Question

the data show the number of vacation days used by a sample of 20 employees in a recent year. use technology to answer parts (a) and (b).
a. find the data sets first, second, and third quartiles.
b. draw a box - and - whisker plot that represents the data set.
0 6 9 5 5 4 2 3 1 2 1 6 5 3 2 9 6 3 7 10

a. find the three quartiles.
$q_1 = \square$
$q_2 = \square$
$q_3 = \square$
(type integers or decimals. do not round.)
b. choose the correct answer plot below. note that different technologies will produce slightly different results.
○ a.
○ b.
○ c.

Explanation:

Step1: Sort the data

First, sort the data set: \(0,1,1,2,2,2,3,3,3,4,5,5,5,6,6,6,7,9,9,10\)

Step2: Find \(Q_2\) (the median)

Since \(n = 20\) (even), \(Q_2=\frac{x_{10}+x_{11}}{2}\). Here \(x_{10} = 4\) and \(x_{11}=5\), so \(Q_2=\frac{4 + 5}{2}=4.5\)

Step3: Find \(Q_1\)

The lower - half of the data is \(0,1,1,2,2,2,3,3,3,4\). Since \(n_1=10\) (even), \(Q_1=\frac{x_{5}+x_{6}}{2}\). Here \(x_{5}=2\) and \(x_{6}=2\), so \(Q_1 = 2\)

Step4: Find \(Q_3\)

The upper - half of the data is \(5,5,5,6,6,6,7,9,9,10\). Since \(n_2 = 10\) (even), \(Q_3=\frac{x_{15}+x_{16}}{2}\). Here \(x_{15}=6\) and \(x_{16}=6\), so \(Q_3=6\)

Answer:

\(Q_1 = 2\), \(Q_2=4.5\), \(Q_3 = 6\)