Chapter 13.2, Problem 42E

a.

To determine

To construct: a box and whisker plot for each data set.

Answer to Problem 42E

The box and whisker plot for original design

  

The box and whisker plot for new design

  

Explanation of Solution

Given:

Original Design

15.1, 78.3, 56.3, 68.9, 30.6, 27.2, 12.5, 42.7, 72.7, 20.2, 53.0, 13.5, 11.0, 18.4, 85.2, 10.8, 38.3, 85.1, 10.0, 12.6

New Design

55.8, 71.5, 25.6, 19.0, 23.1, 37.2, 60.0, 35.33, 18.9, 80.5, 46.7, 31.1, 67.9, 23.5, 99.5, 54.0, 23.2, 45.5, 24.8, 87.8

Concept Used:

The formulas to compute the lower $\left(Q_1\right)$ and upper $\left(Q_3\right)$ quartiles are:

  $\begin{array}{c}{Q}_1={\left(\frac{n+1}{4}\right)}^{\text{th}}\text{term}\\ Q_3=3{\left(\frac{n+1}{4}\right)}^{\text{th}}\text{term}\end{array}$

Where, n is the number of observations.

Median will be 3

  $\frac{{\left(\frac{n}{2}\right)}^{\text{th}}\text{term}\text{+}{\left(\frac{n}{2}+1\right)}^{\text{th}}\text{term}}{2}$

Lower quartile will be the median of lower half.

Upper quartile will be the median of upper half.

Calculation:

Ordered data of Original Design

10.0, 10.8, 11.0, 12.5, 12.6, 13.5, 15.1, 18.4, 20.2, 27.2, 30.6, 38.3, 42.7, 53.0, 56.3, 68.9, 72.7, 78.3, 85.1, 85.2

Here n is 20, minimum value is 10.0 and maximum value is 85.2.

Median will be $\frac{27.2+30.6}{2}=28.9$

Lower quartile will be $\frac{12.6+13.5}{2}=13.05$

Upper quartile will be $\frac{56.3+68.9}{2}=62.6$

Median will be $\frac{{\left(\frac{n}{2}\right)}^{\text{th}}\text{term}\text{+}{\left(\frac{n}{2}+1\right)}^{\text{th}}\text{term}}{2}$

The box plot for the provided data set can be constructed as:

  

Ordered data of new Design

18.9, 19.0, 23.1 23.2, 23.5, 24.8, 25.6, 31.1, 35.3, 37.2, 45.5, 46.7, 54.0, 55.8, 60.0, 67.9, 71.5, 80.5, 87.8, 99.5

Here n is 20, minimum value is 18.9 and maximum value is 99.5.

Median will be $\frac{37.2+45.5}{2}=41.35$

Lower quartile will be $\frac{23.5+24.8}{2}=24.15$

Upper quartile will be $\frac{60.0+67.9}{2}=63.95$

The box plot for the provided data set can be constructed as:

  

b.

To determine

To Explain: the difference between the given box and whisker plots constructed.

Answer to Problem 42E

Explanation of Solution

Given:

Original Design

15.1, 78.3, 56.3, 68.9, 30.6, 27.2, 12.5, 42.7, 72.7, 20.2, 53.0, 13.5, 11.0, 18.4, 85.2, 10.8, 38.3, 85.1, 10.0, 12.6

New Design

55.8, 71.5, 25.6, 19.0, 23.1, 37.2, 60.0, 35.33, 18.9, 80.5, 46.7, 31.1, 67.9, 23.5, 99.5, 54.0, 23.2, 45.5, 24.8, 87.8

The box and whisker plot for original design

  

The box and whisker plot for new design

  

Interpretation:

The above constructed box plot shows that the box and whisker plot of original design is more spread out with the median 28.9 then the box and whisker plot of new design.

It can also be concluded that the life of original design is lower than the lifetime of new design.