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Going for it on fourth-down in the NFL. Refer to the
Chapter 12, Problem 55E(choose chapter or problem)
Going for it on fourth-down in the NFL. Refer to the Chance(Winter 2009) study of fourth-down decisions by coaches in the National Football League (NFL), Exercise 11.63(p. 638). Recall that statisticians at California State University,Northridge, fit a straight-line model for predicting the number of points scored (y) by a team that has a first-down with a given number of yards (x) from the opposing goalline. A second model fit to data collected on five NFL teams from a recent season was the quadratic regression model \(E(y)=B_{0}+B_{1} x+B_{1} x^{2}\). The regression yielded the following results: \(\hat{y}=6.13+.141 x-.0009 x^{2}, R^{2}=.226\).
a. If possible, give a practical interpretation of each of the \(\beta\) estimates in the model.
b. Give a practical interpretation of the coefficient of determination, \(R^2\).
c. In Exercise 11.63, the coefficient of correlation for the straight-line model was reported as \(R^2\) = .18.Does this statistic alone indicate that the quadratic model is a better fit than the straight-line model? Explain.
d. What test of hypothesis would you conduct to determine if the quadratic model is a better fit than the straight-line model?
Questions & Answers
QUESTION:
Going for it on fourth-down in the NFL. Refer to the Chance(Winter 2009) study of fourth-down decisions by coaches in the National Football League (NFL), Exercise 11.63(p. 638). Recall that statisticians at California State University,Northridge, fit a straight-line model for predicting the number of points scored (y) by a team that has a first-down with a given number of yards (x) from the opposing goalline. A second model fit to data collected on five NFL teams from a recent season was the quadratic regression model \(E(y)=B_{0}+B_{1} x+B_{1} x^{2}\). The regression yielded the following results: \(\hat{y}=6.13+.141 x-.0009 x^{2}, R^{2}=.226\).
a. If possible, give a practical interpretation of each of the \(\beta\) estimates in the model.
b. Give a practical interpretation of the coefficient of determination, \(R^2\).
c. In Exercise 11.63, the coefficient of correlation for the straight-line model was reported as \(R^2\) = .18.Does this statistic alone indicate that the quadratic model is a better fit than the straight-line model? Explain.
d. What test of hypothesis would you conduct to determine if the quadratic model is a better fit than the straight-line model?
ANSWER:Step 1 of 4
(a)
Interpretation:
The estimated value of \(\beta_{0}\) is 6.13. Here, the independent variables cannot be implemented whether the number of yards and the quadratic term of number of yards are increased or decreased. Thus, the estimated value remains the constant (6.13) in the model.
The estimated value of \(\beta_{1}\) is 0.141. Here, if the number of points scored has been increased for every one unit increase in the number of yards, but this is no longer slope of the line because the quadratic term is presented in the model. Thus, \(\beta_{1}\) has simply a location parameter in the model.
In the given model, \(\beta_{2}=-0.0009\), if the number of points scored has been decreased for every one unit increase in the quadratic term of number of yards. Thus, if the quadratic term increases, the predicted value decreases in the model.