Question

mickey mantle played for the new york yankees for 18 years. here is the number of home runs by mantle per year: 13, 23, 21, 27, 37, 52, 34, 42, 31, 40, 54, 30, 15, 35, 19, 23, 22, 18. compute the five - number summary of the data and determine which boxplot would best represent the data. a. boxplot with number line 0 - 55, b. boxplot with number line 0 - 55, c. boxplot with number line 0 - 55, d. boxplot with number line 0 - 55

Explanation:

Step1: Order the data

First, we order the data from smallest to largest: \(13, 15, 18, 19, 21, 22, 23, 23, 27, 30, 31, 34, 35, 37, 40, 42, 52, 54\) (Wait, wait, original data: 13,23,21,27,37,52,34,42,31,40,54,30,15,35,19,23,22,18. Let's count the number of data points: 18 values. Let's re - order correctly:

13, 15, 18, 19, 21, 22, 23, 23, 27, 30, 31, 34, 35, 37, 40, 42, 52, 54. Wait, no, 18 data points: let's list all:

13, 15, 18, 19, 21, 22, 23, 23, 27, 30, 31, 34, 35, 37, 40, 42, 52, 54. Wait, 18 numbers. Now, the five - number summary consists of minimum, \(Q_1\) (first quartile), median (\(Q_2\)), \(Q_3\) (third quartile), and maximum.

Step2: Find the minimum and maximum

Minimum (\(min\)): The smallest value in the data set. From the ordered data, \(min = 13\)

Maximum (\(max\)): The largest value in the data set. From the ordered data, \(max = 54\)

Step3: Find the median (\(Q_2\))

Since there are \(n = 18\) (even number of data points), the median is the average of the \(\frac{n}{2}\)-th and \((\frac{n}{2}+ 1)\)-th values. \(\frac{n}{2}=\frac{18}{2}=9\) and \(\frac{n}{2}+1 = 10\). The 9th value is 27 and the 10th value is 30. So median \(Q_2=\frac{27 + 30}{2}=\frac{57}{2}=28.5\)

Step4: Find \(Q_1\) (first quartile)

\(Q_1\) is the median of the lower half of the data. The lower half of the data consists of the first 9 values: \(13, 15, 18, 19, 21, 22, 23, 23, 27\). Since \(n_{lower}=9\) (odd), the median of the lower half is the \(\frac{9 + 1}{2}=5\)-th value. The 5th value in the lower half is 21. So \(Q_1 = 21\)

Step5: Find \(Q_3\) (third quartile)

\(Q_3\) is the median of the upper half of the data. The upper half of the data consists of the last 9 values: \(30, 31, 34, 35, 37, 40, 42, 52, 54\). Since \(n_{upper}=9\) (odd), the median of the upper half is the \(\frac{9+1}{2}=5\)-th value. The 5th value in the upper half is 37. So \(Q_3 = 37\)

So the five - number summary is: Minimum = 13, \(Q_1 = 21\), Median = 28.5, \(Q_3 = 37\), Maximum = 54.

Now, let's analyze the boxplots:

- The box in a boxplot spans from \(Q_1\) to \(Q_3\), the line inside the box is the median, and the whiskers extend to the minimum and maximum.

Looking at the options:

Option a: Let's check the positions. The minimum should be 13, \(Q_1 = 21\), median ~28.5, \(Q_3 = 37\), maximum 54.

Option b: Incorrect quartile positions.

Option c: Incorrect quartile positions.

Option d: Incorrect quartile positions.

Wait, maybe I made a mistake in ordering. Let's re - order the data again carefully:

Original data: 13,23,21,27,37,52,34,42,31,40,54,30,15,35,19,23,22,18.

Let's list them in order:

13, 15, 18, 19, 21, 22, 23, 23, 27, 30, 31, 34, 35, 37, 40, 42, 52, 54. Yes, that's correct (18 numbers).

Wait, maybe the median calculation: for \(n = 18\), the median is the average of the 9th and 10th terms. 9th term: 27, 10th term: 30. So median is \((27 + 30)/2=28.5\). Correct.

\(Q_1\): lower half is first 9 terms: 13,15,18,19,21,22,23,23,27. The median of these 9 terms is the 5th term, which is 21. Correct.

\(Q_3\): upper half is last 9 terms: 30,31,34,35,37,40,42,52,54. The median of these 9 terms is the 5th term, which is 37. Correct.

Now, let's look at the boxplots:

In a boxplot, the box is between \(Q_1\) and \(Q_3\), median inside the box, whiskers to min and max.

Looking at the options:

Option a: The left whisker should go to 13, \(Q_1\) around 21, median around 28.5, \(Q_3\) around 37, right whisker to 54. Let's check the positions on the number line (0 - 55). 13 is on the left, 21, 28.5, 37, 54. So option a seems to match.

Wait, maybe I made a mistake in \(Q_1\) or \(Q_3\). Let's use the formula for quartiles. Another way to calculate quartiles:

For \(n = 18\):

- Position of \(Q_1=\frac{n + 1}{4}=\frac{18+1}{4}=4.75\). So \(Q_1\) is the 4th term + 0.75*(5th term - 4th term). 4th term is 19, 5th term is 21. So \(Q_1=19+0.75*(21 - 19)=19 + 1.5 = 20.5\). Wait, maybe my initial method was wrong.

Wait, different methods of calculating quartiles can give slightly different results. Let's use the method where we split the data into lower half (first 9) and upper half (last 9) when \(n\) is even? No, when \(n\) is even, \(n = 18\), the lower half is the first 9 observations and the upper half is the last 9 observations.

Wait, the data set has 18 values. Let's index them from 1 to 18:

1:13, 2:15, 3:18, 4:19, 5:21, 6:22, 7:23, 8:23, 9:27, 10:30, 11:31, 12:34, 13:35, 14:37, 15:40, 16:42, 17:52, 18:54.

Median (Q2) is the average of the 9th and 10th values: (27 + 30)/2 = 28.5. Correct.

Q1: the median of the first 9 values (indices 1 - 9). The median of 9 values is the 5th value (index 5): 21.

Q3: the median of the last 9 values (indices 10 - 18). The median of 9 values is the 5th value in this sub - set. The sub - set is [30,31,34,35,37,40,42,52,54], the 5th value is 37 (index 14 in the original data).

Now, let's check the boxplots:

Option a: The left whisker is at 13, \(Q_1\) around 21, median around 28.5, \(Q_3\) around 37, right whisker at 54. This matches our five - number summary.

Answer:

The five - number summary is Minimum = 13, \(Q_1 = 21\), Median = 28.5, \(Q_3 = 37\), Maximum = 54. The boxplot that best represents the data is option a.