Chapter 13.2, Problem 40E

To determine

To Find: the lower and upper quartiles and sketch a box and whisper plot for the provided data.

Answer to Problem 40E

The required lower and upper quartiles are 21.8 and 34.9 respectively.

The box plot for the provided data set is:

  

Explanation of Solution

Given:

The provided data set is:

20.1
43.4
34.9
23.9
33.5
24.1
22.5
42.4
25.7
17.4
23.8
33.3
17.3
36.4
21.8

Formula used:

The formulas to compute the lower $\left(Q_1\right)$ and upper $\left(Q_3\right)$ quartiles are:

  $\begin{array}{c}{Q}_1={\left(\frac{n+1}{4}\right)}^{\text{th}}\text{term}\\ Q_3=3{\left(\frac{n+1}{4}\right)}^{\text{th}}\text{term}\end{array}$

Where, n is the number of observations.

Calculation:

Here n is 15

The lower and upper quartiles could be calculated as:

  $\begin{array}{c}{Q}_1={\left(\frac{n+1}{4}\right)}^{\text{th}}\text{term}\\ ={\left(\frac{15+1}{4}\right)}^{\text{th}}\text{term}\\ =4^{\text{th}}\text{term}\\ =21.8\end{array}$

Similarly,

  $\begin{array}{c}{Q}_3=3{\left(\frac{n+1}{4}\right)}^{\text{th}}\text{term}\\ =3{\left(\frac{15+1}{4}\right)}^{\text{th}}\text{term}\\ ={12}^{\text{th}}\text{term}\\ =34.9\end{array}$

Thus, the required lower and upper quartiles are 21.8 and 34.9 respectively.

Graph:

The box plot for the provided data set can be constructed as:

  

Interpretation:

The above constructed box plot shows that the median is closer to the bottom of the plot which makes the distribution skewed to the right. It implies that the data has not been taken from the normally distributed population.