ALL CHAPTERS COVERED
SOLUTIONS MANUAL
, Contents
1. Introduction: Defining the Role of 11. Correlation and Regression:
Statistics in Business 1 Measuring and Predicting
Relationships 27
2. Data Structures: Classifying the
Various Types of Data Sets 2 12. Multiple Regression: Predicting
One Variable from Several
3. Histograms: Looking at the Others 31
Distribution of the Data 3
13. Report Writing: Communicating
4. Landmark Summaries: Interpreting the Results of a Multiple
Typical Values and Percentiles 6 Regression 34
5. Variability: Dealing with Diversity 10 14. Time Series: Understanding
Changes Over Time 35
6. Probability: Understanding
Random Situations 13 15. ANOVA: Testing for Differences
among Many Samples, and
7. Random Variables: Working with Much More 38
Uncertain Numbers 16
16. Nonparametrics: Testing with
8. Random Sampling: Planning Ahead Ordinal Data or Nonnormal
for Data Gathering 18 Distributions 42
9. Confidence Intervals: Admitting 17. Chi-Squared Analysis: Testing
that Estimates are not Exact 20 for Patterns in Qualitative Data 45
10. Hypothesis Testing: Deciding 18. Quality Control: Recognizing and
between Reality and Coincidence 22 Managing Variation 47
iii
, Chapter 1
Introduction
Defining the Role of Statistics in Business
Odd Problem Solutions
1. Exercise for student.
3. Exercise for student.
5. Exercise for student.
7. Designing the study (to produce the needed numbers).
9. Exploring the data (looking through the accounting
information).
11. Designing the study (while you would like, ultimately, to
estimate, the initial phase involves design because no
data are yet available).
13. Modeling the data (identifying the model and its para-
meters before the model is used for estimation and
hypothesis testing).
15. Exploring the data (looking at the data).
17. Hypothesis testing (deciding whether or not the defect
rate has increased).
19. Modeling the data (developing a system of equations
with model parameters).
Practical Business Statistics, Student Solutions Manual. DOI: 10.1016/B978-0-12-387722-2.00001-9 1
© 2012 Andrew F. Siegel. Published by Elsevier, Inc. All rights reserved.
, Chapter 2
Data Structures
Classifying the Various Types of Data Sets
advertising, (b) last year’s spending for radio advertising,
Odd Problem Solutions
and (c) last year’s spending for newspaper advertising.
1. Exercise for the student. Elementary units are competitors.
3. Exercise for the student.
17. a. Bivariate.
5. a. Exercise for the student.
b. Cross-sectional.
b. Exercise for the student.
19. a. Qualitative.
7. Multivariate analysis could be used to predict one vari-
b. Ordinal.
able (your profit) based on several others (competitors’
21. a. Models (of cell phones) are elementary units.
performance, state of the economy, and time of year).
b. Multivariate.
9. This is a multivariate cross-sectional data set consisting
of secondary data. c. Cross-sectional.
11. a. The individual employee is the elementary unit for d. Nominal.
this data set. e. Quantitative.
b. Multivariate. f. Ordinal.
c. Salary and years of experience are quantitative. 23. Ordinal.
Gender and education are qualitative. 25. a. Vacuum cleaners are the elementary units.
d. Education is ordinal qualitative because the cate- b. Multivariate.
gories can be put in a meaningful order from least c. Quantitative: price and weight. Qualitative: quality
to most education. and type.
e. Cross-sectional. d. Quality is ordinal. Type is nominal.
13. a. Months are the elementary units. e. Cross-sectional.
b. Bivariate. 27. a. The elementary units are days.
c. Both of these variables are quantitative. b. Multivariate.
d. Time-series. c. All variables are quantitative.
15. Multivariate cross-sectional data. All variables are quan- d. No variables are qualitative.
titative. Variables are (a) last year’s spending for TV e. Time-series.
Practical Business Statistics, Student Solutions Manual. DOI: 10.1016/B978-0-12-387722-2.00002-0 2
© 2012 Andrew F. Siegel. Published by Elsevier, Inc. All rights reserved.
, Chapter 3
Histograms
Looking at the Distribution of the Data
Odd Problem Solutions b. Typically, CREF has invested $0 to $200 million in
each firm, with one exceptions around $350 million.
1. Approximately normal. c. The distribution is skewed (with a long tail toward
3. Skewed (with a long tail towards higher values). high values).
5. a. 2 contracts. d. The following table gives the base 10 logarithm
b. 3 contracts. of the dollar amounts, so that the first number is
c. No, you can only tell that it lost more than 50% of log10(2,035,000) = 6.309. Other correct answers are
its value but not more than 100%. possible, either using base e logarithm or else using
d. 24 contracts. the numbers in thousands as given: for the first com-
e. Skewed (with a long tail towards high values) with 3 pany listed, you would find log10(2,035) = 3.309,
outliers. loge(2,035,000) = 14.526, or loge(2,035) = 7.618.
7. a. 6
5 Market Log10 of
Number of companies
Value Market
4 Company (Thousands) Value
3 AAR Corp. 2,035 6.309
Alliant Techsystems, Inc. 5,133 6.710
2 Armor Holdings, Inc. 1,758 6.245
BAE Systems PLC 31,984 7.505
1 Boeing Co. 364,299 8.561
Echostar Communications Corp. 14,464 7.160
0 Empresa Brasileira
−15 −5 5 15 25 de Aeronautica S.A. 317 5.501
3-month percentage price change
General Dynamics Corp. 150,671 8.178
b. 0 General Motors Corp. 183,967 8.265
9 Heico Corp. 740 5.869
4 Hexcel Corp. 1,162 6.065
3 Kaman Corp. 2,141 6.331
4
7 2 Lockheed Martin Corp. 81,234 7.910
1 7 6 Moog, Inc. 745 5.872
+ + + + Motient Corp. 784 5.894
−0 0 1 2 Northrop Grumman Corp. 29,878 7.475
3-month percentage price change
Orbital Sciences Corp. 770 5.886
c. Shares have appreciated typically by about 10% to Panamsat Corp. 4,861 6.687
20%. Pegasus Communications Corp. 4,640 6.667
Perkinelmer, Inc. 28,371 7.453
9. a.
Number of companies
20 Precision Cast Parts Corp. 9,822 6.992
Raytheon Co. A 31,952 7.504
15 Raytheon Co. B 25,787 7.411
Remec, Inc. 2,147 6.332
10 Rolls-Royce PLC 40,110 7.603
Smith Group PLC 9,263 6.967
5
Teledyne Technologies, Inc. 4,009 6.603
0 Thales (Ex Thomson CFS) 45,169 7.655
0 100,000 200,000 300,000 400,000 Triumph Group, Inc. 2,875 6.459
Portfolio value ($ thousands) Zodiac S.A. 13,429 7.128
Practical Business Statistics, Student Solutions Manual. DOI: 10.1016/B978-0-12-387722-2.00003-2 3
© 2012 Andrew F. Siegel. Published by Elsevier, Inc. All rights reserved.
,4 Student Solutions Manual
e. 7 19. a. 12
Number of companies
6 10
Number of firms
5 8
4
6
3
2 4
1 2
0 0
4 5 6 7 8 9 10 0 20 40 60 80
Log of portfolio value Revenue loss in millions
Note that the scale may differ according to the type b. The distribution is skewed (toward high values), and
of logarithm used and whether dollars or thousands shows two large gaps with three outliers. In particu-
are used, but the shape of the distribution will be lar, 13 of the 16 data values are crammed into the
basically the same in each case. first two columns of the display.
f. Approximately normal and fairly symmetric. 21. a. 32
11. a. 8,000 28
Number of donots
7,000 24
Frequency
6,000 20
5,000
16
4,000
3,000 12
2,000 8
1,000 4
0
0 20 40 60 80 100 120 0
0 20 40
Number of gifts Number of defective motors
in each batch of 250
b. Very skewed towards high values.
b. The distribution is markedly skewed (with a long tail
13. 5 1
towards high values), with two extreme outliers.
0 8 7
8 7 2 8 c. The outliers are at 25 & 41.
6 6 3 4 d. Outliers removed
9 5 8 2 3 1 12
3 9 7 4 7 1
Frequency
10 20 30 40 50 60 70 8
Average hospital charge
4
15. a. 8
Number of firms
6 0
0 2 4 6 8
Number of defective motors
4 in each batch of 250
2 e. In two particular batches, a large numbers of motors
were rejected for poor quality. Otherwise quality
0 has been approximately normally distributed,
$0 $1 $2 $3 $4 extending from about 0 to 8 rejections, and with a
CEO compensation (millions)
typical value approximately 4.
b. Approximately normal, possibly with an outlier. 23.
17. a. 6
Number of companies
18
16 5
Frequency
14
12 4
10 3
8
6 2
4
2 1
0 0
0 50,000 100,000 150,000 9% 10% 11%
Net income ($ thousands) Interest rate
b. Skewed with an outlier (possibly two outliers). Approximately normal.
,Chapter | 3 Histograms 5
25. a. 29. a. 9
12 8
10 7
Frequency
Frequency 6
8 5
6 4
3
4 2
2 1
0 0
−20% −10% 0% 10% 20% 30% 40%
0 50 100 150 200 250
Change in value of dollar
Salary (thousands)
b. Approximately normal with an outlier.
31. a. 0 4
b. Skewed (with a long tail towards high values).
1 4
c. Typical salaries were from about $40,000 to about 1 3
$100,000, with some even higher. The skewness 1 2
indicates that lower salaries are more likely than 1 1 8
higher salaries in general. 5 2 1 8
0 6 2 0 6
1 7 3 0 5
27. 10 1 8 3 0 5
+ + + + + +
−10 −0 −0 0 0 10
8
Percent change in DJIA companies
Frequency
6 b.
Number of companies
10
4 8
6
2
4
0 2
3 4 5 0
Price (dollars) −20% −10% 0% 10% 20%
Percent change in DJIA companies
Approximately normal. c. Approximately normal.
, Chapter 4
Landmark Summaries
Interpreting Typical Values and Percentiles
Odd Problem Solutions i. The 90th percentile is 30 defects per day.
1. a. Average is 15.6 defects per day. j. The percentile ranking is approximately 87%, as can
b. Median is 14 defects per day. be seen from the cumulative distribution function,
c. 5
as follows:
100%
4 90%
Number of days
80%
3 70% 29 defects is the
Percent of total
60% 87th percentile
2 50%
40%
1 30%
20%
0 10%
0 10 20 30 40
0%
Daily number of defective cars 0 10 20 30
Number of defects
d. Mode is 7.5 defects per day. (With quantitative data 90th percentile
the mode is defined as the value at the highest point is 30 defects
of the histogram, perhaps as the midpoint of the 3. The mean (also called the average) should be used because
highest bar). With a different histogram (different total amounts are important. While skewness can be a
bar widths, for example) a different value of the problem, there is only a small amount of skewness.
mode could be found. 5. a.
e. Lower quartile is 6, upper quartile is 24.5 defects
Number of countries
15
per day.
f. Smallest is 0, largest is 34 defects per day. 10
g. 35
Number of defects per day
5
30
25 0
0% 10% 20% 30% 40%
20
Vat rate
15
Approximately normal (perhaps slightly skewed
10
with a longer tail towards low values).
5 b. 18.18% average.
0 c. 19.6% median.
h. d. They are close to one another, as we expect for an
100%
90% approximately normal distribution.
80% e. 100%
Percent of total
70%
Cumulative percent
80%
60%
50% 60%
40%
30% 40%
20%
20%
10%
0% 0%
0 10 20 30 0% 10% 20% 30%
Number of defects Vat rate
Practical Business Statistics, Student Solutions Manual. DOI: 10.1016/B978-0-12-387722-2.00004-4 6
© 2012 Andrew F. Siegel. Published by Elsevier, Inc. All rights reserved.
,Chapter | 4 Landmark Summaries 7
f. 15.0% and 22.0% (these are the 20 th and 80 th c. 7
percentiles respectively).
6
g. About the 23rd percentile, for a VAT tax of 16%).
Number of samples
7. a. Beta = 0.981. This is a weighted average of the betas 5
(2.4, 0.6, and 1.2) weighted according to the market 4
values (shares times price: 3,500; 17,600, and 7,950).
3
b. Selling all shares of Spec. Comp. will realize $35 ×
100 = $3,500. This will buy $3,500/53 = 66 shares 2
of Dependable Conglomerate. Added to the origi- 1
nal, gives 216 shares of Dependable Conglomerate.
0
The new portfolio would consist of: 40 60 80 100 120 140 160 180 200 220 240
200 shares Conservative Industries Breaking strength in pounds
at $88 per share, beta = 0:6 Median Average
216 shares Dependable Conglomerate
The average and the median are not the same. The
at $53 per share, beta = 1:2
average is larger than the median due to skewness
The beta of this new portfolio, 0.836, has indeed of the distribution.
decreased from 0.981.
9. $13.80, computed as the weighted average. d. 100%
11. a. 80%
Cumulative percent
Per capita
income
60%
Median 40%
household
income
20%
$0 $50,000 $100,000 $150,000 $200,000
Income 0%
75 100 125 150 175 200
b. The distribution of median household income is Breaking strength (pounds)
generally higher than that of per capita income. e. The 10th percentile is 79 pounds, i.e. 10% have a
(This may be explained by the fact that the income breaking strength of 79 pounds or less. The 90th
of a household often combines the income of more percentile is 177 pounds.
than one person). f. No, these supplies do not qualify. These supplies
13. Overall market penetration of 19.75%, using a weighted can provide a breakage value of 100 pounds or
average. more only about 50% of the time. Since manage-
15. Per capita tax burden is $2,288.46, the weighted aver- ment would like the breakage value to be less
age per-capita-tax, weighted by population. than 100 pounds a maximum of 10% of the time,
17. The upper quartile is 75. you would compare to the 10th percentile. This
19. a. The five number summary is $49,489, $59,872, would need to be at least 100 pounds for these
$84,777, $163,220, and $189,058. conditions to be met.
b. 25. a. The average share is 8.19%. The median share
is 4.90%.
$0 $50,000 $100,000 $150,000 $200,000
b. 100%
Revenues 90%
80%
Cumulative percent
21. a. Average loan fee is 1.61%. 70%
b. Median loan fee is 1.75%. 60%
c. The mode is 2.00%, with 7 out of the 14 cases 50%
40%
having this value. Alternately, the midpoint of the
30%
highest bar of a histogram might be used to find a 20%
value close to 2%. 10%
d. It might be argued that the mode is the most useful 0%
description of the typical loan fee, since half the 0% 5% 10% 15% 20% 25% 30%
Share
banks are charging this amount. However, the mode
is also the largest value, an extreme value, and so c. The 80th percentile is 17.2%
one of the other summaries might be preferable. 27. a. The average is −1.09%.
23. a. The average is 115.35. b. On average the dollar weakened (because the average
b. The median is 100. percent change is negative).
, 8 Student Solutions Manual
c. The median is −2.80%. The average is higher d. The five-number summary is $317,000; $2,035,000;
primarily due to the outlier (Brazil). $7,198,000; $31,952,000; $364,299,000.
d. 30% e.
25%
20%
Percent change
15%
10%
5%
0%
−5%
−10%
$0 $100,000 $200,000 $300,000 $400,000
29. a. The average yield is 5.22%. Portfolio value ($ thousands)
b. The median yield is 5.21%. There are two suggestions of skewness here: (1) the
c. The lower quartile is 5.13%. The upper quartile is median is not at the center of the box and (2) the
5.27%. line to the largest value is much longer than
d. The five-number summary is 4.91%, 5.13%, 5.21%, the line to the smallest value.
5.27%, 5.83%. f. Yes. With skewness like this (with a long tail
e. towards large values) we expect the average to be
larger than the median.
33. a. 42.1% (this is 8/19).
b. Law of Desire.
c. They are the same.
35. a. There are 5 outliers: Oracle Corp at 56.81, Bmc
Software Inc at 10.9, Nuance Communications Inc
at 9.91, Rosetta Stone Inc at 9.51, Sybase Inc at
9.29. These are all larger than 1.5 times the inter-
4.8% 5.0% 5.2% 5.4% 5.6% 5.8% 6.0%
quartile range, which is 8.915.
Yield
f. There are two upper outliers: MI StrgcFnd Itd Rf Rv
at 5.64% and HghlndCoHlthFcs FL hospRv at
5.83%. Here is the detailed box plot:
0 10 20 30 40 50 60
CEO compensation ($ millions)
b. Sybase Rosetta stone Nuance BMC
0 2 4 6 8 10 12
CEO compensation ($ millions)
4.8% 5.0% 5.2% 5.4% 5.6% 5.8% 6.0%
Yield 37. a. $4,000,000
Archer-Daniels-Midland
CEO compensation
g. 100%
$3,000,000
75%
$2,000,000 Campbell soup
50%
$1,000,000
25%
Wrigley
$0
0%
4.8% 5.0% 5.2% 5.4% 5.6% 5.8% 6.0% b. $850,000.
Yield
39. a. The average is 4.39. The median is 3.
h. 5.40% is the 90th percentile. b. The average is now 2.92. The median is still 3.
i. The 60th percentile is 5.2% (or 5.23%). c. The average is very sensitive to the presence of out-
31. a. The average is $36,483,900. liers. Removing them reduced the average value
b. The median is $7,198,000. from 4.39 to 2.92; this is a sizable percentage change
c. The average is larger than the median. for the average value. The median is unchanged by
SOLUTIONS MANUAL
, Contents
1. Introduction: Defining the Role of 11. Correlation and Regression:
Statistics in Business 1 Measuring and Predicting
Relationships 27
2. Data Structures: Classifying the
Various Types of Data Sets 2 12. Multiple Regression: Predicting
One Variable from Several
3. Histograms: Looking at the Others 31
Distribution of the Data 3
13. Report Writing: Communicating
4. Landmark Summaries: Interpreting the Results of a Multiple
Typical Values and Percentiles 6 Regression 34
5. Variability: Dealing with Diversity 10 14. Time Series: Understanding
Changes Over Time 35
6. Probability: Understanding
Random Situations 13 15. ANOVA: Testing for Differences
among Many Samples, and
7. Random Variables: Working with Much More 38
Uncertain Numbers 16
16. Nonparametrics: Testing with
8. Random Sampling: Planning Ahead Ordinal Data or Nonnormal
for Data Gathering 18 Distributions 42
9. Confidence Intervals: Admitting 17. Chi-Squared Analysis: Testing
that Estimates are not Exact 20 for Patterns in Qualitative Data 45
10. Hypothesis Testing: Deciding 18. Quality Control: Recognizing and
between Reality and Coincidence 22 Managing Variation 47
iii
, Chapter 1
Introduction
Defining the Role of Statistics in Business
Odd Problem Solutions
1. Exercise for student.
3. Exercise for student.
5. Exercise for student.
7. Designing the study (to produce the needed numbers).
9. Exploring the data (looking through the accounting
information).
11. Designing the study (while you would like, ultimately, to
estimate, the initial phase involves design because no
data are yet available).
13. Modeling the data (identifying the model and its para-
meters before the model is used for estimation and
hypothesis testing).
15. Exploring the data (looking at the data).
17. Hypothesis testing (deciding whether or not the defect
rate has increased).
19. Modeling the data (developing a system of equations
with model parameters).
Practical Business Statistics, Student Solutions Manual. DOI: 10.1016/B978-0-12-387722-2.00001-9 1
© 2012 Andrew F. Siegel. Published by Elsevier, Inc. All rights reserved.
, Chapter 2
Data Structures
Classifying the Various Types of Data Sets
advertising, (b) last year’s spending for radio advertising,
Odd Problem Solutions
and (c) last year’s spending for newspaper advertising.
1. Exercise for the student. Elementary units are competitors.
3. Exercise for the student.
17. a. Bivariate.
5. a. Exercise for the student.
b. Cross-sectional.
b. Exercise for the student.
19. a. Qualitative.
7. Multivariate analysis could be used to predict one vari-
b. Ordinal.
able (your profit) based on several others (competitors’
21. a. Models (of cell phones) are elementary units.
performance, state of the economy, and time of year).
b. Multivariate.
9. This is a multivariate cross-sectional data set consisting
of secondary data. c. Cross-sectional.
11. a. The individual employee is the elementary unit for d. Nominal.
this data set. e. Quantitative.
b. Multivariate. f. Ordinal.
c. Salary and years of experience are quantitative. 23. Ordinal.
Gender and education are qualitative. 25. a. Vacuum cleaners are the elementary units.
d. Education is ordinal qualitative because the cate- b. Multivariate.
gories can be put in a meaningful order from least c. Quantitative: price and weight. Qualitative: quality
to most education. and type.
e. Cross-sectional. d. Quality is ordinal. Type is nominal.
13. a. Months are the elementary units. e. Cross-sectional.
b. Bivariate. 27. a. The elementary units are days.
c. Both of these variables are quantitative. b. Multivariate.
d. Time-series. c. All variables are quantitative.
15. Multivariate cross-sectional data. All variables are quan- d. No variables are qualitative.
titative. Variables are (a) last year’s spending for TV e. Time-series.
Practical Business Statistics, Student Solutions Manual. DOI: 10.1016/B978-0-12-387722-2.00002-0 2
© 2012 Andrew F. Siegel. Published by Elsevier, Inc. All rights reserved.
, Chapter 3
Histograms
Looking at the Distribution of the Data
Odd Problem Solutions b. Typically, CREF has invested $0 to $200 million in
each firm, with one exceptions around $350 million.
1. Approximately normal. c. The distribution is skewed (with a long tail toward
3. Skewed (with a long tail towards higher values). high values).
5. a. 2 contracts. d. The following table gives the base 10 logarithm
b. 3 contracts. of the dollar amounts, so that the first number is
c. No, you can only tell that it lost more than 50% of log10(2,035,000) = 6.309. Other correct answers are
its value but not more than 100%. possible, either using base e logarithm or else using
d. 24 contracts. the numbers in thousands as given: for the first com-
e. Skewed (with a long tail towards high values) with 3 pany listed, you would find log10(2,035) = 3.309,
outliers. loge(2,035,000) = 14.526, or loge(2,035) = 7.618.
7. a. 6
5 Market Log10 of
Number of companies
Value Market
4 Company (Thousands) Value
3 AAR Corp. 2,035 6.309
Alliant Techsystems, Inc. 5,133 6.710
2 Armor Holdings, Inc. 1,758 6.245
BAE Systems PLC 31,984 7.505
1 Boeing Co. 364,299 8.561
Echostar Communications Corp. 14,464 7.160
0 Empresa Brasileira
−15 −5 5 15 25 de Aeronautica S.A. 317 5.501
3-month percentage price change
General Dynamics Corp. 150,671 8.178
b. 0 General Motors Corp. 183,967 8.265
9 Heico Corp. 740 5.869
4 Hexcel Corp. 1,162 6.065
3 Kaman Corp. 2,141 6.331
4
7 2 Lockheed Martin Corp. 81,234 7.910
1 7 6 Moog, Inc. 745 5.872
+ + + + Motient Corp. 784 5.894
−0 0 1 2 Northrop Grumman Corp. 29,878 7.475
3-month percentage price change
Orbital Sciences Corp. 770 5.886
c. Shares have appreciated typically by about 10% to Panamsat Corp. 4,861 6.687
20%. Pegasus Communications Corp. 4,640 6.667
Perkinelmer, Inc. 28,371 7.453
9. a.
Number of companies
20 Precision Cast Parts Corp. 9,822 6.992
Raytheon Co. A 31,952 7.504
15 Raytheon Co. B 25,787 7.411
Remec, Inc. 2,147 6.332
10 Rolls-Royce PLC 40,110 7.603
Smith Group PLC 9,263 6.967
5
Teledyne Technologies, Inc. 4,009 6.603
0 Thales (Ex Thomson CFS) 45,169 7.655
0 100,000 200,000 300,000 400,000 Triumph Group, Inc. 2,875 6.459
Portfolio value ($ thousands) Zodiac S.A. 13,429 7.128
Practical Business Statistics, Student Solutions Manual. DOI: 10.1016/B978-0-12-387722-2.00003-2 3
© 2012 Andrew F. Siegel. Published by Elsevier, Inc. All rights reserved.
,4 Student Solutions Manual
e. 7 19. a. 12
Number of companies
6 10
Number of firms
5 8
4
6
3
2 4
1 2
0 0
4 5 6 7 8 9 10 0 20 40 60 80
Log of portfolio value Revenue loss in millions
Note that the scale may differ according to the type b. The distribution is skewed (toward high values), and
of logarithm used and whether dollars or thousands shows two large gaps with three outliers. In particu-
are used, but the shape of the distribution will be lar, 13 of the 16 data values are crammed into the
basically the same in each case. first two columns of the display.
f. Approximately normal and fairly symmetric. 21. a. 32
11. a. 8,000 28
Number of donots
7,000 24
Frequency
6,000 20
5,000
16
4,000
3,000 12
2,000 8
1,000 4
0
0 20 40 60 80 100 120 0
0 20 40
Number of gifts Number of defective motors
in each batch of 250
b. Very skewed towards high values.
b. The distribution is markedly skewed (with a long tail
13. 5 1
towards high values), with two extreme outliers.
0 8 7
8 7 2 8 c. The outliers are at 25 & 41.
6 6 3 4 d. Outliers removed
9 5 8 2 3 1 12
3 9 7 4 7 1
Frequency
10 20 30 40 50 60 70 8
Average hospital charge
4
15. a. 8
Number of firms
6 0
0 2 4 6 8
Number of defective motors
4 in each batch of 250
2 e. In two particular batches, a large numbers of motors
were rejected for poor quality. Otherwise quality
0 has been approximately normally distributed,
$0 $1 $2 $3 $4 extending from about 0 to 8 rejections, and with a
CEO compensation (millions)
typical value approximately 4.
b. Approximately normal, possibly with an outlier. 23.
17. a. 6
Number of companies
18
16 5
Frequency
14
12 4
10 3
8
6 2
4
2 1
0 0
0 50,000 100,000 150,000 9% 10% 11%
Net income ($ thousands) Interest rate
b. Skewed with an outlier (possibly two outliers). Approximately normal.
,Chapter | 3 Histograms 5
25. a. 29. a. 9
12 8
10 7
Frequency
Frequency 6
8 5
6 4
3
4 2
2 1
0 0
−20% −10% 0% 10% 20% 30% 40%
0 50 100 150 200 250
Change in value of dollar
Salary (thousands)
b. Approximately normal with an outlier.
31. a. 0 4
b. Skewed (with a long tail towards high values).
1 4
c. Typical salaries were from about $40,000 to about 1 3
$100,000, with some even higher. The skewness 1 2
indicates that lower salaries are more likely than 1 1 8
higher salaries in general. 5 2 1 8
0 6 2 0 6
1 7 3 0 5
27. 10 1 8 3 0 5
+ + + + + +
−10 −0 −0 0 0 10
8
Percent change in DJIA companies
Frequency
6 b.
Number of companies
10
4 8
6
2
4
0 2
3 4 5 0
Price (dollars) −20% −10% 0% 10% 20%
Percent change in DJIA companies
Approximately normal. c. Approximately normal.
, Chapter 4
Landmark Summaries
Interpreting Typical Values and Percentiles
Odd Problem Solutions i. The 90th percentile is 30 defects per day.
1. a. Average is 15.6 defects per day. j. The percentile ranking is approximately 87%, as can
b. Median is 14 defects per day. be seen from the cumulative distribution function,
c. 5
as follows:
100%
4 90%
Number of days
80%
3 70% 29 defects is the
Percent of total
60% 87th percentile
2 50%
40%
1 30%
20%
0 10%
0 10 20 30 40
0%
Daily number of defective cars 0 10 20 30
Number of defects
d. Mode is 7.5 defects per day. (With quantitative data 90th percentile
the mode is defined as the value at the highest point is 30 defects
of the histogram, perhaps as the midpoint of the 3. The mean (also called the average) should be used because
highest bar). With a different histogram (different total amounts are important. While skewness can be a
bar widths, for example) a different value of the problem, there is only a small amount of skewness.
mode could be found. 5. a.
e. Lower quartile is 6, upper quartile is 24.5 defects
Number of countries
15
per day.
f. Smallest is 0, largest is 34 defects per day. 10
g. 35
Number of defects per day
5
30
25 0
0% 10% 20% 30% 40%
20
Vat rate
15
Approximately normal (perhaps slightly skewed
10
with a longer tail towards low values).
5 b. 18.18% average.
0 c. 19.6% median.
h. d. They are close to one another, as we expect for an
100%
90% approximately normal distribution.
80% e. 100%
Percent of total
70%
Cumulative percent
80%
60%
50% 60%
40%
30% 40%
20%
20%
10%
0% 0%
0 10 20 30 0% 10% 20% 30%
Number of defects Vat rate
Practical Business Statistics, Student Solutions Manual. DOI: 10.1016/B978-0-12-387722-2.00004-4 6
© 2012 Andrew F. Siegel. Published by Elsevier, Inc. All rights reserved.
,Chapter | 4 Landmark Summaries 7
f. 15.0% and 22.0% (these are the 20 th and 80 th c. 7
percentiles respectively).
6
g. About the 23rd percentile, for a VAT tax of 16%).
Number of samples
7. a. Beta = 0.981. This is a weighted average of the betas 5
(2.4, 0.6, and 1.2) weighted according to the market 4
values (shares times price: 3,500; 17,600, and 7,950).
3
b. Selling all shares of Spec. Comp. will realize $35 ×
100 = $3,500. This will buy $3,500/53 = 66 shares 2
of Dependable Conglomerate. Added to the origi- 1
nal, gives 216 shares of Dependable Conglomerate.
0
The new portfolio would consist of: 40 60 80 100 120 140 160 180 200 220 240
200 shares Conservative Industries Breaking strength in pounds
at $88 per share, beta = 0:6 Median Average
216 shares Dependable Conglomerate
The average and the median are not the same. The
at $53 per share, beta = 1:2
average is larger than the median due to skewness
The beta of this new portfolio, 0.836, has indeed of the distribution.
decreased from 0.981.
9. $13.80, computed as the weighted average. d. 100%
11. a. 80%
Cumulative percent
Per capita
income
60%
Median 40%
household
income
20%
$0 $50,000 $100,000 $150,000 $200,000
Income 0%
75 100 125 150 175 200
b. The distribution of median household income is Breaking strength (pounds)
generally higher than that of per capita income. e. The 10th percentile is 79 pounds, i.e. 10% have a
(This may be explained by the fact that the income breaking strength of 79 pounds or less. The 90th
of a household often combines the income of more percentile is 177 pounds.
than one person). f. No, these supplies do not qualify. These supplies
13. Overall market penetration of 19.75%, using a weighted can provide a breakage value of 100 pounds or
average. more only about 50% of the time. Since manage-
15. Per capita tax burden is $2,288.46, the weighted aver- ment would like the breakage value to be less
age per-capita-tax, weighted by population. than 100 pounds a maximum of 10% of the time,
17. The upper quartile is 75. you would compare to the 10th percentile. This
19. a. The five number summary is $49,489, $59,872, would need to be at least 100 pounds for these
$84,777, $163,220, and $189,058. conditions to be met.
b. 25. a. The average share is 8.19%. The median share
is 4.90%.
$0 $50,000 $100,000 $150,000 $200,000
b. 100%
Revenues 90%
80%
Cumulative percent
21. a. Average loan fee is 1.61%. 70%
b. Median loan fee is 1.75%. 60%
c. The mode is 2.00%, with 7 out of the 14 cases 50%
40%
having this value. Alternately, the midpoint of the
30%
highest bar of a histogram might be used to find a 20%
value close to 2%. 10%
d. It might be argued that the mode is the most useful 0%
description of the typical loan fee, since half the 0% 5% 10% 15% 20% 25% 30%
Share
banks are charging this amount. However, the mode
is also the largest value, an extreme value, and so c. The 80th percentile is 17.2%
one of the other summaries might be preferable. 27. a. The average is −1.09%.
23. a. The average is 115.35. b. On average the dollar weakened (because the average
b. The median is 100. percent change is negative).
, 8 Student Solutions Manual
c. The median is −2.80%. The average is higher d. The five-number summary is $317,000; $2,035,000;
primarily due to the outlier (Brazil). $7,198,000; $31,952,000; $364,299,000.
d. 30% e.
25%
20%
Percent change
15%
10%
5%
0%
−5%
−10%
$0 $100,000 $200,000 $300,000 $400,000
29. a. The average yield is 5.22%. Portfolio value ($ thousands)
b. The median yield is 5.21%. There are two suggestions of skewness here: (1) the
c. The lower quartile is 5.13%. The upper quartile is median is not at the center of the box and (2) the
5.27%. line to the largest value is much longer than
d. The five-number summary is 4.91%, 5.13%, 5.21%, the line to the smallest value.
5.27%, 5.83%. f. Yes. With skewness like this (with a long tail
e. towards large values) we expect the average to be
larger than the median.
33. a. 42.1% (this is 8/19).
b. Law of Desire.
c. They are the same.
35. a. There are 5 outliers: Oracle Corp at 56.81, Bmc
Software Inc at 10.9, Nuance Communications Inc
at 9.91, Rosetta Stone Inc at 9.51, Sybase Inc at
9.29. These are all larger than 1.5 times the inter-
4.8% 5.0% 5.2% 5.4% 5.6% 5.8% 6.0%
quartile range, which is 8.915.
Yield
f. There are two upper outliers: MI StrgcFnd Itd Rf Rv
at 5.64% and HghlndCoHlthFcs FL hospRv at
5.83%. Here is the detailed box plot:
0 10 20 30 40 50 60
CEO compensation ($ millions)
b. Sybase Rosetta stone Nuance BMC
0 2 4 6 8 10 12
CEO compensation ($ millions)
4.8% 5.0% 5.2% 5.4% 5.6% 5.8% 6.0%
Yield 37. a. $4,000,000
Archer-Daniels-Midland
CEO compensation
g. 100%
$3,000,000
75%
$2,000,000 Campbell soup
50%
$1,000,000
25%
Wrigley
$0
0%
4.8% 5.0% 5.2% 5.4% 5.6% 5.8% 6.0% b. $850,000.
Yield
39. a. The average is 4.39. The median is 3.
h. 5.40% is the 90th percentile. b. The average is now 2.92. The median is still 3.
i. The 60th percentile is 5.2% (or 5.23%). c. The average is very sensitive to the presence of out-
31. a. The average is $36,483,900. liers. Removing them reduced the average value
b. The median is $7,198,000. from 4.39 to 2.92; this is a sizable percentage change
c. The average is larger than the median. for the average value. The median is unchanged by
