Q. 3.182 - Elementary Statistics

Question

Women Students. The US Department of Education sponsors a report on educational institutions, including colleges and universities, titled Digest of Education Statistics. Among many of the statistics provided are the numbers of men and women enrolled in $2 year$ and $4 year$ degree-granting institutions. During one year, the percentage of full-time enrolled students that were women, for each of the $50$states and the District of Columbia, is as presented on the WeissStats site.

a. obtain and interpret the quartiles. 

b. determine and interpret the interquartile range.

e. find and interpret the five-number summary

d. identify potential outliers, if any.

e. obtain and interpret boxplot.

Step-by-Step Solution

Answer

a) The quartiles are $53$, $55$, $56$

(b) The interquartile range is, $3$

(c) Five-number summary is, $48, 53, 55, 56, 58$

(d) Potential outliers are, $48$

(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,

$$\begin{gathered} 48, 48, 50, 52, 52, 52, 52, 52, 52, 53, 53, 53, 53, 53, 53, 53, 53, 54, 54, \\ 54, 54, 54, 54, 54, 54, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 56, 56, 56, \\ 56, 56, 56, 56, 56, 57, 57, 57, 57, 57, 58, 58, 58 \end{gathered}$$

Part (a) Step 2: Simplify

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

     $Q_2=55$

So, roughly $50\%$ of the states have a percentage of full-time enrolled students that were women below $55\%$

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

     $Q_1=53$

So, roughly $25\%$ of the states have a percentage of full-time enrolled students that were women below $53\%$

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

      $Q_3=56$

So, roughly $75\%$ of the states have a percentage of full-time enrolled students that were women below $56\%$

Part (b) Step 1: Given information

We need to find out the interquartile range 

Part (b) Step 2: Simplify

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

     $IQR=Q_3-Q_1=56-53=3$

The middle $50\%$ of the states have a percentage varying by $3\%$

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$=48$, first quartile$=53$, second quartile$=55$, third quartile$=56$ and maximum$=58$. 

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,

        $$\begin{gathered} Q_3+1.5 IQR=56+\left(1.5\times 3\right)=60.5 \\ {Q}_1-1.5 IQR=53-\left(1.5\times 3\right)=48.5 \end{gathered}$$

Hence, there is one outlier i.e. $48$ because it does not lie between $48.5$ and $60.5$

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.