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Refer to the Baseball 2018 data given below, which report information on the 30 Major League Baseball teams for the 2018 season.  Let the number of games won be the dependent variable and the following variables be independent variables:  team batting average, team earned run average (ERA), number of home runs and whether the team plays in the American or National league (American League is 1 and National League is 0). a.  Develop a correlation matrix.   (i)  Which independent variables have strong or weak correlations with the dependent variable.   (ii)  Do you see any problems with multicollinearity?  Explain your answer. b.  Use Excel to determine the multiple regression equation.   (i)  Write out the regression equation and determine its practical application (i.e., interpret the equation).   (ii)  Report and interpret the R-square. c. Conduct a global test on the set of independent variables.  Interpret. d.  Conduct a test of hypothesis on each of the independent variables.  Would you consider deleting any of the variables?   (i)  If so, which ones? (ii)  If so, what is your new equation? e.  Develop a histogram of the residuals from the final regression equation developed in part (d-ii).  Is it reasonable to conclude that the normality assumption has been met?  Why or Why not? f.  Plot the residuals against the fitted values from the final regression equation developed in part (d-ii).  Plot the residuals on the vertical axis and the fitted values on the horizontal axis.  What regression assumption is supported?  Why is it supported?       Team League  ($ mil) HR BA Wins ERA Opened mil $ bil Arizona Diamondbacks National 143.32 176 0.235 82 3.72 1998 2.242695 1.21 Atlanta Braves National 130.6 175 0.257 90 3.75 2017 2.555781 1.625 Baltimore Orioles American 127.63 188 0.239 47 5.18 1992 1.564192 1.2 Boston Red Sox American 227.4 208 0.268 108 3.75 1912 2.895575 2.8 Chicago Cubs National 194.26 167 0.258 95 3.65 1914 3.181089 2.9 Chicago White Sox American 71.84 182 0.241 62 4.84 1991 1.608817 1.5 Cincinnati Reds National 100.31 172 0.254 67 4.63 2003 1.629356 1.01 Cleveland Indians American 142.8 216 0.259 91 3.77 1994 1.926701 1.045 Colorado Rockies National 143.97 210 0.256 91 4.33 1995 3.01588 1.1 Detroit Tigers American 130.96 135 0.241 64 4.58 2000 1.85697 1.225 Houston Astros American 163.52 205 0.255 103 3.11 2000 2.980549 1.65 Kansas City Royals American 129.94 155 0.245 58 4.94 1973 1.665107 1.015 Los Angeles Angels American 173.78 214 0.242 80 4.15 1966 3.020216 1.8 Los Angeles Dodgers National 199.58 235 0.25 92 3.38 1962 3.8575 3 Miami Marlins National 91.82 128 0.237 63 4.76 2012 0.811104 1 Milwaukee Brewers National 108.98 218 0.252 96 3.73 2001 2.850875 1.03 Minnesota Twins American 115.51 166 0.25 78 4.5 2010 1.959197 1.15 New York Mets National 150.19 170 0.234 77 4.07 2009 2.224995 2.1 New York Yankees American 179.6 267 0.249 100 3.78 2009 3.482855 4 Oakland Athletics American 80.32 227 0.252 97 3.81 1966 1.573616 1.02 Philadelphia Phillies National 104.3 186 0.234 80 4.14 2004 2.158124 1.7 Pittsburgh Pirates National 91.03 157 0.254 82 4 2001 1.465316 1.26 San Diego Padres National 101.34 162 0.235 66 4.4 2004 2.168536 1.27 San Francisco Giants American 205.67 176 0.254 89 4.13 2000 2.299489 2.85 Seattle Mariners National 160.99 133 0.239 73 3.95 1999 3.156185 1.45 St. Louis Cardinals National 163.78 205 0.249 88 3.85 2006 3.403587 1.9 Tampa Bay Rays American 68.81 150 0.258 90 3.74 1990 1.154973 0.9 Texas Rangers American 140.63 194 0.24 67 4.92 1994 2.107107 1.6 Toronto Blue Jays American 150.95 217 0.244 73 4.85 1989 2.325281 1.35 Washington Nationals National 181.38 191 0.254 82 4.04 2008 2.529604 1.675

Refer to the Baseball 2018 data given below, which report information on the 30 Major League Baseball teams for the 2018 season.  Let the number of games won be the dependent variable and the following variables be independent variables:  team batting average, team earned run average (ERA), number of home runs and whether the team plays in the American or National league (American League is 1 and National League is 0). a.  Develop a correlation matrix.   (i)  Which independent variables have strong or weak correlations with the dependent variable.   (ii)  Do you see any problems with multicollinearity?  Explain your answer. b.  Use Excel to determine the multiple regression equation.   (i)  Write out the regression equation and determine its practical application (i.e., interpret the equation).   (ii)  Report and interpret the R-square. c. Conduct a global test on the set of independent variables.  Interpret. d.  Conduct a test of hypothesis on each of the independent variables.  Would you consider deleting any of the variables?   (i)  If so, which ones? (ii)  If so, what is your new equation? e.  Develop a histogram of the residuals from the final regression equation developed in part (d-ii).  Is it reasonable to conclude that the normality assumption has been met?  Why or Why not? f.  Plot the residuals against the fitted values from the final regression equation developed in part (d-ii).  Plot the residuals on the vertical axis and the fitted values on the horizontal axis.  What regression assumption is supported?  Why is it supported?       Team League  ($ mil) HR BA Wins ERA Opened mil $ bil Arizona Diamondbacks National 143.32 176 0.235 82 3.72 1998 2.242695 1.21 Atlanta Braves National 130.6 175 0.257 90 3.75 2017 2.555781 1.625 Baltimore Orioles American 127.63 188 0.239 47 5.18 1992 1.564192 1.2 Boston Red Sox American 227.4 208 0.268 108 3.75 1912 2.895575 2.8 Chicago Cubs National 194.26 167 0.258 95 3.65 1914 3.181089 2.9 Chicago White Sox American 71.84 182 0.241 62 4.84 1991 1.608817 1.5 Cincinnati Reds National 100.31 172 0.254 67 4.63 2003 1.629356 1.01 Cleveland Indians American 142.8 216 0.259 91 3.77 1994 1.926701 1.045 Colorado Rockies National 143.97 210 0.256 91 4.33 1995 3.01588 1.1 Detroit Tigers American 130.96 135 0.241 64 4.58 2000 1.85697 1.225 Houston Astros American 163.52 205 0.255 103 3.11 2000 2.980549 1.65 Kansas City Royals American 129.94 155 0.245 58 4.94 1973 1.665107 1.015 Los Angeles Angels American 173.78 214 0.242 80 4.15 1966 3.020216 1.8 Los Angeles Dodgers National 199.58 235 0.25 92 3.38 1962 3.8575 3 Miami Marlins National 91.82 128 0.237 63 4.76 2012 0.811104 1 Milwaukee Brewers National 108.98 218 0.252 96 3.73 2001 2.850875 1.03 Minnesota Twins American 115.51 166 0.25 78 4.5 2010 1.959197 1.15 New York Mets National 150.19 170 0.234 77 4.07 2009 2.224995 2.1 New York Yankees American 179.6 267 0.249 100 3.78 2009 3.482855 4 Oakland Athletics American 80.32 227 0.252 97 3.81 1966 1.573616 1.02 Philadelphia Phillies National 104.3 186 0.234 80 4.14 2004 2.158124 1.7 Pittsburgh Pirates National 91.03 157 0.254 82 4 2001 1.465316 1.26 San Diego Padres National 101.34 162 0.235 66 4.4 2004 2.168536 1.27 San Francisco Giants American 205.67 176 0.254 89 4.13 2000 2.299489 2.85 Seattle Mariners National 160.99 133 0.239 73 3.95 1999 3.156185 1.45 St. Louis Cardinals National 163.78 205 0.249 88 3.85 2006 3.403587 1.9 Tampa Bay Rays American 68.81 150 0.258 90 3.74 1990 1.154973 0.9 Texas Rangers American 140.63 194 0.24 67 4.92 1994 2.107107 1.6 Toronto Blue Jays American 150.95 217 0.244 73 4.85 1989 2.325281 1.35 Washington Nationals National 181.38 191 0.254 82 4.04 2008 2.529604 1.675

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.6: Applications And The Perron-frobenius Theorem
Problem 22EQ
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Refer to the Baseball 2018 data given below, which report information on the 30 Major League Baseball teams for the 2018 season.  Let the number of games won be the dependent variable and the following variables be independent variables:  team batting average, team earned run average (ERA), number of home runs and whether the team plays in the American or National league (American League is 1 and National League is 0).
a.  Develop a correlation matrix.  
(i)  Which independent variables have strong or weak correlations with the dependent variable.  
(ii)  Do you see any problems with multicollinearity?  Explain your answer.
b.  Use Excel to determine the multiple regression equation.  
(i)  Write out the regression equation and determine its practical application (i.e., interpret the equation).  
(ii)  Report and interpret the R-square.
c. Conduct a global test on the set of independent variables.  Interpret.
d.  Conduct a test of hypothesis on each of the independent variables.  Would you consider deleting any of the variables?  
(i)  If so, which ones?
(ii)  If so, what is your new equation?
e.  Develop a histogram of the residuals from the final regression equation developed in part (d-ii).  Is it reasonable to conclude that the normality assumption has been met?  Why or Why not?
f.  Plot the residuals against the fitted values from the final regression equation developed in part (d-ii).  Plot the residuals on the vertical axis and the fitted values on the horizontal axis.  What regression assumption is supported?  Why is it supported?
 
 
 
Team League  ($ mil) HR BA Wins ERA Opened mil $ bil
Arizona Diamondbacks National 143.32 176 0.235 82 3.72 1998 2.242695 1.21
Atlanta Braves National 130.6 175 0.257 90 3.75 2017 2.555781 1.625
Baltimore Orioles American 127.63 188 0.239 47 5.18 1992 1.564192 1.2
Boston Red Sox American 227.4 208 0.268 108 3.75 1912 2.895575 2.8
Chicago Cubs National 194.26 167 0.258 95 3.65 1914 3.181089 2.9
Chicago White Sox American 71.84 182 0.241 62 4.84 1991 1.608817 1.5
Cincinnati Reds National 100.31 172 0.254 67 4.63 2003 1.629356 1.01
Cleveland Indians American 142.8 216 0.259 91 3.77 1994 1.926701 1.045
Colorado Rockies National 143.97 210 0.256 91 4.33 1995 3.01588 1.1
Detroit Tigers American 130.96 135 0.241 64 4.58 2000 1.85697 1.225
Houston Astros American 163.52 205 0.255 103 3.11 2000 2.980549 1.65
Kansas City Royals American 129.94 155 0.245 58 4.94 1973 1.665107 1.015
Los Angeles Angels American 173.78 214 0.242 80 4.15 1966 3.020216 1.8
Los Angeles Dodgers National 199.58 235 0.25 92 3.38 1962 3.8575 3
Miami Marlins National 91.82 128 0.237 63 4.76 2012 0.811104 1
Milwaukee Brewers National 108.98 218 0.252 96 3.73 2001 2.850875 1.03
Minnesota Twins American 115.51 166 0.25 78 4.5 2010 1.959197 1.15
New York Mets National 150.19 170 0.234 77 4.07 2009 2.224995 2.1
New York Yankees American 179.6 267 0.249 100 3.78 2009 3.482855 4
Oakland Athletics American 80.32 227 0.252 97 3.81 1966 1.573616 1.02
Philadelphia Phillies National 104.3 186 0.234 80 4.14 2004 2.158124 1.7
Pittsburgh Pirates National 91.03 157 0.254 82 4 2001 1.465316 1.26
San Diego Padres National 101.34 162 0.235 66 4.4 2004 2.168536 1.27
San Francisco Giants American 205.67 176 0.254 89 4.13 2000 2.299489 2.85
Seattle Mariners National 160.99 133 0.239 73 3.95 1999 3.156185 1.45
St. Louis Cardinals National 163.78 205 0.249 88 3.85 2006 3.403587 1.9
Tampa Bay Rays American 68.81 150 0.258 90 3.74 1990 1.154973 0.9
Texas Rangers American 140.63 194 0.24 67 4.92 1994 2.107107 1.6
Toronto Blue Jays American 150.95 217 0.244 73 4.85 1989 2.325281 1.35
Washington Nationals National 181.38 191 0.254 82 4.04 2008 2.529604 1.675
Share Commer League 0.688646 2.022367 0.340515 0.736314 -3.4765 4.853788-3.4765 4.853787928 Dependent variable = Number games won(wins) Independent variables= team batting average, team earned run average (ERA), number of home runs and whether the team plays in the American or National league (American League is 1 and National League is 0). HR 0.09311 0.031856 2.922852 0.007262 0.027502 0.158718 0.027502 0.158717998 )Regression Equation Sensitivity Step 2 a) The correlation matrix is given below Number games won= 22.06117+(494.1202) Batting average +(-19.6055)Earned run average+ home runs (0.688646)League+(0.09311)number of ВА ERA League HR Wins BА 1 The coefficient in the regression equation implies how much the mean of the dependent variable varies for a one-unit change in the independent variable while the other factors are kept constant. ERA -0.4248 1 League 0.184303 0.24187 The sign of the coefficient implies whether the independent and dependent are positively correlated or negatively correlated. If the sign is positive, they are positively correlated and if the sign is negative, they are negatively correlated. HR 0.332278 -0.33829 0.225937 1 Wins 0.66438 -0.87343 -0.03916 0.540622 1 )Batting average has a stronger positive correlation (0.66438) with the dependent variable (wins). ii)R-square=0.905794 91% percentage of the model explains the variation in the dependent variable. Earned run average has a strong negative correlation (-0.87343) with the dependent variable (wins). Step 4 c)Global test: Ho:ß1=B2=ß3=B4=0Ho:B1=B2=B3=B4=0 ii)The batting average and the Earned run average have multicollinearity as their correlation value is (-0.4248). Significance F df SS MS F Step 3 Regression Residual 4 5689.263 1422.316 60.09408 1.85E-12 b) Multiple linear regression model 25 591.7038 23.66815 Total 29 6280.967 The output is given below since the significance value is approximately zero. the null hypothesis is rejected. Regression Statistics Multiple R 0.951732 R Square Adjusted R Square Standard It means that the independent variables explain the model well. 0.905794 The regression model with an independent variable fits the data better than the model without the independent variable. 0.890721 4.864992 Error + Observations 30 Standard Error Lower Upper 95% 95% Lower Coefficients t Stat P-value Upper 95.0% 95.0% Intercept 22.06117 34.2046 0.644977 0.52481748.3845 92.50686-48.3845 92.50685714 ВА 494.1202 118.0685 4.18503 0.000308 250.9536 737.2869 250.9536 737.2868921 ERA -19.6055 2.2074 -8.88173 3.32E-09 -24.1518 -15.0593 -24.1518 -15.05930404
Transcribed Image Text:Share Commer League 0.688646 2.022367 0.340515 0.736314 -3.4765 4.853788-3.4765 4.853787928 Dependent variable = Number games won(wins) Independent variables= team batting average, team earned run average (ERA), number of home runs and whether the team plays in the American or National league (American League is 1 and National League is 0). HR 0.09311 0.031856 2.922852 0.007262 0.027502 0.158718 0.027502 0.158717998 )Regression Equation Sensitivity Step 2 a) The correlation matrix is given below Number games won= 22.06117+(494.1202) Batting average +(-19.6055)Earned run average+ home runs (0.688646)League+(0.09311)number of ВА ERA League HR Wins BА 1 The coefficient in the regression equation implies how much the mean of the dependent variable varies for a one-unit change in the independent variable while the other factors are kept constant. ERA -0.4248 1 League 0.184303 0.24187 The sign of the coefficient implies whether the independent and dependent are positively correlated or negatively correlated. If the sign is positive, they are positively correlated and if the sign is negative, they are negatively correlated. HR 0.332278 -0.33829 0.225937 1 Wins 0.66438 -0.87343 -0.03916 0.540622 1 )Batting average has a stronger positive correlation (0.66438) with the dependent variable (wins). ii)R-square=0.905794 91% percentage of the model explains the variation in the dependent variable. Earned run average has a strong negative correlation (-0.87343) with the dependent variable (wins). Step 4 c)Global test: Ho:ß1=B2=ß3=B4=0Ho:B1=B2=B3=B4=0 ii)The batting average and the Earned run average have multicollinearity as their correlation value is (-0.4248). Significance F df SS MS F Step 3 Regression Residual 4 5689.263 1422.316 60.09408 1.85E-12 b) Multiple linear regression model 25 591.7038 23.66815 Total 29 6280.967 The output is given below since the significance value is approximately zero. the null hypothesis is rejected. Regression Statistics Multiple R 0.951732 R Square Adjusted R Square Standard It means that the independent variables explain the model well. 0.905794 The regression model with an independent variable fits the data better than the model without the independent variable. 0.890721 4.864992 Error + Observations 30 Standard Error Lower Upper 95% 95% Lower Coefficients t Stat P-value Upper 95.0% 95.0% Intercept 22.06117 34.2046 0.644977 0.52481748.3845 92.50686-48.3845 92.50685714 ВА 494.1202 118.0685 4.18503 0.000308 250.9536 737.2869 250.9536 737.2868921 ERA -19.6055 2.2074 -8.88173 3.32E-09 -24.1518 -15.0593 -24.1518 -15.05930404
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