Question

wayne gretzky, a retired professional hockey player, played 20 seasons in the national hockey league (nhl) from 1980 through 1999. the accompanying table shows the number of games in which gretzky played during a sample of 16 of his seasons in the nhl. complete parts (a) through (e) below.
click the icon to view a sample of the games played data.
a. obtain and interpret the quartiles.
determine the quartiles.
q1 =
q2 =
q3 =
(type integers or decimals. do not round.)
interpret the quartiles. choose the correct answer below.
a. the quartiles suggest that the average number of games played in a season is q2.
b. the quartiles suggest that gretzky played less than q1 games in 25% of the seasons, between q1 and q2 games in 25% of the seasons, between q2 and q3 games in 25% of the seasons, and more than more than q3 games in 25% of the seasons.
c. the quartiles suggest that gretzky played between q1 and q3 games in all of the seasons.
d. the quartiles suggest that gretzky played less than q1 games in 33% of the seasons, between q1 and q3 games in 33% of the seasons, and more than q3 games in 33% of the seasons.
b. determine and interpret the interquartile range.
the interquartile range is
(type an integer or a decimal. do not round.)
interpret the interquartile range. choose the correct answer below.
a. the data span roughly the iqr.
b. the approximate difference between each quartile is the iqr.
c. the number of games played in the middle 50% of seasons spans roughly the iqr.
d. the average of the first quartile and the third quartile is the iqr.
c. find and interpret the five - number summary.
determine the five - number summary.

Explanation:

Step1: Arrange data in ascending order

Let the data - set of the number of games played in 16 seasons be \(x_1,x_2,\cdots,x_{16}\). First, sort the data from smallest to largest.

Step2: Calculate the position of \(Q_1\)

The position of \(Q_1\) is \(L_{Q_1}=\frac{n + 1}{4}\), where \(n = 16\). So \(L_{Q_1}=\frac{16+1}{4}=4.25\). \(Q_1\) is the value such that \(Q_1=(1 - 0.25)x_4+0.25x_5\) (using linear - interpolation for non - integer positions).

Step3: Calculate the position of \(Q_2\)

The position of \(Q_2\) (the median) is \(L_{Q_2}=\frac{n + 1}{2}\). For \(n = 16\), \(L_{Q_2}=\frac{16 + 1}{2}=8.5\). So \(Q_2=(1 - 0.5)x_8+0.5x_9\).

Step4: Calculate the position of \(Q_3\)

The position of \(Q_3\) is \(L_{Q_3}=\frac{3(n + 1)}{4}\). For \(n = 16\), \(L_{Q_3}=\frac{3\times(16 + 1)}{4}=12.75\). So \(Q_3=(1 - 0.75)x_{12}+0.75x_{13}\).

Step5: Calculate the inter - quartile range (IQR)

\(IQR=Q_3 - Q_1\).

Step6: Determine the five - number summary

The five - number summary consists of the minimum value (the first value in the sorted data set), \(Q_1\), \(Q_2\), \(Q_3\), and the maximum value (the last value in the sorted data set).

Since the data values are not given, assume the sorted data set is \(x_1,x_2,\cdots,x_{16}\).
Let's assume the sorted data set for illustration purposes (but in a real - case, you need to use the actual data): Suppose \(x_1 = 40,x_2 = 45,x_3 = 50,x_4 = 55,x_5 = 60,x_6 = 65,x_7 = 70,x_8 = 75,x_9 = 80,x_{10}=85,x_{11}=90,x_{12}=95,x_{13}=100,x_{14}=105,x_{15}=110,x_{16}=115\).
\(L_{Q_1}=\frac{16 + 1}{4}=4.25\), \(Q_1=(1 - 0.25)\times55+0.25\times60 = 56.25\).
\(L_{Q_2}=\frac{16+1}{2}=8.5\), \(Q_2=(1 - 0.5)\times75+0.5\times80 = 77.5\).
\(L_{Q_3}=\frac{3\times(16 + 1)}{4}=12.75\), \(Q_3=(1 - 0.75)\times95+0.75\times100 = 98.75\).
\(IQR=Q_3 - Q_1=98.75 - 56.25 = 42.5\).
The five - number summary: Minimum \(=40\), \(Q_1 = 56.25\), \(Q_2 = 77.5\), \(Q_3 = 98.75\), Maximum \(=115\).

Interpretation of quartiles:
The quartiles divide the data set into four equal parts. The first quartile \(Q_1\) represents the 25th percentile, the second quartile \(Q_2\) (the median) represents the 50th percentile, and the third quartile \(Q_3\) represents the 75th percentile. So, the correct interpretation of quartiles is that Gretzky played less than \(Q_1\) games in 25% of the seasons, between \(Q_1\) and \(Q_2\) games in 25% of the seasons, between \(Q_2\) and \(Q_3\) games in 25% of the seasons, and more than \(Q_3\) games in 25% of the seasons. So the answer for the interpretation of quartiles is B.
Interpretation of the inter - quartile range:
The inter - quartile range (\(IQR\)) represents the range of the middle 50% of the data. So the correct interpretation of the inter - quartile range is that the number of games played in the middle 50% of seasons spans roughly the \(IQR\). So the answer for the interpretation of \(IQR\) is C.

Answer:

a. (Since no data is given, we can't give exact values for \(Q_1,Q_2,Q_3\)). Interpretation of quartiles: B.
b. (Since no data is given, we can't give an exact value for \(IQR\)). Interpretation of \(IQR\): C.
c. (Since no data is given, we can't give exact values for the five - number summary). The five - number summary consists of the minimum value, \(Q_1\), \(Q_2\), \(Q_3\), and the maximum value. The interpretation is that it gives a summary of the distribution of the data set, with the minimum and maximum showing the range of the data, and \(Q_1,Q_2,Q_3\) dividing the data into four equal - sized parts (in terms of the number of data points).