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In: Statistics and Probability

3. Use "MLB_Salaries" data in Chapter3.xlsx to answer the following questions. For questions that require Excel,...

3. Use "MLB_Salaries" data in Chapter3.xlsx to answer the following questions. For questions that require Excel, include the appropriate output (copy + paste) along with an explanation. Data description: An article in The Wall Street Journal (July 11, 2008) outlined a number of reasons as to why the 16 teams in Major League Baseball’s National League (NL) are inferior to the 14 teams in the American League (AL). One reason for the imbalance pointed to the disparity in opening-day payrolls: the average AL payroll is greater than the NL average. A portion of the data showing opening-day payroll (in $) for each team is shown in the accompanying table.

Questions:

a. Discuss the mean and the median of AL and NL opening-day salaries and comment on skewness.

b. Compare the range and the standard deviation of AL and NL opening-day salaries.

c. Use these summary measures to comment on the findings in The Wall Street Journal.

American League Payroll National League Payroll
New York Yankees 206333389 Chicago Cubs 146609000
Boston Red Sox 162447333 Philadelphia Phillies 141928379
Detroit Tigers 122864928 New York Mets 134422942
Chicago White Sox 105530000 San Francisco Giants 98641333
Los Angeles Angels 104963866 Los Angeles Dodgers 95358016
Minnesota Twins 97559166 St. Louis Cardinals 93540751
Seattle Mariners 86510000 Houston Astros 92355500
Baltimore Orioles 81612500 Atlanta Braves 84423666
Tampa Bay Rays 71923471 Colorado Rockies 84227000
Kansas City Royals 71405210 Milwaukee Brewers 81108278
Toronto Blue Jays 62234000 Cincinnati Reds 71761542
Cleveland Indians 61203966 Washington Nationals 61400000
Texas Rangers 55250544 Arizona Diamondbacks 60718166
Oakland Athletics 51654900 Florida Marlins 57034719
San Diego Padres 37799300
Pittsburgh Pirates 34943000

Solutions

Expert Solution

American League Payroll National League Payroll
New York Yankees 206333389 Chicago Cubs 146609000
Boston Red Sox 162447333 Philadelphia Phillies 141928379
Detroit Tigers 122864928 New York Mets 134422942
Chicago White Sox 105530000 San Francisco Giants 98641333
Los Angeles Angels 104963866 Los Angeles Dodgers 95358016
Minnesota Twins 97559166 St. Louis Cardinals 93540751
Seattle Mariners 86510000 Houston Astros 92355500
Baltimore Orioles 81612500 Atlanta Braves 84423666
Tampa Bay Rays 71923471 Colorado Rockies 84227000
Kansas City Royals 71405210 Milwaukee Brewers 81108278
Toronto Blue Jays 62234000 Cincinnati Reds 71761542
Cleveland Indians 61203966 Washington Nationals 61400000
Texas Rangers 55250544 Arizona Diamondbacks 60718166
Oakland Athletics 51654900 Florida Marlins 57034719
San Diego Padres 37799300
Pittsburgh Pirates 34943000
Total 1341493273 1376271592

a.

Mean of American League Payroll = 1341493273 / 14 = 95820948.07

Mean of National League Payroll = 1376271592 / 16 = 86016974.5

Now, we arrange the data in ascending order as :

S.No. American League Payroll S.No. National League Payroll
1 51654900 Oakland Athletics 1 Pittsburgh Pirates 34943000
2 55250544 Texas Rangers 2 San Diego Padres 37799300
3 61203966 Cleveland Indians 3 Florida Marlins 57034719
4 62234000 Toronto Blue Jays 4 Arizona Diamondbacks 60718166
5 71405210 Kansas City Royals 5 Washington Nationals 61400000
6 71923471 Tampa Bay Rays 6 Cincinnati Reds 71761542
7 81612500 Baltimore Orioles 7 Milwaukee Brewers 81108278
8 86510000 Seattle Mariners 8 Colorado Rockies 84227000
9 97559166 Minnesota Twins 9 Atlanta Braves 84423666
10 104963866 Los Angeles Angels 10 Houston Astros 92355500
11 105530000 Chicago White Sox 11 St. Louis Cardinals 93540751
12 122864928 Detroit Tigers 12 Los Angeles Dodgers 95358016
13 162447333 Boston Red Sox 13 San Francisco Giants 98641333
14 206333389 New York Yankees 14 New York Mets 134422942
15 Philadelphia Phillies 141928379
16 Chicago Cubs 146609000

Median = [ (n+1)/2 ]th value

Median for American League Payroll = [ (14+1)/2 ]th value = 7.5th value = 7th value + ( 8th value - 7th value)*0.5

= 81612500 + ( 86510000 - 81612500)*0.5 = 84061250

Median for National League Payroll = [ (16+1)/2 ]th value = 8.5th value = 8th value + ( 9th value - 8th value)*0.5

= 84227000+ ( 84423666- 84227000)*0.5 = 84325333

b.

Range for American League Payroll = 206333389 - 51654900 = 154678489

Range for National League Payroll = 146609000 - 34943000 = 111666000

Sample Standard deviation is obtained using the formula :

We get :

Sample Standard deviation for American League Payroll = 43803002.27

Sample Standard deviation for National League Payroll = 33403170.09

c.

We can see that Mean of American League Payroll is greater than Mean of National League Payroll.

Median of American League Payroll is approximately equal to Median of National League Payroll.

Sample Standard deviation for American League Payroll is greater than Sample Standard deviation for National League Payroll.

Hence, one of the findings in The Wall Street Journal is that the average AL payroll is greater than the NL average which is observed from the given data.

orchestra answered 1 year ago
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