Chapter 3.3, Problem 31E

Hazardous waste: Following is a list of the number of hazardous waste sites in each of the 50 states of the United States in a recent year. The list has been sorted into numerical order.

1. Find the first and third quartiles of these data.
2. Find the median of these data.
3. Find the upper and lower outlier boundaries.
4. Are there any outliers? If so, list them.
5. Construct a boxplot for these data.
6. Describe the shape of this distribution.
7. What is the 30th percentile?
8. What is the 85th percentile?
9. The state of Georgia has 16 hazardous waste sites. What percentile is this?

To determine

To find the first and third quartiles of the given data.

Answer to Problem 31E

First quartile = 10.5 and third quartile = 32

Explanation of Solution

Formula:

First quartile $Q_1=\frac{1}{4}{(n+1)}^{th}value$

Third quartile $Q_3=\frac{3}{4}{(n+1)}^{th}value$

Calculation:

Given data sorted in ascending order:

$x$
0	9	14	20	40
1	9	14	21	48
2	11	14	25	49
2	12	14	26	49
3	12	15	29	52
5	12	15	30	67
6	12	15	32	86
9	12	16	32	97
9	13	19	32	97
9	13	19	38	116

Here, n = 50

First need to find First quartile and third quartile

First Quartile:

  $\begin{array}{l}$ Q_1=\frac{1}{4}{(50+1)}^{th}value$$ Q_1=\frac{1}{4}{(51)}^{th}value$$ Q_1={(12.75)}^{th}value$Q_1=9+0.75\times (11-9)\\ Q_1=10.5\end{array}$

First quartile is 10.5

Third quartile:

  $Q_3=\frac{3}{4}{(n+1)}^{th}value$

  $Q_3=\frac{3}{4}{(50+1)}^{th}value$

  $Q_3=\frac{3}{4}{(51)}^{th}value$

   $Q_3={(38.25)}^{th}value$

  $Q_3=32+\frac{1}{4}\times (38-38)$

  $Q_3=32$

Third quartile is 32

To determine

To find median of the data.

Answer to Problem 31E

Median is 15

Explanation of Solution

Formula:

  $median={\left(\frac{n+1}{2}\right)}^{th}value$

Calculation:

  $\begin{array}{l}median={\left(\frac{50+1}{2}\right)}^{th}value\\ median={\left(25.5\right)}^{th}value\\ Median=\frac{15+15}{2}\\ Median=15\end{array}$

To determine

To find upper and lower outlier boundaries.

Answer to Problem 31E

Lower outlier boundary is -21.5

Upper outlier boundary is 64.25

Explanation of Solution

  $\begin{array}{l}{Q}_1=10.5\\ Q_3=32\end{array}$

Formula:

IQR:

  $IQR=Q_3-Q_1$

Calculation:

  $\begin{array}{l}IQR=Q_3-Q_1\hfill \\ IQR=\text{ 32-10}\text{.5}\hfill \\ IQR=\text{ 21}\text{.5}\hfill \end{array}$

Therefore,

  $\begin{array}{l}Loweroutlierlimit=Q_1-1.5\times IQR\hfill \\ Loweroutlierlimit=\text{ 10}\text{.5-1}\text{.5}\times \text{21}\text{.5}\hfill \\ Loweroutlierlimit=\text{ -21}\text{.5}\hfill \\ And\hfill \\ Upperoutlierlimit=Q3+1.5\times IQR\hfill \\ Upperoutlierlimit=\text{ 32+1}\text{.5}\times \text{21}\text{.5}\hfill \\ Upperoutlierlimit=\text{ 64}\text{.25}\hfill \end{array}$

To determine

To find the list of outliers.

Answer to Problem 31E

The outlier points are 67,86,97,97,116

Explanation of Solution

Outliers are those values which are less than $Q_1-1.5\times IQR$ and greater than $Q_3+1.5\times IQR$ .

Here,

Lower outlier boundary is -21.5

Upper outlier boundary is 64.5

There is no any value less than -21.5. But there are five values larger than 64.5 which are:

67,86,97,97,116

Hence outlier points are 67,86,97,97,116

To determine

To construct a boxplot from given data

Explanation of Solution

Boxplot from given data:

  

To determine

To find shape of the distribution

Answer to Problem 31E

The shape of the distribution is positively-skewed.

Explanation of Solution

Here, median is close to first quarter. Hence the shape of the distribution is positively-skewed.

To determine

To find 30^th percentile

Answer to Problem 31E

30^th percentile is 12

Explanation of Solution

  $\begin{array}{l}PercentilePosition=\frac{(n+1)P}{100}\\ PercentilePosition=\frac{(50+1)\times 30}{100}\\ PercentilePosition=15.3\end{array}$

Therefore,

  $P_{30}=12+0.3\times (12-12)=12$

To determine

To find 85^th percentile

Answer to Problem 31E

85^th percentile is 49

Explanation of Solution

  $\begin{array}{l}PercentilePosition=\frac{(n+1)P}{100}\\ PercentilePosition=\frac{(48+1)\times 85}{100}\\ PercentilePosition=43.35\end{array}$

Therefore,

  $P_{85}=49+0.35\times (49-49)=49$

To determine

To find percentile rank of Georgia.

Answer to Problem 31E

Percentile rank of Georgia at 16 is 54%

Explanation of Solution

Formula:

  $Percentilerankofx=\frac{no.ofvaluesbelowx}{n}\times 100$

Calculation:

  $\begin{array}{l}Percentilerankof16=\frac{27}{50}\times 100\\ Percentilerankof16=54\%\end{array}$