Chapter 4, Problem 13E

a.

To determine

Find the first quartile.

Find the third quartile.

Answer to Problem 13E

The first quartile is 33.25.

The third quartile is 50.25.

Explanation of Solution

Calculation:

Location of percentile:

The formula for percentile is,

$L_P=\left(n+1\right)\frac{P}{100}$

In the formula, P denotes the location of the percentile and n denotes the total number of observations.

First quartile:

The first quartile represents the 25% of observation lies below first quartile. That is $P=25$. There are 30 observations in the dataset arranged in ascending order.

Substitute, $P=25,n=30$ in the percentile formula.

$\begin{array}{c}{L}_{25}=\left(30+1\right)\frac{25}{100}\\ =\frac{31\times 25}{100}\\ =\frac{775}{100}\\ =7.75\end{array}$

The position of first quartile is 7.75^th value in the dataset.

$\begin{array}{c}\text{First quartile}=7^{\text{th}}\text{value}+\left(8^{\text{th}}\text{value}-7^{\text{th}}\text{value}\right)0.75\\ =31+\left(34-31\right)0.75\\ =31+2.25\\ =33.25\end{array}$

Hence, the first quartile is 33.25.

Third quartile:

The third quartile represents the 75% of observation lies above third quartile. That is $P=75$.

Substitute, $P=75,n=30$ in the percentile formula.

$\begin{array}{c}{L}_{75}=\left(30+1\right)\frac{75}{100}\\ =\frac{31\times 75}{100}\\ =\frac{2,325}{100}\\ =23.25\end{array}$

The position of third quartile is 23.25^th value in the dataset.

$\begin{array}{c}\text{Third quartile}={23}^{\text{rd}}\text{value}+\left({24}^{\text{th}}\text{value}-{23}^{\text{rd}}\text{value}\right)0.25\\ =50+\left(51-50\right)0.25\\ =50+0.25\\ =50.25\end{array}$

Hence, the third quartile is 50.25.

b.

To determine

Find the second decile.

Find the eighth decile.

Answer to Problem 13E

The second decile is 27.8.

The eighth decile is 52.6.

Explanation of Solution

Calculation:

Location of decile:

The deciles divide the total observation into 10 equal parts. The formula for decile is,

$D_i=\left(n+1\right)\frac{i}{10}$

In the formula, i denotes the location of the decile and n denotes the total number of observations.

Second decile:

Substitute, $i=2,n=30$ in the percentile formula.

$\begin{array}{c}{D}_2=\left(30+1\right)\frac{2}{10}\\ =\frac{31\times 2}{10}\\ =\frac{62}{10}\\ =6.2\end{array}$

The position of second decile is 6.2^th value in the dataset.

$\begin{array}{c}{D}_2=6^{\text{th}}\text{value}+\left(7^{\text{th}}\text{value}-6^{\text{th}}\text{value}\right)0.20\\ =27+\left(31-27\right)0.20\\ =27+0.8\\ =27.8\end{array}$

Hence, the second decile is 27.8.

Eighth decile:

Substitute, $i=8,n=30$ in the percentile formula.

$\begin{array}{c}{D}_8=\left(30+1\right)\frac{8}{10}\\ =\frac{31\times 8}{10}\\ =\frac{248}{10}\\ =24.8\end{array}$

The position of eighth decile is 24.8^th value in the dataset.

$\begin{array}{c}{D}_8={24}^{\text{th}}\text{value}+\left({25}^{\text{th}}\text{value}-{24}^{\text{th}}\text{value}\right)0.80\\ =51+\left(53-51\right)0.80\\ =51+1.6\\ =52.6\end{array}$

Hence, the eighth decile is 52.6.

c.

To determine

Find the 67^th percentile.

Answer to Problem 13E

The 67^th percentile is 47.

Explanation of Solution

Calculation:

Substitute, $P=67,n=30$ in the percentile formula.

$\begin{array}{c}{L}_{67}=\left(30+1\right)\frac{67}{100}\\ =\frac{31\times 67}{100}\\ =\frac{2,077}{100}\\ =20.77\end{array}$

The position of 67^th percentile is 20.77^th value in the dataset.

$\begin{array}{c}{67}^{th}\text{percentile}={20}^{\text{th}}\text{value}+\left({21}^{\text{th}}\text{value}-{20}^{\text{th}}\text{value}\right)0.77\\ =47+\left(47-47\right)0.77\\ =47\end{array}$

Hence, the 67^th percentile is 47.