Question

1. home runs 2012 here is a \back - to - back\ stemplot that shows two data sets at once—one going to the left, one to the right. the display compares the number of home runs for major league baseball teams in the national league and the american league during the 2012 season. team home runs national league american league 3 10 6 11 1 12 977 13 116 96 14 9 98 15 65 16 35 20 17 5 18 7 4 19 58 2 20 0 21 14 22 23 24 5 key: 2 | 20 | 0 means 202 hr for a team in the nl and 200 for a team in the al a) use the data in the stemplot to construct comparative boxplots. b) write a few sentences comparing the distributions in home runs for teams in the two leagues.

Explanation:

Step1: Organize data from stemplot

For National League: 103, 116, 121, 137, 137, 139, 146, 149, 158, 159, 165, 166, 170, 172, 184, 192, 202, 210, 220, 230, 240
For American League: 101, 113, 116, 121, 131, 131, 136, 149, 150, 163, 165, 175, 187, 195, 198, 200, 214, 245

Step2: Calculate five - number summary for each league

National League:

• Minimum: 103
• First quartile ($Q_1$): Find the median of the lower half. There are 20 data points. The lower half has 10 points. The median of the lower half is the average of the 5th and 6th ordered values. The 5th value is 137 and the 6th is 139, so $Q_1=\frac{137 + 139}{2}=138$
• Median ($Q_2$): Since there are 20 data points, the median is the average of the 10th and 11th ordered values. The 10th value is 159 and the 11th is 165, so $Q_2=\frac{159+165}{2}=162$
• Third quartile ($Q_3$): Find the median of the upper half. The upper - half has 10 points. The median of the upper half is the average of the 15th and 16th ordered values. The 15th value is 192 and the 16th is 202, so $Q_3=\frac{192 + 202}{2}=197$
• Maximum: 240

American League:

• Minimum: 101
• First quartile ($Q_1$): There are 18 data points. The lower half has 9 points. The 5th value of the lower half is 131, so $Q_1 = 131$
• Median ($Q_2$): Since there are 18 data points, the median is the average of the 9th and 10th ordered values. The 9th value is 163 and the 10th is 165, so $Q_2=\frac{163+165}{2}=164$
• Third quartile ($Q_3$): The upper - half has 9 points. The 5th value of the upper half is 198, so $Q_3 = 198$
• Maximum: 245

Step3: Construct boxplots

Draw two parallel number lines. Mark the five - number summary values for each league on the respective number lines and draw the box (from $Q_1$ to $Q_3$ with a line at $Q_2$) and the whiskers (from the box to the minimum and maximum).

Step4: Compare distributions

The American League has a slightly lower minimum (101 vs 103) and a slightly higher maximum (245 vs 240) than the National League. The median of the American League (164) is slightly higher than the median of the National League (162). The inter - quartile range (IQR) for the National League is $Q_3 - Q_1=197 - 138 = 59$, and for the American League is $Q_3 - Q_1=198 - 131 = 67$, so the spread of the middle 50% of the data is larger in the American League. Overall, the distributions are somewhat similar in shape, but the American League has a slightly wider spread and a slightly higher central tendency.

Answer:

a) Boxplots are constructed using the five - number summary values calculated above for each league.
b) The American League has a slightly lower minimum, a slightly higher maximum, a slightly higher median and a larger inter - quartile range compared to the National League. The distributions are somewhat similar in shape.