Question

The Great White Shark. In an article titled "Great White. Deep Trouble" (National Geographic, Vol. $197\left(4\right)$, pp. $2-29$). Peter Benchley-the author of JAWS-discussed various aspects of the Great White Shark Carcharodon carcharias). Data on the number of pups borne in a lifetime by each of $80$Great White Shark females are provided on the WeissStats site.

a. obtain and interpret the quartiles.

b. determine and interpret the interquartile range.

c. find and interpret the five-number summary

d. identify potential outliers, if any.

e. obtain and interpret boxplot.

Short answer

(a) The quartiles are $6$, $7$, $8$

(b) The interquartile range is, $2$

(c) Five-number summary is, $3$, $6$, $7$, $8$, $12$

(d) Potential outliers is,$12$

(e) The required boxplot is given below.

Step-by-step solution

Part (a) Step 1: Given information

We are given that,

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

$$\begin{gathered} 3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6, \\ 7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8, \\ 8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,11,11,12 \end{gathered}$$

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 even, the median is the average of two middle values:-

$Q_2=\frac{7+7}{2}=\frac{14}{2}=7$

So, roughly $50\%$the number of pups born below or equal to $7$pups.

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

localid="1650704820538" $Q_1=6$

So, roughly localid="1650704845630" $25\%$the number of pups born below or equal to localid="1650704865831" $6$pups

Now, the third quartile is the median of values below localid="1650704888683" $Q_2$

localid="1650704910684" $Q_3=8$

So, roughly localid="1650704932507" $75\%$the number of pups born below or equal to localid="1650704948281" $8$pups

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 $Q_3$

$IQR=Q_3-Q_1=8-6=2$

The middle $50\%$of the no. of pups born vary about $2$pups.

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$=3$, first quartile$Q_1=6$, second quartile $Q_2=7$, third quartile $Q_3=8$and maximum$=12$.

Part (d) Step 1: Given information

We need to find out the potential outliers.

Part (d) Step 2: Simplify

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

Therefore,

$$\begin{gathered} Q_3+1.5IQR=8+\left(1.5\times 2\right)=11 \\ {Q}_1-1.5IQR=6-\left(1.5\times 2\right)=3 \end{gathered}$$

Hence, there is one outlier i.e. $12$because it does not lie between $3$and$11$

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.