Passing and Winning in the NFL. 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 (Yds/Att) and the percentage of games won (WinPct) for a random sample of 10 NFL teams for the 2011 season (NFL website).

$$\begin{array}{lcc}\text { Team } & \text { Yds/Att } & \text { WinPct } \\ \text { Arizona Cardinals } & 6.5 & 50 \\ \text { Atlanta Falcons } & 7.1 & 63 \\ \text { Carolina Panthers } & 7.4 & 38 \\ \text { Chicago Bears } & 6.4 & 50 \\ \text { Dallas Cowboys } & 7.4 & 50 \\ \text { New England Patriots } & 8.3 & 81 \\ \text { Philadelphia Eagles } & 7.4 & 50 \\ \text { Seattle Seahawks } & 6.1 & 44 \\ \text { St. Louis Rams } & 5.2 & 13 \\ \text { Tampa Bay Buccaneers } & 6.2 & 25\end{array}$$

d. Provide an interpretation for the slope of the estimated regression equation.

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 conference (Conf), average number of passing yards per attempt (Yds/Att), the number of interceptions thrown per attempt (Int/Att), and the percentage of games won (Win%) for a random sample of 16 NFL teams for the 2011 season (NFL website, February 12, 2012).

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.

b. Develop the estimated regression equation that could be used to predict the percentage of games won given the number of interceptions thrown per attempt.

c. 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 and the number of interceptions thrown per attempt.

d. The average number of passing yards per attempt for the Kansas City Chiefs was 6.2 and the number of interceptions thrown per attempt was .036. Use the estimated regression equation developed in part (c) to predict the percentage of games won by the Kansas City Chiefs. (Note: For the 2011 season the Kansas City Chiefs’record was 7 wins and 9 losses.) Compare your prediction to the actual percentage of games won by the Kansas City Chiefs

Passing and Winning in the NFL. 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 (Yds/Att) and the percentage of games won (WinPct) for a random sample of 10 NFL teams for the 2011 season (NFL website).

$$\begin{array}{lcc}\text { Team } & \text { Yds/Att } & \text { WinPct } \\ \text { Arizona Cardinals } & 6.5 & 50 \\ \text { Atlanta Falcons } & 7.1 & 63 \\ \text { Carolina Panthers } & 7.4 & 38 \\ \text { Chicago Bears } & 6.4 & 50 \\ \text { Dallas Cowboys } & 7.4 & 50 \\ \text { New England Patriots } & 8.3 & 81 \\ \text { Philadelphia Eagles } & 7.4 & 50 \\ \text { Seattle Seahawks } & 6.1 & 44 \\ \text { St. Louis Rams } & 5.2 & 13 \\ \text { Tampa Bay Buccaneers } & 6.2 & 25\end{array}$$

c. 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.

Passing and Winning in the NFL. 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 (Yds/Att) and the percentage of games won (WinPct) for a random sample of 10 NFL teams for the 2011 season (NFL website).

$$\begin{array}{lcc}\text { Team } & \text { Yds/Att } & \text { WinPct } \\ \text { Arizona Cardinals } & 6.5 & 50 \\ \text { Atlanta Falcons } & 7.1 & 63 \\ \text { Carolina Panthers } & 7.4 & 38 \\ \text { Chicago Bears } & 6.4 & 50 \\ \text { Dallas Cowboys } & 7.4 & 50 \\ \text { New England Patriots } & 8.3 & 81 \\ \text { Philadelphia Eagles } & 7.4 & 50 \\ \text { Seattle Seahawks } & 6.1 & 44 \\ \text { St. Louis Rams } & 5.2 & 13 \\ \text { Tampa Bay Buccaneers } & 6.2 & 25\end{array}$$

b. What does the scatter diagram developed in part (a) indicate about the relationship between the two variables?

Passing and Winning in the NFL. 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 (Yds/Att) and the percentage of games won (WinPct) for a random sample of 10 NFL teams for the 2011 season (NFL website).

$$\begin{array}{lcc} \text { Team } & \text { Yds/Att } & \text { WinPct } \\ \text { Arizona Cardinals } & 6.5 & 50 \\ \text { Atlanta Falcons } & 7.1 & 63 \\ \text { Carolina Panthers } & 7.4 & 38 \\ \text { Chicago Bears } & 6.4 & 50 \\ \text { Dallas Cowboys } & 7.4 & 50 \\ \text { New England Patriots } & 8.3 & 81 \\ \text { Philadelphia Eagles } & 7.4 & 50 \\ \text { Seattle Seahawks } & 6.1 & 44 \\ \text { St. Louis Rams } & 5.2 & 13 \\ \text { Tampa Bay Buccaneers } & 6.2 & 25 \end{array}$$

Develop a scatter diagram with the number of passing yards per attempt on the horizontal axis and the percentage of games won on the vertical axis.

The average number of passing yards per attempt for the Kansas City Chiefs was 6.2 and the number of interceptions thrown per attempt was .036. Use the estimated regression equation developed in part ( c) to predict the percentage of games won by the Kansas City Chiefs. (Note: For this season the Kansas City Chiefs' record was 7 wins and 9 losses.) Compare your prediction to the actual percentage of games won by the Kansas City Chiefs.

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 conference (Conf), average number of passing yards per attempt (Yds/Att), the number of interceptions thrown per attempt (Int/Att), and the percentage of games won (Win%) for a random sample of 16 NFL teams for the 2011 season (NFL Web site, February 12, 2012)

$$\begin{array}{lcccc} \text { Team } & \text { Conf } & \text { Yds/Att } & \text { Int/Att } & \text { Win } \% \\ \text { Arizona Cardinals } & \text { NFC } & 6.5 & 0.042 & 50.0 \\ \text { Atlanta Falcons } & \text { NFC } & 7.1 & 0.022 & 62.5 \\ \text { Carolina Panthers } & \text { NFC } & 7.4 & 0.033 & 37.5 \\ \text { Cincinnati Bengals } & \text { AFC } & 6.2 & 0.026 & 56.3 \\ \text { Detroit Lions } & \text { NFC } & 7.2 & 0.024 & 62.5 \\ \text { Green Bay Packers } & \text { NFC } & 8.9 & 0.014 & 93.8 \\ \text { Houston Texans } & \text { AFC } & 7.5 & 0.019 & 62.5 \\ \text { Indianapolis Colts } & \text { AFC } & 5.6 & 0.026 & 12.5 \\ \text { Jacksonville Jaguars } & \text { AFC } & 4.6 & 0.032 & 31.3 \\ \text { Minnesota Vikings } & \text { NFC } & 5.8 & 0.033 & 18.8 \\ \text { New England Patriots } & \text { AFC } & 8.3 & 0.020 & 81.3 \\ \text { New Orleans Saints } & \text { NFC } & 8.1 & 0.021 & 81.3 \\ \text { Oakland Raiders } & \text { AFC } & 7.6 & 0.044 & 50.0 \\ \text { San Francisco 49ers } & \text { NFC } & 6.5 & 0.011 & 81.3 \\ \text { Tennessee Titans } & \text { AFC } & 6.7 & 0.024 & 56.3 \\ \text { Washington Redskins } & \text { NFC } & 6.4 & 0.041 & 31.3 \end{array}$$

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. What proportion of variation in the sample values of proportion of games won does this model explain?

Predicting NFL Wins. 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 conference (Conf), average number of passing yards per attempt (Yds/Att), the number of interceptions thrown per attempt (Int/Att), and the percentage of games won (Win%) for a random sample of 16 NFL teams for one full season.

$$\begin{array}{llccc} \text { Team } & \text { Conf } & \text { Yds/Att } & \text { Int/Att } & \text { Win\% } \\ \text { Arizona Cardinals } & \text { NFC } & 6.5 & .042 & 50.0 \\ \text { Atlanta Falcons } & \text { NFC } & 7.1 & .022 & 62.5 \\ \text { Carolina Panthers } & \text { NFC } & 7.4 & .033 & 37.5 \\ \text { Cincinnati Bengals } & \text { AFC } & 6.2 & .026 & 56.3 \\ \text { Detroit Lions } & \text { NFC } & 7.2 & .024 & 62.5 \\ \text { Green Bay Packers } & \text { NFC } & 8.9 & .014 & 93.8 \\ \text { Houston Texans } & \text { AFC } & 7.5 & .019 & 62.5 \\ \text { Indianapolis Colts } & \text { AFC } & 5.6 & .026 & 12.5 \\ \text { Jacksonville Jaguars } & \text { AFC } & 4.6 & .032 & 31.3 \\ \text { Minnesota Vikings } & \text { NFC } & 5.8 & .033 & 18.8 \\ \text { New England Patriots } & \text { AFC } & 8.3 & .020 & 81.3 \\ \text { New Orleans Saints } & \text { NFC } & 8.1 & .021 & 81.3 \\ \text { Oakland Raiders } & \text { AFC } & 7.6 & .044 & 50.0 \\ \text { San Francisco 49ers } & \text { NFC } & 6.5 & .011 & 81.3 \\ \text { Tennessee Titans } & \text { AFC } & 6.7 & .024 & 56.3 \\ \text { Washington Redskins } & \text { NFC } & 6.4 & .041 & 31.3 \end{array}$$

c. 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 and the number of interceptions thrown per attempt.

The owner of Showtime Movie Theaters, Inc., would like to estimate weekly gross revenue as a function of advertising expenditures. Historical data for a sample of eight weeks follow.

$$\begin{matrix} \text{Weekly} & \text{Television} & \text{Newspaper}\\ \text{Gross Revenue} & \text{Advertising} & \text{Advertising}\\ \text{(\$1000s)} & \text{(\$1000s)} & \text{(\$1000s)}\\ \hline \text{96} & \text{5.0} & \text{1.5}\\ \text{90} & \text{2.0} & \text{2.0}\\ \text{95} & \text{4.0} & \text{1.5}\\ \text{92} & \text{2.5} & \text{2.5}\\ \text{95} & \text{3.0} & \text{3.3}\\ \text{94} & \text{3.5} & \text{2.3}\\ \text{94} & \text{2.5} & \text{4.2}\\ \text{94} & \text{3.0} & \text{2.5}\\ \end{matrix}$$

Develop an estimated regression equation with the amount of television advertising as the independent variable.

In exercise 6, data were given on the average number of passing yards per attempt (Yds/Att), the number of interceptions thrown per attempt (Int/Att), and the percentage of games won (Win%) for a random sample of 16 National Football League (NFL) teams for the 2011 season (NFL website, February 12, 2012).

a. Did the estimated regression equation that uses only the average number of passing yards per attempt as the independent variable to predict the percentage of games won provide a good fit?

b. Discuss the benefit of using both the average number of passing yards per attempt and the number of interceptions thrown per attempt to predict the percentage of games won.

Determine whether each player has an equal chance of winning given of the following games.
a. Toss a fair coin. If a head appears, I win; if a tail appears, you lose.
b. Toss a fair coin. If a head appears, I win; otherwise, you win.
c. Toss a fair die numbered 1 through 6. If 1 appears, I win; if 6 appears, you win.
d. Toss a fair die numbered 1 through 6. If an even number appears, I win; if an odd number appears, you win.
e. Toss a fair die numbered 1 through 6. If a number greater than or equal to 3 appears, I win; otherwise, you win.
f. Toss two fair dice numbered 1 through 6. If a 1 appears on each die, I win; if a 6 appears on each die, you win.
g. Toss two fair dice numbered 1 through 6. If the sum is 3, I win; if the sum is 2, you win.

Predicting NFL Wins. 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 conference (Conf), average number of passing yards per attempt (Yds/Att), the number of interceptions thrown per attempt (Int/Att), and the percentage of games won (Win%) for a random sample of 16 NFL teams for one full season.

$$\begin{array}{llccc} \text { Team } & \text { Conf } & \text { Yds/Att } & \text { Int/Att } & \text { Win\% } \\ \text { Arizona Cardinals } & \text { NFC } & 6.5 & .042 & 50.0 \\ \text { Atlanta Falcons } & \text { NFC } & 7.1 & .022 & 62.5 \\ \text { Carolina Panthers } & \text { NFC } & 7.4 & .033 & 37.5 \\ \text { Cincinnati Bengals } & \text { AFC } & 6.2 & .026 & 56.3 \\ \text { Detroit Lions } & \text { NFC } & 7.2 & .024 & 62.5 \\ \text { Green Bay Packers } & \text { NFC } & 8.9 & .014 & 93.8 \\ \text { Houston Texans } & \text { AFC } & 7.5 & .019 & 62.5 \\ \text { Indianapolis Colts } & \text { AFC } & 5.6 & .026 & 12.5 \\ \text { Jacksonville Jaguars } & \text { AFC } & 4.6 & .032 & 31.3 \\ \text { Minnesota Vikings } & \text { NFC } & 5.8 & .033 & 18.8 \\ \text { New England Patriots } & \text { AFC } & 8.3 & .020 & 81.3 \\ \text { New Orleans Saints } & \text { NFC } & 8.1 & .021 & 81.3 \\ \text { Oakland Raiders } & \text { AFC } & 7.6 & .044 & 50.0 \\ \text { San Francisco 49ers } & \text { NFC } & 6.5 & .011 & 81.3 \\ \text { Tennessee Titans } & \text { AFC } & 6.7 & .024 & 56.3 \\ \text { Washington Redskins } & \text { NFC } & 6.4 & .041 & 31.3 \end{array}$$

b. Develop the estimated regression equation that could be used to predict the percentage of games won given the number of interceptions thrown per attempt.

A multiple regression model is used to predict an NFL team's winning record (Win in %). 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 (Yards Allowed). A portion of the data for the 2009 NFL season is shown in the accompanying table.

$$\begin{array}{|l|c|c|c|c|} \hline \text { Team } & \text { Win } & \text { Rush } & \text { Pass } & \text { Yards Allowed } \\ \hline \text { Arizona Cardinals } & 62.50 & 93.40 & 251.00 & 346.40 \\ \hline \text { Atlanta Falcons } & 56.30 & 117.21 & 223.19 & 348.90 \\ \hline \text { : } & \vdots & \vdots & \vdots & \vdots \\ \hline \text { Washington Redskins } & 25.00 & 94.38 & 218.13 & 319.70 \\ \hline \end{array}$$

a. Estimate the model: Win $=\beta_0+\beta_1$ Rush $+\beta_2$ Pass $+\beta_3$ Yards Allowed $+\varepsilon$.

b. Conduct a test at the $10 \%$ significance level to determine whether the impact of Rush is different from that of Pass in explaining Win, or $\beta_1+\beta_2$.

Predicting NFL Wins. 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 conference (Conf), average number of passing yards per attempt (Yds/Att), the number of interceptions thrown per attempt (Int/Att), and the percentage of games won (Win%) for a random sample of 16 NFL teams for one full season.

$$\begin{array}{llccc} \text { Team } & \text { Conf } & \text { Yds/Att } & \text { Int/Att } & \text { Win\% } \\ \text { Arizona Cardinals } & \text { NFC } & 6.5 & .042 & 50.0 \\ \text { Atlanta Falcons } & \text { NFC } & 7.1 & .022 & 62.5 \\ \text { Carolina Panthers } & \text { NFC } & 7.4 & .033 & 37.5 \\ \text { Cincinnati Bengals } & \text { AFC } & 6.2 & .026 & 56.3 \\ \text { Detroit Lions } & \text { NFC } & 7.2 & .024 & 62.5 \\ \text { Green Bay Packers } & \text { NFC } & 8.9 & .014 & 93.8 \\ \text { Houston Texans } & \text { AFC } & 7.5 & .019 & 62.5 \\ \text { Indianapolis Colts } & \text { AFC } & 5.6 & .026 & 12.5 \\ \text { Jacksonville Jaguars } & \text { AFC } & 4.6 & .032 & 31.3 \\ \text { Minnesota Vikings } & \text { NFC } & 5.8 & .033 & 18.8 \\ \text { New England Patriots } & \text { AFC } & 8.3 & .020 & 81.3 \\ \text { New Orleans Saints } & \text { NFC } & 8.1 & .021 & 81.3 \\ \text { Oakland Raiders } & \text { AFC } & 7.6 & .044 & 50.0 \\ \text { San Francisco 49ers } & \text { NFC } & 6.5 & .011 & 81.3 \\ \text { Tennessee Titans } & \text { AFC } & 6.7 & .024 & 56.3 \\ \text { Washington Redskins } & \text { NFC } & 6.4 & .041 & 31.3 \end{array}$$

a. Develop the estimated regression equation that could be used to predict the percent- age of games won given the average number of passing yards per attempt.