Question

Question #6 & #7 A-C.

Q6. Jeff Sagarin has been providing sports ratings for USA Today since 1985. In baseball, his predicted RPG (runs per game) statistics considers the entire player's offensive statistics and is claimed to be the best measure of a player's true offensive value. A set of sample data was collected for RPG and a variety of offensive statistics for a recent Major League Baseball (MLB) season for members of the New York Yankees. The variables are defined as follows: RPG, predicted runs per game statistics; H, hits; 2B, doubles; 3B, triples; HR, home runs; RBI, runs batted in; BB, bases on balls (walks); SO, strikeouts; SB, stolen bases; CS, caught stealing, OBP, on-base percentage; SLG, slugging percentage; and AVG, batting average. Let the dependent variable be the RPG statistics for the regression analysis. Based on the above information about RPG, the following output is taken from applying Backward Elimination procedure to fit a regression model (consider a to remove = 0.05). Regression Analysis: RPG versus H, 2B, 3B, HR, SO, OBP, AVG Candidate terms: H, 2B, 3B, HR, SO, OBP, AVG

Answer 1

Sol:

6).

(a) Consider the sample correlation coefficients between each pair of variables. The Minitab output is shown:

Correlations: RPG; H; 2B; 3B; HR; RBI; BB; SO; SB; CS; OBP; SLG; AVG TI 2B cs RPG 3B HR RBI BB SO SB H 0,718 2B 0,585 0,914 3B 0,268 0,660 0,620 HR 0,876 0,830 0,716 0,237 RBI 0,808 0,923 0,867 0,372 0,949 BB 0,845 0,817 0,687 0,250 0,904 0,884 SO 0,725 0,853 0,773 0,401 0,823 0,822 0,908 SB 0,287 0,475 0,248 0,295 0,365 0,338 0,311 0,433 CS 0,395 0,697 0,506 0,647 0,476 0,501 0,382 0,531 0,863 OBP 0,902 0,677 0,543 0,292 0,743 0,713 0,781 0,680 0,270 0,388 SLG 0,888 0,823 0,759 0,347 0,925 0,908 0,835 0,790 0,273 0,439 AVG 0,599 0,697 0,588 0,517 0,579 0,630 0,522 0,514 0,370 0,582 SLG SLG AVG OBP 0,815 0,798 0,704 Cell Contents: Pearson correlation

We see that the single best predictor of RPG is OBP, because it has the highest sample correlation coefficient.

Correlations: RPG; H; 2B; 3B; HR; RBI; BB; SO; SB; CS; OBP; SLG; AVG TI 2B cs RPG 3B HR RBI BB SO SB H 0,718 2B 0,585 0,914 3B 0,268 0,660 0,620 HR 0,876 0,830 0,716 0,237 RBI 0,808 0,923 0,867 0,372 0,949 BB 0,845 0,817 0,687 0,250 0,904 0,884 SO 0,725 0,853 0,773 0,401 0,823 0,822 0,908 SB 0,287 0,475 0,248 0,295 0,365 0,338 0,311 0,433 CS 0.395 0,697 0,506 0,647 0,476 0,501 0,382 0,531 0,863 OBP 0,902 0,677 0,543 0,292 0,743 0,713 0,781 0,680 0,270 0,388 SLG 0,888 0,823 0,759 0,347 0,925 0,908 0,835 0,790 0,273 0,439 AVG 0,599 0,697 0,588 0,517 0,579 0,630 0,522 0,514 0,370 0,582 SLG OBP SLG 0,815 AVG 0,798 0,704 Cell Contents: Pearson correlation

Therefore, we construct an estimated regression equation using the variable OBP. Using Minitab, the estimated regression equation is: RPG = -4.05+27.6 OBP The Minitab output is shown:

Regression Analysis: RPG versus OBP The regression equation is RPG = - 4,05 + 27, 6 OBP Predictor Constant ОВР Coef SE Coef T P -4,049 1,006 -4,02 0,001 27, 555 3,103 8,88 0,000 S = 0,956308 R-Sa 81,4% R-Sq(adj) 80,43 Analysis of Variance Source DF SS MS F P Regression 72,108 72,108 78,85 78,85 0,000 Residual Error 18 16,461 0,915 Total 19 88,569 Unusual Observations Obs OBP 15 0,263 RPG Fit SE Fit Residual St Resid 5,150 3,198 0,271 1,952 2,13R R denotes an observation with a large standardized residual.

(b) The stepwise procedure terminated after three steps. The estimated regression equation is: RPG = -0.9808 + 25.1 OBP +0.069 HR - 12.6 AVG The value of R-Sq has been increased from 81.41% to 94.43% and recommended estimated regression equation has an R-Sq(adj) value of 93.38%. The Minitab output is shown:

Stepwise Regression: RPG versus H; 2B; ... Alpha-to-Enter: 0,05 Alpha-to-Remove: 0,05 Response is RPG on 12 predictors, with N = 20 Step Constant 1 2 3 -4,0491 -1,5951 -0,9808 OBP T-Value P-Value 27,6 8,88 0,000 17,2 5,10 0,000 25,1 6,88 0,000 HR T-Value P-value 0,071 4,16 0,001 0,069 5,06 0,000 AVG T-Value P-Value -12,6 -3,23 0,005 S R-Sa R-Sq(adj) Mallows Cp 0,956 81,41 80,38 66,6 0,693 90,78 89,70 27,0 0,556 94,43 93,38 12,8

The stepwise procedure terminated after five steps. The estimated regression equation is: RPG = -0.9088 +32.2 OBP +0.109 HR - 21.5 AVG +0.244 3B -0.0223 BB The value of R-Sq has been increased from 81.41% to 97.20% and recommended estimated regression equation has an R - Sq(adj) value of 96.20%. The Minitab output is shown:

Stepwise Regression: RPG versus H; 2B; ... Alpha-to-Enter: 0,1 Alpha-to-Remove: 0,1 Response is RPG on 12 predictors, with N = 20 Step Constant 1 2 3 -4,0491 -1,5951 -0,9808 4 -0, 6161 5 -0,9088 OBP T-Value P-Value 27,6 8,88 0,000 17,2 5,10 0,000 25,1 6,88 0,000 26,6 7,64 0,000 32,2 9,40 0,000 HR T-Value P-Value 0,071 4,16 0,001 0,069 5,06 0,000 0,068 5,34 0,000 0,109 6,26 0,000 AVG T-Value P-Value -12,6 -3,23 0,005 -16,5 -3,96 0,001 -21,5 -5,65 0,000 3B T-Value P-Value 0,182 1,88 0,079 0,244 2,99 0,010 BB T-Value P-Value -0,0223 -2,92 0,011 S R-Sa R-Sq(adj) Mallows Cp 0,956 81,41 80,38 66,6 0,693 90,78 89,70 27,0 0,556 94,43 93,38 12,8 0,516 95,49 94,29 10,0 0,421 97,20 96,20 4,5

Therefore, the best estimated regression equation is: RPG = -0.9088+32.2 OBP +0.109 HR - 21.5 AVG +0.244 3B -0.0223 BB

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