Question

Suppose you are a rabid football fan and you get into a discussion about the importance of offense (yards made) versus defense (yards allowed) in terms of winning a game. You decide to look at football statistics to provide evidence of which variable is a stronger predictor of wins.

Part a) Develop a simple linear regression that compares wins to yards made. Perform the following diagnostics on this regression: 1) test of significance on the slope; 2) assess the fit of the line using the appropriate statistics; 3) interpret the slope of the equation if the slope is significant. Part b) Develop a simple linear regression that compares wins against yards allowed. Perform the following diagnostics on this regression: 1) test of significance on the slope; 2) assess the fit of the line using the appropriate statistics; 3) interpret the slope of the equation if the slope is significant. Part c) Which explanatory variable provides a better prediction of the response variable? Support your answer briefly by citing the appropriate diagnostics. Note: Use an alpha of .05 for both tests of significance. Be sure to show ALL steps of the hypothesis testing procedure

EXCEL DATA TO USE.

Team	Win	Rush	Pass	Yds Allowed	Yds Made
Arizona Cardinals	62.50	93.40	251.00	346.40	344.40
Atlanta Falcons	56.30	117.21	223.19	348.90	340.40
Baltimore Ravens	56.30	137.51	213.69	305.00	351.20
Buffalo Bills	37.50	116.71	157.19	340.60	273.90
Carolina Panthers	50.00	156.16	174.94	315.80	331.10
Chicago Bears	43.80	93.24	217.06	337.80	310.30
Cincinnati Bengals	62.50	128.48	180.63	301.40	309.10
Cleveland Browns	31.30	130.45	129.75	389.30	260.20
Dallas Cowboys	68.80	131.46	267.94	315.90	399.40
Denver Broncos	50.00	114.71	226.69	315.00	341.40
Detroit Lions	12.50	101.00	198.00	392.10	299.00
Green Bay Packers	68.80	117.85	261.25	284.40	379.10
Houston Texans	56.30	92.23	290.88	324.90	383.10
Indianapolis Colts	87.50	80.91	282.19	339.20	363.10
Jacksonville Jaguars	53.80	126.85	209.75	352.30	336.60
Kansas City Chiefs	25.00	120.58	182.63	388.20	303.20
Miami Dolphins	43.80	139.48	198.13	349.30	337.60
Minnesota Vikings	75.00	119.85	259.75	305.50	379.60
New England Patriots	62.50	120.05	277.25	320.20	397.30
New Orleans Saints	81.30	131.61	272.19	357.80	403.80
New York Giants	50.00	114.81	251.19	324.90	366.00
New York Jets	56.30	172.25	148.75	252.30	321.00
Oakland Raiders	31.30	106.29	159.81	361.90	266.10
Philadelphia Eagles	68.80	102.34	255.56	321.10	357.90
Pittsburgh Steelers	56.30	112.05	259.25	305.30	371.30
Saint Louis Rams	6.30	111.50	167.88	327.00	279.38
San Diego Chargers	81.30	88.94	271.13	326.40	360.06
San Francisco 49ers	50.00	100.00	190.75	356.40	290.75
Seattle Seahawks	31.30	97.86	218.94	372.80	316.80
Tampa Bay Buccaneers	18.80	101.69	185.81	365.60	287.50
Tennessee Titans	50.00	161.96	189.44	365.60	351.40
Washington Redskins	25.00	94.38	218.13	319.70	312.50

Answer 1

a. The regression of wins to yards made regression output is as below.

1) The slope is about 0.386, and the standard error is 0.05838. The null would be , and the alternate hypothesis would be .

The t-statistic is (checking for rounding errors), as given. The critical t-would be , and since (two-way test), we may reject the null. Hence, the slope coefficient is indeed significant (different from zero), which can be matched with the fact that the p-value of the calculated t in the regression is indeed less than 0.05.

2) The fit of the line can be checked with the R-squared (goodness of fit) and the F-statistic (significance of goodness of fit). We have , meaning that 59.31% of the variation in wins is explained by the yards made variable. The F-statistic of 43.72 is significant with its low p-value, suggesting that the R-squared is significant.

3) The slope means that, for a unit increase in the yards made, the wins increases by 0.386 units on average.

b. The regression output is as below.

1) The slope is about -0.3002, and the standard error is 0.1041. The null would be , and the alternate hypothesis would be .

The t-statistic is (checking for rounding errors), as given. The critical t-would be same , and since (two-way test), we may reject the null. Hence, the slope coefficient is indeed significant, which can be matched with the fact that the p-value of the calculated t in the regression is indeed less than 0.05.

2) The fit of the line can be checked with the R-squared (goodness of fit) and the F-statistic (significance of goodness of fit). We have , meaning that 21.71% of the variation in wins is explained by the yards made variable. The F-statistic of 8.32 is significant with its low p-value (less than alpha of 0.05), suggesting that the R-squared is significant.

3) The slope means that, for a unit increase in the yards made, the wins decreases by 0.3002 units on average.

c. The yards made variable is a better at explaining the wins variable, since it have higher R-square. The comparison of two models with respect to their R-square is valid only when the dependent variable is the same, which is indeed the same in this case. Both of the model have significant R-square, but the variation in the wins is explained more by the yards made variable (59.31%) than the yards allowed variable (21.71%).