T-Value 46.51 16.71 P-Value 0.000 0.000 2.135 0.452 ____ _____ 3.075 3.311 0.539 0.541 5.70 _____ 0.000 _____ Regression Equation Hwy MPG = 35.827 - 3.242 Displacement + 0.0 Fuel_Premium + 2.135 Fuel_Regular + 0.0 Drive_A + 3.075 Drive_F + 3.311 Drive_R For 309 cars, highway mileage (Hwy MPG) was recorded along with engine Displacement (in liters), type of Fuel used, and type of Drive train (All-Wheel, Front Wheel, or Rear Wheel). The output (with some deletions) for a multiple regression to predict Hwy MPG from the other variables is shown above. Use this output to answer Questions 3-6. 3. The T-Value for the Regular Fuel variable is: 4. Assuming no violation of the rule against extrapolation, an rear wheel drive car with a 2.5 liter engine displacement which uses premium fuel will be predicted to get: 5. According to the model, controlling for the other variables a rear wheel drive (Indicate all correct answers): A. Will be expected to get 3.311 more highway miles per gallon than a front wheel drive car B. Will be expected to get 3.311 more highway miles per gallon than an all-wheel drive car C. Will be expected to get 0.236 more highway miles per gallon than a front wheel drive car D. Will be expected to get 0.236 less highway miles per gallon than a front wheel drive car. 6. According to this model, controlling for the other variables, a car using regular gas (select all that are correct - for significance, use = 0.05): I. Will be expected to get significantly lower highway mileage than a car using premium II. Will be expected to get significantly higher highway mileage than a car using premium III. Will be expected to get 2.135 more highway miles per gallon than a car using premium IV. Will be expected to get 2.135 fewer highway miles per gallon than a car using premium A. B. C. D. I and IV only II and III only III only IV only One-way ANOVA: Hwy MPG versus Drive Analysis of Variance Source Drive Error Total DF 2 306 308 Adj SS 4542 5446 9988 Adj MS 2270.79 17.80 F-Value 127.59 P-Value 0.000 Tukey Pairwise Comparisons Grouping Information Using the Tukey Method and 95% Confidence Drive N Mean Grouping F 167 33.353 A A 61 25.787 B R 81 25.568 B Means that do not share a letter are significantly different. Using the same Highway Mileage dataset as in Questions 3-6, we are interested in a regression model to predict Hwy MPG from type of Drive. To prepare for this, we first do the 1-way ANOVA shown above. Assume that the conditions for a valid 1-Way ANOVA are satisfied. Use this to answer Questions 7-10. 7. From just the P-Value of this ANOVA we can conclude that, at any typical level: 8. In the Tukey simultaneous 95% confidence intervals for this ANOVA, which interval(s) will include 0? I. Front - All II. Front - Rear III. All - Rear A. B. C. D. E. II only III only I and II only II and III only I and III only. 9. In the regression of Hwy MPG versus Drive, if we use Rear Drive as the reference category, the slope of the All-Wheel Drive predictor will be: 10. In the regression of Hwy MPG versus Drive using Rear Wheel as the reference, will the slope of Front Wheel be significant at 5%? For a recent season, several variables were recorded for 125 professional golfers. We are interested in using multiple regression to predict Earnings ($) from the predictors indicated in the backward elimination output (with some deletions) below. Use this output to answer Questions 11 & 12. Regression Analysis: Earnings ($) versus DrDist, DrAccu, GIR, Sand Saves, Scrambling Backward Elimination of Terms Candidate terms: DrDist, DrAccu, GIR, Sand Saves, Scrambling Constant DrDist DrAccu GIR Sand Saves Scrambling ------Step 1----Coef P -11280801 12222 0.448 -58196 0.026 107488 0.008 48630 0.009 66373 0.114 -----Step 2----Coef P -7264460 936403 21.64% 18.35% 12.93% 6.00 934761 _____% _____% 13.60% 4.58 S R-sq R-sq(adj) R-sq(pred) Mallows' Cp to remove = 0.05 -71841 121557 46404 58872 -----Step 3----Coef P -4296110 0.000 0.001 0.012 0.149 939054 19.87% 17.88% 14.14% 4.69 11. If Step 3 produces the final model, which of the predictors below will not be in it? A. DrDist B. GIR C. Sand Saves D. Scrambling E. Both A and D. 12. For the model at Step 2, which of the following must be correct? A. The multiple R-squared is less than 21.64% B. The Adjusted R-squared is greater than 18.35% C. Both A and B. D. None of the above. Refer again to the data for Questions 11 & 12. Not satisfied with these predictors, we add a predictor to our dataset called Bounce Back. Below is Best Subsets output using this new dataset. Use it to answer Questions 13- 15. Best Subsets Regression: Earnings ($) versus DrDist, DrAccu, ... Response is Earnings ($) Vars 1 2 3 4 5 6 R-Sq 8.1 15.4 19.9 21.3 22.4 22.6 R-Sq (adj) 7.4 14.1 17.9 18.6 19.2 18.6 R-Sq (pred) 5.0 11.1 14.1 13.6 13.1 12.2 Mallows Cp 19.0 9.9 5.1 5.0 5.2 7.0 S 997373 960722 939054 934761 931759 934739 D r D i s t S a n d S c r a m b l i n g B o u n c e D r S A a B c G v a c I e c u R s k X X X X X X X X X X X X X X X X X X X X X 13. Based on this output, the predictor with the highest correlation in absolute value with Earnings is: 14. By the criterion of Best Subsets Regression, the best model for this dataset has how many predictors? 15. For the 5-predictor model shown in the output: A. If Bounce Back is replaced with DrDist, the adjusted R-squared will increase. B. The slope of Bounce Back must be significant at 5%. C. The slope of Bounce Back must be negative. D. None of the above