Chapter 10.5, Problem 17PPS

a.

To determine

To Graph a box-and-whisker plot.

Explanation of Solution

Given information: Given data is,

L’s times
4.5	2.8
1.8	1.8
5.1	2.2
2.0	3.9
2.6	2.3
4.8	3.3
2.4	2.4
2.2	3.2

and

C’s times
2.3	5.7
5.8	3.8
4.8	4.2
3.3	5.0
5.2	4.3
4.6	5.5
3.6	4.9
2.4	5.2

Graph:

  

  

Interpretation:

L’s Times box-and-whisker graph is negatively skewed.

C’s Times box-and-whisker graph is positively skewed.

b.

To determine

To describe the center and spread the data using five-number summary.

Answer to Problem 17PPS

The L’s Times five number summary is,

1.8, 2.2, 2.5, 3.9, 5.1

The C’s Times five number summary is,

2.3, 3.6, 4.7, 5.2, 5.8

Explanation of Solution

Given information: Given data is,

L’s times
4.5	2.8
1.8	1.8
5.1	2.2
2.0	3.9
2.6	2.3
4.8	3.3
2.4	2.4
2.2	3.2

and

C’s times
2.3	5.7
5.8	3.8
4.8	4.2
3.3	5.0
5.2	4.3
4.6	5.5
3.6	4.9
2.4	5.2

Formula used: Median is,

  $M=\frac{{(\frac{n}{2})}^{th}+{(\frac{n}{2}+1)}^{th}}{2}term$

Lower quartile is,

  $Q1=(n+1)\frac{1}{4}$

Upper quartile is,

  $Q3=(n+1)\frac{3}{4}$

Calculation:

Re-arranging the data in an ascending order,

L’s Times

1.8,1.8,2.0,2.2,2.2,2.3,2.4,2.4,2.6,2.8,3.2,3.3,3.9,4.5,4.8,5.1

C’s Times

2.3,2.4,3.3,3.6,3.8,4.2,4.3,4.6,4.8,4.9,5.0,5.2,5.2,5.5,5.7,5.8

For L’s Times

For median,

Substituting the values,

  $M=\frac{{(\frac{16}{2})}^{th}+{(\frac{16}{2}+1)}^{th}}{2}$

On solving,

  $M=\frac{8^{th}+9^{th}}{2}term$

Substituting the values,

  $M=\frac{2.4+2.6}{2}$

On solving,

  $M=2.5$

Hence, Median is 2.5

For lower quartile,

Substituting the values,

  $Q1=(16+1)\frac{1}{4}$

On solving,

  $Q1=4.25$

Rounding off,

  $\approx 4$

Substituting the value,

  $Q1=2.2$

Hence, Lower quartile is 2.2

For higher quartile,

Substituting the values,

  $Q3=(16+1)\frac{3}{4}$

On solving,

  $Q3=12.75$

Rounding off,

  $\approx 13$

Substituting the values,

  $Q3=3.9$

Hence, higher quartile is 3.9

Minimum is the smallest value of dataset.

Hence, minimum is 1.8.

Maximum is the largest value of the dataset.

Hence, maximum is 5.1.

The five number summary is,

1.8, 2.2, 2.5, 3.9, 5.1

For L’s Times

For median,

Substituting the values,

  $M=\frac{{(\frac{16}{2})}^{th}+{(\frac{16}{2}+1)}^{th}}{2}$

On solving,

  $M=\frac{8^{th}+9^{th}}{2}term$

Substituting the values,

  $M=\frac{4.6+4.8}{2}$

On solving,

  $M=4.7$

Hence, Median is 4.7

For lower quartile,

Substituting the values,

  $Q1=(16+1)\frac{1}{4}$

On solving,

  $Q1=4.25$

On rounding off,

  $\approx 4$

Substituting the value,

Lower quartile is 3.6

For higher quartile,

Substituting the values,

  $Q3=(16+1)\frac{3}{4}$

On solving,

  $Q3=12.75$

Rounding off,

  $\approx 13$

Substituting the value,

Higher quartile is 5.2

Minimum is the smallest value of dataset.

Hence, minimum is 2.3.

Maximum is the largest value of the dataset.

Hence, maximum is 5.8.

The five number summary is,

2.3, 3.6, 4.7, 5.2, 5.8

Five number summary is used as it is less time consuming and more efficient.