Question

The National Football League (NFL) records a variety of performance data for individuals and teams. To investigate the importance of passing on the percentage of games won by a team, the following data show the average number of passing yards per attempt (Yards/Attempt) and the percentage of games won (WinPct) for a random sample of 10 NFL teams for the 2011 season.†

Team	Yards/Attempt	WinPct
Arizona Cardinals	6.5	50
Atlanta Falcons	7.1	63
Carolina Panthers	7.4	38
Chicago Bears	6.4	50
Dallas Cowboys	7.4	50
New England Patriots	8.3	81
Philadelphia Eagles	7.4	50
Seattle Seahawks	6.1	44
St. Louis Rams	5.2	13
Tampa Bay Buccaneers	6.2	25

A. Develop the estimated regression equation that could be used to predict the percentage of games won given the average number of passing yards per attempt. (Round your numerical values to three decimal places.)

B. For the 2011 season, suppose the average number of passing yards per attempt for a certain NFL team was 6.4. Use the estimated regression equation developed in part (c) to predict the percentage of games won by that NFL team. (Note: For the 2011 season, suppose this NFL team's record was 8 wins and 8 losses. Round your answer to the nearest integer.)

Answer 1

Baed on the given data, before running a regression model, we may make the following assumptions for the test:

- Linearity: We assume that the predictor (independent variable) - Passing yards per attempt is linearly related to the response (dependent) variable - Percentage of games won - Normality: We assume that he data is normally distributed - Independence: We assume that the observations are independent of one another - Homogeneity of variance: We assume that the Variance is homogenous

A. On running a linear regression model, we may obtain the fitted regression line, expressed in the form:

where Predicted percentage of games won = Estimated Intercept coefficient    = Estimated Slope coefficient x = Passing yards per attempt

The slope coefficient can be computed using the formula:

Substituting the values:

And the intercept coefficient can be estimated using the formula:

= 46.4 - (17.175) 6.8

= -70.391

Hence, the fitted regression equation can be obtained as:

B. The fitted regression may be used to predict the Percentage of games won for a given value of Passing yards per attempt, by substituting the required x value in the regression equation:

For x = 6.4,

= 39.529%

It is given that in 2011, the actual percentage of games won was 8 / (8+8) = 8 / 16 = 50%. We find that the predicted percentage is not that close to the actual value. This implies that the regression line is only a moderate fit to the data. If we measure the goodness of fit : Coefficient of determination R2:

Substituting the values, we get which implies that the variation in Passing yards per attempt explains about 65.8% of the total variation in Percentage of games won, which suggests a moderately good fit, hence, the difference in prediction from the actual.