SOLUTIONS TO EXERCISES
About these solutions
The solutions that follow were prepared by Jackie Miller, although some of these have their
roots in solutions prepared by Bill Notz, David Moore, and Darryl Nester, for the previous
editions, and modified (or not) by me for the seventh edition. Should you discover any errors or
have any comments about these solutions (or the odd answers, in the back of the text), please
report them to me:
Jackie Miller
The Ohio State University
404 Cockins Hall
1958 Neil Avenue
Columbus, OH 43210
e-mail:
In putting these solutions together, I used the following software:
• For typesetting: Microsoft Word (with MathType6 by Design Science)
• For graphs: Adobe Illustrator
• For statistical analysis: Minitab, Excel, and Texas Instruments graphing calculators (especially
the TI-84). Some of the solutions in later chapters include output from Minitab.
Using the table of random digits
Grading SRSs chosen from Table A is complicated by the fact that students can find some
creative ways to (mis)use the table. Some approaches are not mistakes, but may lead to different
students having different “right” answers. Correct answers will vary based on:
• the line in the table on which they begin (you may want to specify one, if the text does
not).
• whether they start with, e.g., 00 or 01.
• whether they assign labels across the rows or down the columns (nearly all lists in the
text are alphabetized down the columns).
Some approaches can potentially lead to wrong answers. Mistakes to watch out for include the
following:
• They may forget that all labels must be the same length (e.g., assigning labels like 0, 1, 2,
..., 9, 10,... rather than 00, 01, 02, ...).
• In assigning multiple labels, they may not give the same number of labels to all units. For
example, if there are 30 units, they may try to use up all the two-digit numbers, thus
assigning four labels to the first 10 units and only three to the remaining 20.
As an alternative to using the random digits in Table A, students can pick a random sample by
generating (pseudo)random numbers. Many, if not all, calculators have a seed value that
,294 Solutions to Exercises
determines the sequence of random numbers that it produces. Rather than pointing students to a
particular line of Table A, you could specify a seed value for generating random numbers, so that
all students would obtain the same results (if all are using the same calculator).
On a TI-84, for example, after executing the command 0→rand, the rand command will produce
the sequence (rounded to four decimals) 0.9436, 0.9083, 0.1467, ..., while 1→rand initiates the
sequence 0.7456, 0.8559, 0.2254, .... So to choose, say, an SRS of size 10 from 30 subjects, use
the command 0→rand to set the seed, and then type 1+30∗rand, and press ENTER repeatedly.
Ignoring the decimal portion of the resulting numbers, this produces the sample
29, 28, 5, 16, 13, 23, 2, 11, 30, 7
(Generally, to generate random numbers from 1 to n, use the command 1+ n∗rand and ignore the
decimal portion of the result.)
Using statistical software
While Statistics: Concepts and Controversies is written so that most problems can be done with
a calculator capable of finding means and standard deviations, the use of more sophisticated
calculators or computer software (if readily available) can make some computations, especially
in later chapters, much more convenient. In choosing technology, be aware of the following
considerations:
• Standard deviations: Students may easily get confused by software that gives both the so-
called “sample standard deviation” (the one used in the text) and the “population standard
deviation” (dividing by n rather than n – 1). Symbolically, the former is usually given as s
and the later as σ (sigma), but the distinction is not always clear. For example, many
computer spreadsheets have a command such as “STDEV(...)” to compute a standard
deviation, but you may need to check the manual to find out which kind it is. As a quick
check: for the numbers 1, 2, 3, s = 1 while σ = 0.8165. In general, if two values are given,
the larger one is s and the smaller is σ. If only one value is given, and it is the wrong one,
use the relationship .
• Stemplots: In creating a stemplot, some software packages might round to the nearest
integer, for example, or might simply truncate. For example, if our data set ranges from
20 to 99, then using the digits 2 through 9 as stems, the number 59.6 might be represented
as either a 0 on stem 6 (with rounding) or a 9 on stem 5 (using truncation). Minitab opts
for truncation over rounding, so all the solutions in this guide show truncated-data
stemplots (except for exercises that instructed students to round). This usually makes
little difference in the overall appearance of the stemplot.
I have come to prefer truncation over rounding for two reasons. First, there is less
chance of making a mistake using truncation (if doing the task by hand). More important,
truncating keeps the shape of the stemplot similar to the shape of a histogram of the same
data. In the example mentioned above, if we were to make a histogram, with intervals 20
,Solutions Part I: Chapters 1 to 9 295
to 29.9, 30 to 39.9, etc. (a fairly natural choice), the number 59.6 would fall in the 50s
interval, so the truncated stemplot (which placed the leaf 9 on the stem 5) would be the
best match for the histogram.
• Quartiles and five-number summaries: Methods of computing quartiles vary among
different packages. Some use the approach given in the text (that is, Q1 is the median of
all the numbers below the location of the overall median, etc.), while others use a more
complicated approach. For the numbers 1, 2, 3, 4, for example, we would want Q1 = 1.5
and Q3 = 2.5, but Minitab reports these as 1.25 and 2.75, respectively.
In these solutions (and the odd-numbered answers in the back of the text), I opted to
report five-number summaries as they would be found using the text’s method.
• Boxplots: Some programs that draw boxplots use the convention that the “whiskers”
extend to the lower and upper deciles (the 10th and 90th percentiles) rather than to the
minimum and maximum. The interpretation of the plot is essentially unchanged, but be
prepared to adapt to the software available to your students.
While the decile method is merely different from that given in the text, some methods
are (in the opinions of some) just plain wrong. Some early graphing calculators drew
“box charts,” which have a centerline at the mean (not the median), and a box extending
from to . I know of no statistics text that uses that method, and I hope that
such graphing calculators are no longer manufactured (or used).
, Solutions Part I: Chapters 1 to 9 297
6.1 Part I Solutions
Chapter 1 Solutions
1.1. Population: The population is not explicitly defined. Presumably it is all adults in the United
States. However, because this is the National Annenberg Election Survey, it may be that the
population is intended to be all adults in the United States who are registered voters. Sample:
The sample is the 1,345 randomly selected adults.
1.2. This is an observational study. The research examined student comments but no treatment
was applied to the students.
1.3. (a) The individuals are cars (or “motor vehicles”). (b) The variables are make/model, vehicle
type, transmission type, number of cylinders, city MPG, and highway MPG. The last three of
these are numerical.
1.4. (a) The individuals are baseball players. (b) There are four variables: team, position, age,
salary. The last two of these are numerical. (c) Age is given in years; salary is given in thousands
of dollars per year. Beckett’s annual salary was $6,667,000 (almost 6.7 million dollars).
1.5. Answers may vary. One possibility is whether a household participates in the recycling
program. Another possibility would be the percentage of a household’s recyclable waste that is
actually recycled. (This may sound like a good variable until one actually thinks about how one
would gather such information ....)
Note: The variable here is not, e.g., “the percentage of households participating,”
because that is not a characteristic of an individual.
1.6. The population is pregnant and breast-feeding women. The sample consists of the 21 women
who returned the surveys. Only 21/340*100% = 6.2% of the women who were contacted
responded.
1.7. The variable was if a person favors the death penalty for a person convicted of murder. The
population is probably adult Americans. The sample was 1,010 adult Americans.
1.8. It is not an experiment; it is a sample survey in which information is gathered without
imposing any treatment. The variables are gender and political party voted for in the last
congressional election.
1.9. Exact descriptions of the populations may vary. (a) All adult, presumably all adult
Americans. (b) All video adapter cables in the lot. (c) All U.S. households.
1.10. Exact descriptions of the populations may vary. (a) Teenagers (or “tenth-graders,” but from
the description of the situation, the researcher would like information about all teens). (b) The
About these solutions
The solutions that follow were prepared by Jackie Miller, although some of these have their
roots in solutions prepared by Bill Notz, David Moore, and Darryl Nester, for the previous
editions, and modified (or not) by me for the seventh edition. Should you discover any errors or
have any comments about these solutions (or the odd answers, in the back of the text), please
report them to me:
Jackie Miller
The Ohio State University
404 Cockins Hall
1958 Neil Avenue
Columbus, OH 43210
e-mail:
In putting these solutions together, I used the following software:
• For typesetting: Microsoft Word (with MathType6 by Design Science)
• For graphs: Adobe Illustrator
• For statistical analysis: Minitab, Excel, and Texas Instruments graphing calculators (especially
the TI-84). Some of the solutions in later chapters include output from Minitab.
Using the table of random digits
Grading SRSs chosen from Table A is complicated by the fact that students can find some
creative ways to (mis)use the table. Some approaches are not mistakes, but may lead to different
students having different “right” answers. Correct answers will vary based on:
• the line in the table on which they begin (you may want to specify one, if the text does
not).
• whether they start with, e.g., 00 or 01.
• whether they assign labels across the rows or down the columns (nearly all lists in the
text are alphabetized down the columns).
Some approaches can potentially lead to wrong answers. Mistakes to watch out for include the
following:
• They may forget that all labels must be the same length (e.g., assigning labels like 0, 1, 2,
..., 9, 10,... rather than 00, 01, 02, ...).
• In assigning multiple labels, they may not give the same number of labels to all units. For
example, if there are 30 units, they may try to use up all the two-digit numbers, thus
assigning four labels to the first 10 units and only three to the remaining 20.
As an alternative to using the random digits in Table A, students can pick a random sample by
generating (pseudo)random numbers. Many, if not all, calculators have a seed value that
,294 Solutions to Exercises
determines the sequence of random numbers that it produces. Rather than pointing students to a
particular line of Table A, you could specify a seed value for generating random numbers, so that
all students would obtain the same results (if all are using the same calculator).
On a TI-84, for example, after executing the command 0→rand, the rand command will produce
the sequence (rounded to four decimals) 0.9436, 0.9083, 0.1467, ..., while 1→rand initiates the
sequence 0.7456, 0.8559, 0.2254, .... So to choose, say, an SRS of size 10 from 30 subjects, use
the command 0→rand to set the seed, and then type 1+30∗rand, and press ENTER repeatedly.
Ignoring the decimal portion of the resulting numbers, this produces the sample
29, 28, 5, 16, 13, 23, 2, 11, 30, 7
(Generally, to generate random numbers from 1 to n, use the command 1+ n∗rand and ignore the
decimal portion of the result.)
Using statistical software
While Statistics: Concepts and Controversies is written so that most problems can be done with
a calculator capable of finding means and standard deviations, the use of more sophisticated
calculators or computer software (if readily available) can make some computations, especially
in later chapters, much more convenient. In choosing technology, be aware of the following
considerations:
• Standard deviations: Students may easily get confused by software that gives both the so-
called “sample standard deviation” (the one used in the text) and the “population standard
deviation” (dividing by n rather than n – 1). Symbolically, the former is usually given as s
and the later as σ (sigma), but the distinction is not always clear. For example, many
computer spreadsheets have a command such as “STDEV(...)” to compute a standard
deviation, but you may need to check the manual to find out which kind it is. As a quick
check: for the numbers 1, 2, 3, s = 1 while σ = 0.8165. In general, if two values are given,
the larger one is s and the smaller is σ. If only one value is given, and it is the wrong one,
use the relationship .
• Stemplots: In creating a stemplot, some software packages might round to the nearest
integer, for example, or might simply truncate. For example, if our data set ranges from
20 to 99, then using the digits 2 through 9 as stems, the number 59.6 might be represented
as either a 0 on stem 6 (with rounding) or a 9 on stem 5 (using truncation). Minitab opts
for truncation over rounding, so all the solutions in this guide show truncated-data
stemplots (except for exercises that instructed students to round). This usually makes
little difference in the overall appearance of the stemplot.
I have come to prefer truncation over rounding for two reasons. First, there is less
chance of making a mistake using truncation (if doing the task by hand). More important,
truncating keeps the shape of the stemplot similar to the shape of a histogram of the same
data. In the example mentioned above, if we were to make a histogram, with intervals 20
,Solutions Part I: Chapters 1 to 9 295
to 29.9, 30 to 39.9, etc. (a fairly natural choice), the number 59.6 would fall in the 50s
interval, so the truncated stemplot (which placed the leaf 9 on the stem 5) would be the
best match for the histogram.
• Quartiles and five-number summaries: Methods of computing quartiles vary among
different packages. Some use the approach given in the text (that is, Q1 is the median of
all the numbers below the location of the overall median, etc.), while others use a more
complicated approach. For the numbers 1, 2, 3, 4, for example, we would want Q1 = 1.5
and Q3 = 2.5, but Minitab reports these as 1.25 and 2.75, respectively.
In these solutions (and the odd-numbered answers in the back of the text), I opted to
report five-number summaries as they would be found using the text’s method.
• Boxplots: Some programs that draw boxplots use the convention that the “whiskers”
extend to the lower and upper deciles (the 10th and 90th percentiles) rather than to the
minimum and maximum. The interpretation of the plot is essentially unchanged, but be
prepared to adapt to the software available to your students.
While the decile method is merely different from that given in the text, some methods
are (in the opinions of some) just plain wrong. Some early graphing calculators drew
“box charts,” which have a centerline at the mean (not the median), and a box extending
from to . I know of no statistics text that uses that method, and I hope that
such graphing calculators are no longer manufactured (or used).
, Solutions Part I: Chapters 1 to 9 297
6.1 Part I Solutions
Chapter 1 Solutions
1.1. Population: The population is not explicitly defined. Presumably it is all adults in the United
States. However, because this is the National Annenberg Election Survey, it may be that the
population is intended to be all adults in the United States who are registered voters. Sample:
The sample is the 1,345 randomly selected adults.
1.2. This is an observational study. The research examined student comments but no treatment
was applied to the students.
1.3. (a) The individuals are cars (or “motor vehicles”). (b) The variables are make/model, vehicle
type, transmission type, number of cylinders, city MPG, and highway MPG. The last three of
these are numerical.
1.4. (a) The individuals are baseball players. (b) There are four variables: team, position, age,
salary. The last two of these are numerical. (c) Age is given in years; salary is given in thousands
of dollars per year. Beckett’s annual salary was $6,667,000 (almost 6.7 million dollars).
1.5. Answers may vary. One possibility is whether a household participates in the recycling
program. Another possibility would be the percentage of a household’s recyclable waste that is
actually recycled. (This may sound like a good variable until one actually thinks about how one
would gather such information ....)
Note: The variable here is not, e.g., “the percentage of households participating,”
because that is not a characteristic of an individual.
1.6. The population is pregnant and breast-feeding women. The sample consists of the 21 women
who returned the surveys. Only 21/340*100% = 6.2% of the women who were contacted
responded.
1.7. The variable was if a person favors the death penalty for a person convicted of murder. The
population is probably adult Americans. The sample was 1,010 adult Americans.
1.8. It is not an experiment; it is a sample survey in which information is gathered without
imposing any treatment. The variables are gender and political party voted for in the last
congressional election.
1.9. Exact descriptions of the populations may vary. (a) All adult, presumably all adult
Americans. (b) All video adapter cables in the lot. (c) All U.S. households.
1.10. Exact descriptions of the populations may vary. (a) Teenagers (or “tenth-graders,” but from
the description of the situation, the researcher would like information about all teens). (b) The
