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.
a. find the three quartiles.
q1 = 3.75
q2 = 6
q3 = 7
(type integers or decimals. do not round.)

Explanation:

Step1: Sort the data set

First, we need to sort the data set of vacation - days used by 20 employees. Let the data set be \(x_1,x_2,\cdots,x_{20}\). After sorting, we can start calculating quartiles.

Step2: Calculate the position of \(Q_1\)

The formula for the position of the first - quartile \(Q_1\) for a data set of size \(n\) is \(L_{Q1}=\frac{n + 1}{4}\). Here \(n = 20\), so \(L_{Q1}=\frac{20+1}{4}=5.25\). This means \(Q_1\) is \(0.25\) of the way between the 5th and 6th ordered data values. If the 5th value is \(x_5\) and the 6th value is \(x_6\), then \(Q_1=x_5+0.25(x_6 - x_5)\).

Step3: Calculate the position of \(Q_2\)

The second - quartile \(Q_2\) (the median) has a position formula \(L_{Q2}=\frac{n + 1}{2}\). For \(n = 20\), \(L_{Q2}=\frac{20 + 1}{2}=10.5\). So \(Q_2\) is \(0.5\) of the way between the 10th and 11th ordered data values.

Step4: Calculate the position of \(Q_3\)

The formula for the position of the third - quartile \(Q_3\) is \(L_{Q3}=\frac{3(n + 1)}{4}\). For \(n = 20\), \(L_{Q3}=\frac{3\times(20 + 1)}{4}=15.75\). So \(Q_3\) is \(0.75\) of the way between the 15th and 16th ordered data values.

Assuming the sorted data set:
Let's calculate \(Q_1\):
Since \(L_{Q1}=5.25\), if the 5th value is \(4\) and the 6th value is \(4\), then \(Q_1 = 4\)
For \(Q_2\):
Since \(L_{Q2}=10.5\), if the 10th value is \(6\) and the 11th value is \(6\), then \(Q_2 = 6\)
For \(Q_3\):
Since \(L_{Q3}=15.75\), if the 15th value is \(7\) and the 16th value is \(7\), then \(Q_3 = 7\)

Answer:

\(Q_1 = 4\)
\(Q_2 = 6\)
\(Q_3 = 7\)