Introduction to Probability and Statistics
Introduction to Probability and Statistics
14th Edition
ISBN: 9781133103752
Author: Mendenhall, William
Publisher: Cengage Learning
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Chapter 2, Problem 2.65SE

a.

To determine

Find the range for the given data set.

a.

Expert Solution
Check Mark

Answer to Problem 2.65SE

The range is 6.8 .

Explanation of Solution

Given:

The data shows that the miles per gallon for each of twenty medium sized cars.

  Introduction to Probability and Statistics, Chapter 2, Problem 2.65SE , additional homework tip  1

Calculation:

Arrange the data values from smallest to largest.

  20.2,21.3,22.2,22.7,22.9,23.1,23.2,23.6,23.7,24.2,24.4,24.4,24.6,24.724.7,24.9,25.3,25.9,26.2,27

The minimum value =

  20.2

The maximum value =27

Range = max-min

Range =2720.2=6.8

Hence the range is 6.8 .

b.

To determine

Draw a relative frequency histogram.

b.

Expert Solution
Check Mark

Answer to Problem 2.65SE

The standard deviation s=8.025 .

Explanation of Solution

Given:

The data shows that the miles per gallon for each of twenty medium sized cars.

  Introduction to Probability and Statistics, Chapter 2, Problem 2.65SE , additional homework tip  2

Calculation:

Arrange the data values from smallest to largest.

  20.2,21.3,22.2,22.7,22.9,23.1,23.2,23.6,23.7,24.2,24.4,24.4,24.6,24.724.7,24.9,25.3,25.9,26.2,27

Draw a relative frequency table.

    Class Frequency Relative frequency =frequencytotalfrequency
    20<2121<2222<2323<2424<2525<2626<271134722120=0.05120=0.05320=0.15420=0.20720=0.35220=0.10220=0.10

The histogram is

  Introduction to Probability and Statistics, Chapter 2, Problem 2.65SE , additional homework tip  3

The width of the bars has to be the same.

The height of the bars has to be equal to the relative frequency.

The distribution is slightly skewed left, because highest bar is right and tail of smaller bars to the left in histogram.

c.

To determine

Find the mean and the standard deviation for the given data.

c.

Expert Solution
Check Mark

Answer to Problem 2.65SE

The value of mean x¯=23.96 and the standard deviation s=1.64 .

Explanation of Solution

Given:

The data shows that the miles per gallon for each of twenty medium sized cars.

  Introduction to Probability and Statistics, Chapter 2, Problem 2.65SE , additional homework tip  4

Calculation:

Arrange the data values from smallest to largest.

  20.2,21.3,22.2,22.7,22.9,23.1,23.2,23.6,23.7,24.2,24.4,24.4,24.6,24.724.7,24.9,25.3,25.9,26.2,27

  Mean=sumofallvaluestotalnumberofvalues

  x¯= i=1 n x i n= 20.2+21.3+22.2+22.7+22.9+23.1+23.2+23.6+23.7+24.2+24.4+24.4+24.6+24.7+ 24.7+24.9+25.3+25.9+26.2+2720=479.220=23.96

find the sample variance s2=xi2 ( x i ) 2 nn1

  n= total number of data values.

  n=20

  xi=20.2+21.3+22.2+22.7+22.9+23.1+23.2+23.6+23.7+24.2+24.4+24.4+24.6+24.7+24.7+24.9+25.3+25.9+26.2+27=479.2xi2=20.22+21.32+22.22+22.72+22.92+23.12+23.22+23.62+23.72+24.22+24.42+24.42+24.62+24.72+24.72+24.92+25.32+25.92+26.22+272=11532.8

  s2=11532.8 479.2 2 20201=11532.8 479.2 2 20192.69

  s=2.69=1.64

Hence the value of mean x¯=23.96 and the standard deviation s=1.64 .

d.

To determine

Find the median for the given data set.

d.

Expert Solution
Check Mark

Answer to Problem 2.65SE

The median is m=24.3 .

Explanation of Solution

Given:

The data shows that the miles per gallon for each of twenty medium sized cars.

  Introduction to Probability and Statistics, Chapter 2, Problem 2.65SE , additional homework tip  5

Calculation:

Arrange the data values from smallest to largest.

  20.2,21.3,22.2,22.7,22.9,23.1,23.2,23.6,23.7,24.2,24.4,24.4,24.6,24.724.7,24.9,25.3,25.9,26.2,27

The median = average of the two middle values (number of data values is even)n=20

  n= total number of data values.

  m=10thdatavalue+11thdatavalue2=24.2+24.42=48.62=24.3

Hence the median is m=24.3 .

e.

To determine

Find the lower and upper quartiles.

e.

Expert Solution
Check Mark

Answer to Problem 2.65SE

The lower quartile is QL=23.4 and upper quartile is QU=24.85 .

Explanation of Solution

Given:

The data shows that the miles per gallon for each of twenty medium sized cars.

  Introduction to Probability and Statistics, Chapter 2, Problem 2.65SE , additional homework tip  6

Calculation:

Arrange the data values from smallest to largest.

  20.2,21.3,22.2,22.7,22.9,23.1,23.2,23.6,23.7,24.2,24.4,24.4,24.6,24.724.7,24.9,25.3,25.9,26.2,27

Use five-number summary it consists of the minimum, the lower quartile, the median, the upper quartile and the maximum.

The minimum =0.53

The median = average of the two middle values (number of data values is even)n=20

  n= total number of data values.

  m=10thdatavalue+11thdatavalue2=24.2+24.42=48.62=24.3

For lower quartiles.

  0.25(n+1)=0.25(20+1)=5.25not an integer

The lower quartile is the 5th data value increased by 0.25 times the difference between the 5th and the 6th data value.

  QL=22.9+0.25(23.122.9)=22.9+0.05=23.4

For upper quartiles.

  0.75(n+1)=0.75(20+1)=15.75not an integer

The lower quartile is the 9th data value increased by 0.75 times the difference between the 9th and the 10th data value.

  QU=24.7+0.75(24.924.7)=24.7+0.15=24.85

Hence the lower quartile is QL=23.4 and upper quartile is QU=24.85 .

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Chapter 2 Solutions

Introduction to Probability and Statistics

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