62.Refer to the Baseball 2009 data, which report information on the 30 Major League Baseball teams for the 2009 season.
a.
At the .05 significance level, can we conclude that there is a difference in the mean salary of teams in the American League versus teams in the National League?
b.
At the .05 significance level, can we conclude that there is a difference in the mean home attendance of teams in the American League versus teams in the National League?
c.
Compute the mean and the standard deviation of the number of wins for the 10 teams with the highest salaries. Do the same for the 10 teams with the lowest salaries. At the .05 significance level, is there a difference in the mean number of wins for the two groups
Team League Size Salary Wins Attendance
X1 X2 X4 X5 X6 X7
Baltimore Orioles 1 48876 67.1 64 1.91
Boston Red Sox 1 39928 121.8 95 3.06
Chicago White Sox 1 40615 96.1 79 2.28
Cleveland Indians 1 43345 81.6 65 1.77
Detroit Tigers 1 41782 115.1 86 2.57
Kansas City Royals 1 40793 70.5 65 1.80
Los Angeles Angels 1 45050 113.7 97 3.24
Minnesota Twins 1 40000 65.3 87 2.42
New York Yankees 1 52325 201.5 103 3.72
Oakland Athletics 1 34077 62.3 75 1.41
Seattle Mariners 1 47116 98.9 85 2.20
Tampa Bay Rays 1 36048 63.3 84 1.87
Texas Rangers 1 49115 68.2 87 2.16
Toronto Blue Jays 1 50516 80.5 75 1.88
Arizona Diamondbacks 0 49033 73.5 70 2.13
Atlanta Braves 0 50091 96.7 86 2.37
Chicago Cubs 0 41118 134.8 83 3.17
Cincinnati Reds 0 42059 73.6 78 1.75
Colorado Rockies 0 50445 75.2 92 2.67
Florida Marlins 0 36331 36.8 87 1.46
Houston Astros 0 40950 103.0 74 2.52
Los Angeles Dodgers 0 56000 100.4 95 3.76
Milwaukee Brewers 0 42200 80.2 80 3.04
New York Mets 0 45000 149.4 70 3.15
Philadelphia Phillies 0 43647 113.0 93 3.60
Pittsburgh Pirates 0 38496 48.7 62 1.58
San Diego Padres 0 42445 43.7 75 1.92
San Francisco Giants 0 41503 82.6 88 2.86
St. Louis Cardinals 0 49660 77.6 91 3.34
Washington Nationals 0 41888 60.3 59 1.82
a. To determine if there is a difference in the mean salary of teams in the American League versus teams in the National League at the .05 significance level, we can perform a two-sample t-test.
1. First, separate the data into two groups: American League teams and National League teams.
2. Calculate the mean (average) salary for each group.
3. Calculate the standard deviation for each group.
4. Determine the number of observations (teams) in each group.
5. Use the t-test formula to calculate the t-value:
t = (mean 1 - mean 2) / sqrt((s1^2 / n1) + (s2^2 / n2))
where mean 1 and mean 2 are the means of the two groups,
s1 and s2 are the standard deviations of the two groups, and
n1 and n2 are the number of observations in each group.
6. Calculate the degrees of freedom using the formula:
df = n1 + n2 - 2
where n1 and n2 are the number of observations in each group.
7. Look up the critical t-value in the t-distribution table for the desired significance level (0.05).
8. Compare the calculated t-value with the critical t-value.
- If the calculated t-value is larger than the critical t-value, there is a significant difference.
- If the calculated t-value is smaller than the critical t-value, there is no significant difference.
b. To determine if there is a difference in the mean home attendance of teams in the American League versus teams in the National League at the .05 significance level, we can perform a two-sample t-test.
1. Follow the same steps as in part a to separate the data into two groups: American League teams and National League teams.
2. Calculate the mean (average) home attendance for each group.
3. Calculate the standard deviation for each group.
4. Determine the number of observations (teams) in each group.
5. Use the t-test formula to calculate the t-value, following the same steps as in part a.
6. Calculate the degrees of freedom using the formula, following the same steps as in part a.
7. Look up the critical t-value in the t-distribution table for the desired significance level (0.05).
8. Compare the calculated t-value with the critical t-value, following the same steps as in part a.
c. To compare the mean and standard deviation of the number of wins for the 10 teams with the highest salaries versus the 10 teams with the lowest salaries at the .05 significance level, we can perform an independent samples t-test.
1. Sort the data based on salary, from highest to lowest.
2. Select the top 10 teams with the highest salaries and calculate the mean and standard deviation of their number of wins.
3. Select the bottom 10 teams with the lowest salaries and calculate the mean and standard deviation of their number of wins.
4. Use the t-test formula to calculate the t-value, using the means, standard deviations, and number of observations for each group.
5. Calculate the degrees of freedom using the formula.
6. Look up the critical t-value in the t-distribution table for the desired significance level (0.05).
7. Compare the calculated t-value with the critical t-value.
- If the calculated t-value is larger than the critical t-value, there is a significant difference.
- If the calculated t-value is smaller than the critical t-value, there is no significant difference.