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 2.55SE

a.

To determine

Find the five-number summary for the given data set.

a.

Expert Solution
Check Mark

Answer to Problem 2.55SE

For generic brand.

The minimum =24

  m=26

  QL=25

  QU=27.25

The maximum =28

  IQR=2.25

For Sunmaid.

The minimum =22

  m=26

  QL=24

  QU=28

The maximum =30

  IQR=4

Explanation of Solution

Given:

The given data set is

  Introduction to Probability and Statistics, Chapter 2, Problem 2.55SE , additional homework tip  1

Calculation:

For generic brand.

Arrange the data set from smallest to largest.

  25,25,25,26,26,26,26,26,27,27,28,28,28

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

The minimum =25

The median = average of the two middle values (number of data values is even)n=14

  n= total number of data values.

  m=7thdata value+8thdata value2=26+262=522=26

For lower quartiles.

  0.25(n+1)=0.25(14+1)=3.75not an integer

The lower quartile is the 6th data value increased by 0.75 times the difference between the 3rd and the 4th data value.

  QL=25+0.75(2525)=25+0=25

For upper quartiles.

  0.75(n+1)=0.75(14+1)=11.25not an integer

The lower quartile is the 19th data value increased by 0.25 times the difference between the 11th and the 12th data value.

  QU=27+0.25(2827)=27+0.25=27.25

The maximum =28

  IQR=QUQL=27.2525=2.25

For Sunmaid.

Arrange the data set from smallest to largest.

  22,24,24,24,24,25,25,27,28,28,28,29,30

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

The minimum =22

The median = average of the two middle values (number of data values is even)n=14

  n= total number of data values.

  m=7thdata value+8thdata value2=25+272=522=26

For lower quartiles.

  0.25(n+1)=0.25(14+1)=3.75not an integer

The lower quartile is the 6th data value increased by 0.75 times the difference between the 3rd and the 4th data value.

  QL=24+0.75(2424)=25+0=24

For upper quartiles.

  0.75(n+1)=0.75(14+1)=11.25not an integer

The lower quartile is the 19th data value increased by 0.25 times the difference between the 11th and the 12th data value.

  QU=28+0.25(2828)=28+0=28

The maximum =30

  IQR=QUQL=2824=4

b.

To determine

Draw two box plots on the same horizontal scale.

b.

Expert Solution
Check Mark

Answer to Problem 2.55SE

The boxplot is

  Introduction to Probability and Statistics, Chapter 2, Problem 2.55SE , additional homework tip  2

Explanation of Solution

Given:

The given data set is

  Introduction to Probability and Statistics, Chapter 2, Problem 2.55SE , additional homework tip  3

Calculation:

For generic brand.

Arrange the data set from smallest to largest.

  25,25,25,26,26,26,26,26,27,27,28,28,28

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

The minimum =25

The median = average of the two middle values (number of data values is even)n=14

  n= total number of data values.

  m=7thdata value+8thdata value2=26+262=522=26

For lower quartiles.

  0.25(n+1)=0.25(14+1)=3.75not an integer

The lower quartile is the 6th data value increased by 0.75 times the difference between the 3rd and the 4th data value.

  QL=25+0.75(2525)=25+0=25

For upper quartiles.

  0.75(n+1)=0.75(14+1)=11.25not an integer

The lower quartile is the 19th data value increased by 0.25 times the difference between the 11th and the 12th data value.

  QU=27+0.25(2827)=27+0.25=27.25

The maximum =28

  IQR=QUQL=27.2525=2.25

Outliers it is an observation that are more than 1.5 times the IQR above QU or below QL .

  QU+1.5IQR=27.25+1.5(2.25)=30.625QL1.5IQR=251.5(2.25)=21.625

There are no outliers, because all data values are between 21.625 and 30.625 .

For Sunmaid.

Arrange the data set from smallest to largest.

  22,24,24,24,24,25,25,27,28,28,28,29,30

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

The minimum =22

The median = average of the two middle values (number of data values is even)n=14

  n= total number of data values.

  m=7thdata value+8thdata value2=25+272=522=26

For lower quartiles.

  0.25(n+1)=0.25(14+1)=3.75not an integer

The lower quartile is the 6th data value increased by 0.75 times the difference between the 3rd and the 4th data value.

  QL=24+0.75(2424)=25+0=24

For upper quartiles.

  0.75(n+1)=0.75(14+1)=11.25not an integer

The lower quartile is the 19th data value increased by 0.25 times the difference between the 11th and the 12th data value.

  QU=28+0.25(2828)=28+0=28

The maximum =30

  IQR=QUQL=2824=4

Outliers it is an observation that are more than 1.5 times the IQR above QU or below QL .

  QU+1.5IQR=28+1.5(4)=34QL1.5IQR=241.5(4)=18

There are no outliers, because all data values are between 18 and 34 .

The boxplot starts at the lower quartile, ends at the upper quartile and vertical line at the median.

The whiskers of the boxplot are at the minimum and maximum value (excluding the outliers).

The box plot is

  Introduction to Probability and Statistics, Chapter 2, Problem 2.55SE , additional homework tip  4

c.

To determine

Draw a stem and leaf plots for the given data sets.

c.

Expert Solution
Check Mark

Answer to Problem 2.55SE

The shape of the distributions are symmetric and boxplots confirm the result.

Explanation of Solution

Given:

The given data set is

  Introduction to Probability and Statistics, Chapter 2, Problem 2.55SE , additional homework tip  5

Calculation:

For generic brand.

Arrange the data set from smallest to largest.

  25,25,25,26,26,26,26,26,27,27,28,28,28

Placethe digits of the ones to the left of the vertical line and the digits of the tenths of every data value to the right of the vertical line.

    Stem Leaf
    24252627280000000000000

For Sunmaid.

Arrange the data set from smallest to largest.

  22,24,24,24,24,25,25,27,28,28,28,29,30

Placethe digits of the ones to the left of the vertical line and the digits of the tenths of every data value to the right of the vertical line.

    Stem Leaf
    22232425262728293000000000000000

Both distributionsare symmetric, because most of the data values lie in the middle of the distribution.

The boxplots also confirms that the distribution is symmetric, because the box of the boxplot lies in the middle between the whiskers.

d.

To determine

Check the result for the average number of raisins for the two brands.

d.

Expert Solution
Check Mark

Answer to Problem 2.55SE

The two brands appear to have the same average number of raisins.

Explanation of Solution

Given:

The given data set is

  Introduction to Probability and Statistics, Chapter 2, Problem 2.55SE , additional homework tip  6

Calculation:

For generic brand.

Arrange the data set from smallest to largest.

  25,25,25,26,26,26,26,26,27,27,28,28,28

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

The minimum =25

The median = average of the two middle values (number of data values is even)n=14

  n= total number of data values.

  m=7thdata value+8thdata value2=26+262=522=26

For Sunmaid.

Arrange the data set from smallest to largest.

  22,24,24,24,24,25,25,27,28,28,28,29,30

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

The minimum =22

The median = average of the two middle values (number of data values is even)n=14

  n= total number of data values.

  m=7thdata value+8thdata value2=25+272=522=26

The median of both the brands are identical, the two brands appear to have the same average number of raisins.

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Chapter 2 Solutions

Introduction to Probability and Statistics

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