A multiple regression model is used to predict an NFL team’s winning record (Win

Question

A multiple regression model is used to predict an NFL team’s winning record (Win). For the explanatory variables, the average rushing yards (Rush) and the average passing yards (Pass) are used to capture offense and the average yards allowed are used to capture defense. The data for the 2009 NFL season is given below; the data, labeled Football, can also be found on the text website. Use Table 4. Team Win(%) Rush Pass Yards Allowed Arizona Cardinals 62.5 93.4000 251.0000 346.4 Atlanta Falcons 56.3 117.2125 223.1875 348.9 Baltimore Ravens 56.3 137.5125 213.6875 305.0 Buffalo Bills 37.5 116.7125 157.1875 340.6 Carolina Panthers 50.0 156.1625 174.9375 315.8 Chicago Bears 43.8 93.2375 217.0625 337.8 Cincinnati Bengals 62.5 128.4750 180.6250 301.4 Cleveland Browns 31.3 130.4500 129.7500 389.3 Dallas Cowboys 68.8 131.4625 267.9375 315.9 Denver Broncos 50.0 114.7125 226.6875 315.0 Detroit Lions 12.5 101.0000 198.0000 392.1 Green Bay Packers 68.8 117.8500 261.2500 284.4 Houston Texans 56.3 92.2250 290.8750 324.9 Indianapolis Colts 87.5 80.9125 282.1875 339.2 Jacksonville Jaguars 53.8 126.8500 209.7500 352.3 Kansas City Chiefs 25.0 120.5750 182.6250 388.2 Miami Dolphins 43.8 139.4750 198.1250 349.3 Minnesota Vikings 75.0 119.8500 259.7500 305.5 New England Patriots 62.5 120.0500 277.2500 320.2 New Orleans Saints 81.3 131.6125 272.1875 357.8 New York Giants 50.0 114.8125 251.1875 324.9 New York Jets 56.3 172.2500 148.7500 252.3 Oakland Raiders 31.3 106.2875 159.8125 361.9 Philadelphia Eagles 68.8 102.3375 255.5625 321.1 Pittsburgh Steelers 56.3 112.0500 259.2500 305.3 Saint Louis Rams 6.3 111.5000 167.8750 327.0 San Diego Chargers 81.3 88.9375 271.1250 326.4 San Francisco 49ers 50.0 100.0000 190.7500 356.4 Seattle Seahawks 31.3 97.8625 218.9375 372.8 Tampa Bay Buccaneers 18.8 101.6875 185.8125 365.6 Tennessee Titans 50.0 161.9625 189.4375 365.6 Washington Redskins 25.0 94.3750 218.1250 319.7 SOURCE: NFL website. PictureClick here for the Excel Data File a. Estimate the model: Win = 0 + 1Rush + 2Pass + 3Yards Allowed + . (Negative values should be indicated by a minus sign. Round your answers to 2 decimal places.) 1formula436.mml = + Rush + Pass + Yards Allowed b-1. Choose the appropriate hypotheses to determine whether the impact of Rush is different from that of Pass in explaining Win? H0: 1 2; HA: 1 < 2 H0: 1 2; HA: 1 > 2 H0: 1 = 2; HA: 1 2 b-2. Estimate the restricted model given that the null hypothesis is true? (Negative values should be indicated by a minus sign. Do not round the intermediate calculations. Round your answers to 2 decimal places.) 1formula436.mml = + (Rush + Pass) + Yards Allowed b-3. Calculate the value of the test statistic in order to determine whether the influence of Rush differs from that of Pass. (Round your answer to 2 decimal places.) Test statistic b-4. At the 10% significance level, find the critical value. (Round your answer to 2 decimal places.) Critical value b-5. What is the conclusion to the hypothesis test? Does the influence of Rush differ from that of Pass? Do not reject H0; the influence of Rush does differ from that of Pass. Reject H0; the influence of Rush does not differ from that of Pass. Do not reject H0; the influence of Rush does not differ from that of Pass. Reject H0; the influence of Rush does differ from that of Pass.

Explanation / Answer

Let dependent variable is win percentage.

and there are three independent variables namely rush, pass and yards allowed.

There are total 32 observations.

This is multiple regression problem.

The whole data is,

win   rush   pass   yards allowed
62.5   93.400   251.000   346.0
56.3   117.213   223.188   348.9
56.3   137.513   213.688   305.0
37.5   116.713   157.188   340.6
50.0   156.163   174.938   315.8
43.8   93.238   217.063   337.0
62.5   128.475   180.625   301.4
31.3   130.450   129.750   389.0
68.8   131.463   267.938   315.9
50.0   114.713   226.688   315.0
12.5   101.000   198.000   392.1
68.8   117.850   261.250   284.4
56.3   92.225   290.875   324.9
87.5   80.913   282.188   339.2
53.8   126.850   209.750   352.3
25.0   120.575   182.625   388.2
43.8   139.475   198.125   349.3
75.0   119.850   259.750   305.5
62.5   120.050   277.250   320.2
81.3   131.613   272.188   357.8
50.0   114.813   251.188   324.9
56.3   172.250   148.750   252.3
31.3   106.288   159.813   361.0
68.8   102.338   255.563   321.1
56.3   112.050   259.250   305.3
6.3   111.500   167.875   327.0
81.3   88.938   271.125   326.4
50.0   100.000   190.750   356.4
31.3   97.863   218.938   372.8
18.8   101.688   185.813   365.6
50.0   161.963   189.438   365.6
25.0   94.375   218.125   139.7

Multiple regression we can done using MINITAB.

steps :

Enter all the data in MINITAB sheet --> Stat --> Regression --> Regression --> Response : select y --> Predictors : x1, x2 and x3 --> Result : select second option --> ok --> ok

The output is as follows :

Regression Analysis: win versus rush, pass, yards allowed

The regression equation is
win = - 80.1 + 0.380 rush + 0.387 pass + 0.0044 yards allowed

Predictor Coef SE Coef T P
Constant -80.13 32.52 -2.46 0.020
rush 0.3804 0.1257 3.03 0.005
pass 0.38739 0.06197 6.25 0.000
yards allowed 0.00441 0.05339 0.08 0.935

S = 13.5025 R-Sq = 59.3% R-Sq(adj) = 55.0%

Analysis of Variance

Source DF SS MS F P
Regression 3 7445.6 2481.9 13.61 0.000
Residual Error 28 5104.9 182.3
Total 31 12550.5

Choose the appropriate hypotheses to determine whether the impact of Rush is different from that of Pass in explaining Win?

H0: 1 = 2; HA: 1 2

Here last option is correct.

Here we see that for all the factors test statistic F is 13.61 and P-value = 0.000

Assume alpha = level of significance = 10% = 0.1

P-value < alpha

Reject H0 at 10% level of significance.

Conclusion : At least one of the regression coefficient is differ than 0.

And the critical value we can find by using EXCEL.

syntax is,

=FINV(probability, deg_freedom1, deg_freedom2)

where probability = alpha = 0.1

deg_freedom1 = degrees of freedom for regression = 3

deg_freedom2 = degrees of freedom for residual error = 28

critical value = 2.29

F > critical value

Reject H0; the influence of Rush does differ from that of Pass.

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