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 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. A scatter diagram has 9 points plotted on it. The horizontal axis ranges from 5 to 9 and is labeled: Yards/Attempt. The vertical axis ranges from 0 to 90 and is labeled: Win %. The points are plotted from left to right in an upward, diagonal direction starting in the lower left corner of the diagram and are between 5 to 8.5 on the horizontal axis and between 10 to 85 on the vertical axis. A scatter diagram has 10 points plotted on it. The horizontal axis ranges from 5 to 9 and is labeled: Yards/Attempt. The vertical axis ranges from 0 to 90 and is labeled: Win %. The points appear to be distributed somewhat randomly and are between 5 to 8.5 on the horizontal axis. The 4 leftmost points are between 35 to 65 on the vertical axis and the 5 rightmost points are between 10 to 85 on the vertical axis. A scatter diagram has 9 points plotted on it. The horizontal axis ranges from 5 to 9 and is labeled: Yards/Attempt. The vertical axis ranges from 0 to 90 and is labeled: Win %. The points appear to be distributed somewhat randomly and are between 5 to 8.5 on the horizontal axis. The 4 leftmost points are between 40 to 85 on the vertical axis and the 5 rightmost points are between 10 to 50 on the vertical axis. A scatter diagram has 9 points plotted on it. The horizontal axis ranges from 5 to 9 and is labeled: Yards/Attempt. The vertical axis ranges from 0 to 90 and is labeled: Win %. The points are plotted from left to right in an upward, diagonal direction starting in the lower left corner of the diagram and are between 5 to 8.5 on the horizontal axis and between 5 to 80 on the vertical axis. (b) What does the scatter diagram developed in part (a) indicate about the relationship between the two variables? The scatter diagram indicates a negative linear relationship between average number of passing yards per attempt and the percentage of games won by the team. The scatter diagram indicates no noticeable linear relationship between average number of passing yards per attempt and the percentage of games won by the team. The scatter diagram indicates a positive linear relationship between average number of passing yards per attempt and the percentage of games won by the team. (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. (Round your numerical values to three decimal places.) ŷ = (d) Provide an interpretation for the slope of the estimated regression equation. The slope gives the percentage of games won when the average number of passes per attempt is 0. The slope gives the average number of passes per attempt when the percentage of games won is 0%. The slope gives the change in the percentage of games won for every one yard increase in the average number of passes per attempt. The slope gives the change in the average number of passes per attempt for every one percentage point increase in the percentage of games won. The slope gives the change in the average number of passes per attempt for every one percentage point decrease in the percentage of games won. (e) For the 2011 season, suppose the average number of passing yards per attempt for a certain NFL team was 6.1. 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 7 wins and 9 losses. Round your answer to the nearest integer.) % Compare your prediction to the actual percentage of games won by this NFL team. The predicted value is higher than the actual value. The predicted value is lower than the actual value. The predicted value is identical to the actual value.
Inverse Normal Distribution
The method used for finding the corresponding z-critical value in a normal distribution using the known probability is said to be an inverse normal distribution. The inverse normal distribution is a continuous probability distribution with a family of two parameters.
Mean, Median, Mode
It is a descriptive summary of a data set. It can be defined by using some of the measures. The central tendencies do not provide information regarding individual data from the dataset. However, they give a summary of the data set. The central tendency or measure of central tendency is a central or typical value for a probability distribution.
Z-Scores
A z-score is a unit of measurement used in statistics to describe the position of a raw score in terms of its distance from the mean, measured with reference to standard deviation from the mean. Z-scores are useful in statistics because they allow comparison between two scores that belong to different normal distributions.
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 a
A scatter diagram has 9 points plotted on it. The horizontal axis ranges from 5 to 9 and is labeled: Yards/Attempt. The vertical axis ranges from 0 to 90 and is labeled: Win %. The points are plotted from left to right in an upward, diagonal direction starting in the lower left corner of the diagram and are between 5 to 8.5 on the horizontal axis and between 10 to 85 on the vertical axis.
A scatter diagram has 10 points plotted on it. The horizontal axis ranges from 5 to 9 and is labeled: Yards/Attempt. The vertical axis ranges from 0 to 90 and is labeled: Win %. The points appear to be distributed somewhat randomly and are between 5 to 8.5 on the horizontal axis. The 4 leftmost points are between 35 to 65 on the vertical axis and the 5 rightmost points are between 10 to 85 on the vertical axis.
A scatter diagram has 9 points plotted on it. The horizontal axis ranges from 5 to 9 and is labeled: Yards/Attempt. The vertical axis ranges from 0 to 90 and is labeled: Win %. The points appear to be distributed somewhat randomly and are between 5 to 8.5 on the horizontal axis. The 4 leftmost points are between 40 to 85 on the vertical axis and the 5 rightmost points are between 10 to 50 on the vertical axis.
A scatter diagram has 9 points plotted on it. The horizontal axis ranges from 5 to 9 and is labeled: Yards/Attempt. The vertical axis ranges from 0 to 90 and is labeled: Win %. The points are plotted from left to right in an upward, diagonal direction starting in the lower left corner of the diagram and are between 5 to 8.5 on the horizontal axis and between 5 to 80 on the vertical axis.
(b)
What does the scatter diagram developed in part (a) indicate about the relationship between the two variables?
The scatter diagram indicates a negative linear relationship between average number of passing yards per attempt and the percentage of games won by the team.
The scatter diagram indicates no noticeable linear relationship between average number of passing yards per attempt and the percentage of games won by the team.
The scatter diagram indicates a positive linear relationship between average number of passing yards per attempt and the percentage of games won by the team.
(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. (Round your numerical values to three decimal places.)
ŷ =
(d)
Provide an interpretation for the slope of the estimated regression equation.
The slope gives the percentage of games won when the average number of passes per attempt is 0.
The slope gives the average number of passes per attempt when the percentage of games won is 0%.
The slope gives the change in the percentage of games won for every one yard increase in the average number of passes per attempt.
The slope gives the change in the average number of passes per attempt for every one percentage point increase in the percentage of games won.
The slope gives the change in the average number of passes per attempt for every one percentage point decrease in the percentage of games won.
(e)
For the 2011 season, suppose the average number of passing yards per attempt for a certain NFL team was 6.1. 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 7 wins and 9 losses. Round your answer to the nearest integer.)
%
Compare your prediction to the actual percentage of games won by this NFL team.
The predicted value is higher than the actual value.
The predicted value is lower than the actual value.
The predicted value is identical to the actual value.
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For the 2011 season, suppose the average number of passing yards per attempt for a certain NFL team was 6.1. 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 7 wins and 9 losses. Round your answer to the nearest integer.)
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