Question

1. during the early part of a major league baseball season, many fans and players noticed that the number of home runs being hit seemed unusually large. here are the data on the number of home runs hit by american league and national league teams in the early part of that season:

american league 35 40 43 49 51 54 57 58 58 64 68 68 75 77
national league 29 31 42 46 47 48 48 53 55 55 55 63 63 67
make parallel boxplots to compare the distributions of home runs for the two leagues. describe what you see.

Explanation:

Step1: Find the five - number summary for American League

• Minimum: \(35\)
• First Quartile (\(Q_1\)): The position of \(Q_1=\frac{n + 1}{4}\) where \(n = 14\). \(\frac{14+1}{4}=3.75\). \(Q_1=43+(49 - 43)\times0.75=47.5\)
• Median (\(Q_2\)): The position of the median is \(\frac{n+1}{2}\). \(\frac{14 + 1}{2}=7.5\). \(Q_2=57+(58 - 57)\times0.5 = 57.5\)
• Third Quartile (\(Q_3\)): The position of \(Q_3=\frac{3(n + 1)}{4}\). \(\frac{3\times(14 + 1)}{4}=11.25\). \(Q_3=68+(68 - 68)\times0.25=68\)
• Maximum: \(77\)

Step2: Find the five - number summary for National League

• Minimum: \(29\)
• First Quartile (\(Q_1\)): The position of \(Q_1=\frac{n + 1}{4}\) where \(n = 14\). \(\frac{14+1}{4}=3.75\). \(Q_1=42+(46 - 42)\times0.75=45\)
• Median (\(Q_2\)): The position of the median is \(\frac{n+1}{2}\). \(\frac{14 + 1}{2}=7.5\). \(Q_2=48+(53 - 48)\times0.5=50.5\)
• Third Quartile (\(Q_3\)): The position of \(Q_3=\frac{3(n + 1)}{4}\). \(\frac{3\times(14 + 1)}{4}=11.25\). \(Q_3=55+(63 - 55)\times0.25=57\)
• Maximum: \(67\)

Step3: Analyze the boxplots

• Center: The median of the American League (\(57.5\)) is higher than that of the National League (\(50.5\)). So, on average, American League teams hit more home runs.
• Spread: The inter - quartile range (\(IQR\)) for American League is \(Q_3-Q_1=68 - 47.5 = 20.5\). For National League, \(IQR = 57-45=12\). American League has a greater spread.
• Shape: Both distributions seem to be slightly skewed. American League may have a right - skew (since the distance from median to \(Q_3\) (\(68 - 57.5=10.5\)) is larger than the distance from \(Q_1\) to median (\(57.5 - 47.5 = 10\))). National League may also have a right - skew (distance from median to \(Q_3\) (\(57 - 50.5 = 6.5\)) and from \(Q_1\) to median (\(50.5 - 45=5.5\)))
• Outliers: There are no obvious outliers in the given data (using the \(1.5\times IQR\) rule: For American League, \(Q_1-1.5\times IQR=47.5-1.5\times20.5=16.75\) and \(Q_3 + 1.5\times IQR=68+1.5\times20.5=98.75\); for National League, \(Q_1-1.5\times IQR=45-1.5\times12 = 27\) and \(Q_3+1.5\times IQR=57+1.5\times12=75\))

Answer:

The American League has a higher median (more home runs on average), a greater spread (more variability in the number of home runs), and both distributions seem to be slightly right - skewed. There are no outliers in the data within the \(1.5\times IQR\) rule for either league.