Q. 3.185 - Elementary Statistics

Question

High School Completion. As reported by the US Census Bureau in Educational Attainment in the United States, the percentage of adults in each state and the District of Columbia who have completed high school is provided on the Weiss Stats site.

Step-by-Step Solution

Answer

(a) The quartiles are $84, 87, 90$

(b) The interquartile range is, $6$

(c) Five-number summary is, $80, 84, 87, 90, 92$

(d) There are no Potential outliers.

(e) The required boxplot is given below.

Part (a) Step 1: Given information

We are given that,

Sorted data is given in the Weiss stats which are as follows,

$$\begin{gathered} 80, 80, 81, 82, 82, 82, 82, 83, 83, 83, 84, 84, 84, 84, 84,85, 85, 85, 86, \\ 86,87, 87, 87, 87, 87, 87, 88, 88, 88, 88, 88, 89, 89, 89, 89, 90, 90, 90, \\ 90, 90, 90, 90, 90, 90, 91, 91, 91, 91, 91, 92, 92 \end{gathered}$$

Part (a) Step 2: Simplify

As we know that median is the middle value of a sorted data set. Since the number of data values is odd, the median is:-

    $Q_2=87$

So, roughly $50\%$ the states have percentage of adults in required condition is below $87\%$

Now, the first quartile is the median of values below $Q_2$ i.e.

       $Q_1=84$

So, roughly $25\%$ the states have percentage of adults in required condition is below $84\%$

Now,  the third quartile is the median of values below  $Q_2$

        $Q_3=90$

So, roughly $75\%$ the states have percentage of adults in required condition is below $90\%$

Part (b) Step 1: Given information

We need to find out the interquartile range 

Part (b) Step 2: Simplify

The interquartile range is the difference between $Q_1$ and $Q_3$

     $IQR=Q_3-Q_1=90-84=6$

The middle $50\%$ of the states have percentage of adults in required condition varies by $6\%$

Part (c) Step 1: Given information

We need to find out the five-number summary 

Part (c) Step 2: Explanation

The five-number summary is minimum$=80$ , first quartile$=84$, second quartile$=87$ , third quartile$=90$  and maximum$=92$ 

Part (d) Step 1: Given information

We need to find out the potential outliers. 

Part (d) Step 2: Simplify

An outlier is more than $1.5 IQR$ or greater than $Q_3$ or less than $Q_1$

Therefore,

        $$\begin{gathered} Q_3+\left(1.5 IQR\right)=90+\left(1.5\times 6\right)=99 \\ {Q}_3-\left(1.5 IQR\right)=84-\left(1.5\times 6\right)=75 \end{gathered}$$

Hence, there is no outlier because all data values are in range of outliers

Part (e) Step 1: Given information

We need to find the boxplot which is given below 

Part(e) Step 2: Simplify

The whiskers of the boxplot are at a low and high value. The box starts at $Q_1$and ends at $Q_3$ and has a straight line at the median.