Chapter 10, Problem 26RE

Solution

a.

To Plot: The boxplot of the length of songs data.

The earlier 12 years and later 12 years data boxplot is:

  

  

Given information:

Earlier 12 Years	Later 12 Years
143	136
120	124
120	179
139	131
124	180
144	137
131	156
132	202
148	191
163	197
140	230
177	190

Concept Used:

The boxplot is created in the following steps:

Step 1: Arrange the data in ascending order.

Step 2: Compute the quartiles of the data as $Q_1,Median,Q_3.$

Step 3: Consider the data on y-axis and plot the quartiles to obtain the box.

Step 4: Now compute the lower limit and upper limit to find the length of whiskers as follows,

  $\begin{array}{c}\text{Lower Limt=}{Q}_1-1.5\times IQR\\ \text{Upper Limt=}{Q}_3+1.5\times IQR\end{array}$

Step 5: Plot the limits and connect the points with the box obtained using the quartiles.

Plot:

The data in ascending order is:

Earlier 12 Years	Later 12 Years
120	124
120	131
124	136
131	137
132	156
139	179
140	180
143	190
144	191
148	197
163	202
177	230

The boxplot of earlier 12 years data is:

  $\begin{array}{c}{Q}_1=median(X_1),\text{ where }{X}_1\text{ is upper part for data}\\ =127.5\\ Q_3=median(X_2),\text{ where }{X}_2\text{ is lower part for data}\\ =146\\ IQR=148-128\\ =18.5\\ Median=139.5\end{array}$

  

The boxplot of later 12 years is:

  $\begin{array}{c}{Q}_1=median(X_1),\text{ where }{X}_1\text{ is upper part for data}\\ =136.5\\ Q_3=median(X_2),\text{ where }{X}_2\text{ is lower part for data}\\ =194\\ IQR=194-136.5\\ =57.5\\ Median=179.5\end{array}$

  

b.

To find: The data that has greater interquartile range

The data of later 12 years had greater interquartile range.

Given:

  

The quartiles from the boxtplot are

  $\begin{array}{c}{Q}_1=127.5\\ Q_3=146\\ IQR=18.5\end{array}$

  

The quartiles from the boxtplot are

  $\begin{array}{c}{Q}_1=136.5\\ Q_3=194\\ IQR=57.5\end{array}$

The IQR of the later 12 years data has $IQR=57.5$ which is greater than the earlier 12 years data.