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Book Title: eTextbook: Elementary Statistics
2.6. Interpreting and Understanding Standard Deviation
Section 2.6 Exercises
Section 2.6 Exercises
2.139 Instructions for an essay assignment include the statement “The
length is to be within 25 words of 200.” What values of x, number of words,
satisfy these instructions?
SHOW ANSWER
2.140 The empirical rule indicates that we can expect to find what
proportion of the sample included between the following:
a. x̅ − s and x̅ + s
b. x̅ − 2s and x̅ + 2s
c. x̅ − 3s and x̅ + 3s
2.141 Why is it that the z-score for a value that belongs to a normal
distribution usually lies between −3 and + 3?
SHOW ANSWER
2.142 The mean lifetime of a certain tire is 30,000 miles and the standard
deviation is 2500 miles.
a. If we assume the mileages are normally distributed, approximately
what percentage of all such tires will last between 22,500 and 37,500
miles?
100
b. If we assume nothing about the shape of the distribution,
approximately what percentage of all such tires will last between
22,500 and 37,500 miles?
2.143 The average cleanup time for a crew of a mediumsize firm is 84.0
hours and the standard deviation is 6.8 hours. Assume the empirical rule is
appropriate.
a. What proportion of the time will it take the cleanup crew 97.6 hours or
more to clean the plant?
SHOW ANSWER
b. Within what interval will the total cleanup time fall 95% of the time?
SHOW ANSWER
2.144
a. What proportion of a normal distribution is greater than the mean?
b. What proportion is within 1 standard deviation of the mean?
c. What proportion is greater than a value that is 1 standard deviation
below the mean?
2.145 Using the empirical rule, determine the approximate percentage of a
normal distribution that is expected to fall within the interval described.
a. Less than the mean
SHOW ANSWER
b. Greater than 1 standard deviation above the mean
SHOW ANSWER
c. Less than 1 standard deviation above the mean
SHOW ANSWER
d. Between 1 standard deviation below the mean and 2 standard
deviations above the mean
SHOW ANSWER
2.146 According to the empirical rule, almost all the data should lie
between ( x̅ − 3s ) and ( x̅ + 3s ). The range accounts for all the data.
a. What relationship should hold (approximately) between the standard
deviation and the range?
b. How can you use the results of part a to estimate the standard
deviation in situations when the range is known?
2.147 Chebyshev’s theorem guarantees that what proportion of a
distribution will be included between the following:
a. x̅ − 2s and x̅ + 2s
SHOW ANSWER
b. x̅ − 3s and x̅ + 3s
SHOW ANSWER
2.148 According to Chebyshev’s theorem, what proportion of a distribution
will be within k = 4 standard deviations of the mean?
2.149 Chebyshev’s theorem can be stated in an equivalent form to that
given. For example, to say “at least 75% of the data fall within 2 standard
deviations of the mean” is equivalent to stating “at most, 25% will be more
than 2 standard deviations away from the mean.”
a. At most, what percentage of a distribution will be 3 or more standard
deviations from the mean?
SHOW ANSWER
b. At most, what percentage of a distribution will be 4 or more standard
deviations from the mean?
SHOW ANSWER
2.150 The scores achieved by students in America make the news often,
and all kinds of conclusions are drawn based on these scores. The ACT
Assessment® is designed to assess high school students’ general
educational development and their ability to complete college-level work.
One of the categories tested is Science Reasoning. The mean ACT test score
for all high school graduates in 2008 in Science Reasoning was 20.8 with a
standard deviation of 4.6.
a. According to Chebyshev’s theorem, at least what percent of high school
graduates’ ACT scores in Science Reasoning were between 11.6 and
30.0?
b. If we know that ACT scores are normally distributed, what percent of
ACT Science Reasoning scores were between 11.6 and 30.0?
2.151 According to the U.S. Census Bureau, approximately 45% of the 29
million 18- to 24-year-olds in the United States are enrolled in college. To
more accurately gauge these young voters, an Edgewood College, Madison,
WI, professor conducted a national college on-campus survey of 945 18- to
24-year-olds at 29 colleges in October 19–21, 2004. The survey analyzed
which sources of information most influenced students’ votes. Based upon
a scale of 0 to 100, with 100 being most influential, students said the top
influence was the presidential debates ( mean = 69.05 , standard deviation
= 31.65 ).
a. According to Chebyshev’s theorem, at least what percent of the scores
are between 5.75 and 132.35?
SHOW ANSWER
b. If it is known that these scores are normally distributed, what percent
of these scores are between 5.75 and 132.35?
SHOW ANSWER
c. Explain why the relationship between the interval bounds in parts a
and b and the mean and standard deviation given in the question
suggest that the distribution of scores is not normally distributed.
Include specifics.
2.152 [EX02-152] On the first day of class last semester, 50 students were
asked for the one-way distance from home to college (to the nearest mile).
The resulting data follow:
101
a. Construct a grouped frequency distribution of the data by using 1–4
as the first class.
b. Calculate the mean and the standard deviation.
c. Determine the values of x̅ ± 2s , and determine the percentage of data
within 2 standard deviations of the mean.
2.153 [EX02-153] The Labor Department issued the February 2009 state-bystate unemployment report, and it shows continued declines in the job
market. These are the February 2009 unemployment rates for the 50 states
and DC.
Source: http://blogs.wsj.com/
a. Construct a histogram.
SHOW ANSWER
b. Does the histogram suggest an approximately normal distribution?
c. Find the mean and standard deviation.
SHOW ANSWER
d. Find the percentage of data falling within the three different intervals
about the mean and compare them to the empirical rule. Do the
percentages and the empirical rule agree with your answer to part b?
Explain.
SHOW ANSWER
e. Utilize one of the “testing for normality” Technology Instructions.
Compare the results with your answer to part d.
2.154 [EX02-154] One of the many things the U.S. Census Bureau reports to
the public is the increase in population for various geographic areas within
the country. The percents of increase in population for the 100 fastestgrowing counties with 10,000 or more population in 2008 in the United
States from April 1, 2008, to July 1, 2008, are listed in the following table.
Source: http://www.census.gov/
a. Calculate the mean and standard deviation.
b. Determine the values of x̅ ± 2s and x̅ ± 3s , and determine the
percentage of data within 2 and 3 standard deviations of the mean.
c. Do the percentages found in part b agree with the empirical rule?
What does that mean?
d. Do the percentages found in part b agree with Chebyshev’s theorem?
What does that mean?
e. Construct a histogram and one other graph of your choice. Does the
graph show a distribution that agrees with your answers in parts c and
d? Explain.
f. Utilize one of the “testing for normality” Technology Instructions.
Compare the results with your answer to part c.
2.155 [EX02-155] Each year, NCAA college football fans like to learn about
the up-and-coming freshman class of players. Following are the heights (in
inches) of the nation’s top 100 high school football players for 2009.
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77
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72
74
78
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76
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79
77
74
78
Source: http://www.takkle.com/
a. Construct a histogram and one other graph of your choice that display
the distribution on heights.
SHOW ANSWER
b. Calculate the mean and standard deviation.
SHOW ANSWER
c. Sort the data into a ranked list.
SHOW ANSWER
d. Determine the values of x̅ ± s ; x̅ ± 2s and x̅ ± 3s, and determine the
percentage of data within 1, 2, and 3 standard deviations of the mean.
SHOW ANSWER
e. Do the percentages found in part d agree with the empirical rule?
What does this imply? Explain.
SHOW ANSWER
f. Do the percentages found in part d agree with Chebyshev’s theorem?
What does that mean?
g. Does the graph show a distribution that agrees with your answers in
part e? Explain.
h. Utilize one of the “testing for normality” Technology Instructions.
Compare the results with your answer to part e.
2.156 [EX02-156] Each year, NCAA college football fans like to learn about
the size of the players in the current year’s recruit class. Following are the
weights (in pounds) of the nation’s top 100 high school football players for
2009.
Weight in Pounds
179
226
210
205
225
*** For remainder of data, logon at cengagebrain.com
Source: http://www.takkle.com/
102
a. Construct a histogram and one other graph of your choice that displays
the distribution of weights.
b. Calculate the mean and standard deviation.
c. Sort the data into a ranked list.
d. Determine the values of x̅ ± s , x̅ ± 2s and x̅ ± 3s and determine the
percentage of data within 1, 2, and 3 standard deviations of the mean.
e. Do the percentages found in part d agree with the empirical rule?
What does this imply? Explain.
f. Do the percentages found in part d agree with Chebyshev’s theorem?
What does that mean?
g. Do the graphs show a distribution that agrees with your answers in
part e? Explain.
h. Utilize one of the “testing for normality” Technology Instructions.
Compare the results with your answer to part e.
2.157 The empirical rule states that the 1, 2, and 3 standard deviation
intervals about the mean will contain 68%, 95%, and 99.7%, respectively.
a. Use the computer or calculator commands to randomly generate a
sample of 100 data from a normal distribution with mean 50 and
standard deviation 10. Construct a histogram using class boundaries
that are multiples of the standard deviation 10; that is, use boundaries
from 10 to 90 in intervals of 10 (see the commands). Calculate the
mean and the standard deviation using the commands found; then
inspect the histogram to determine the percentage of the data that fell
within each of the 1, 2, and 3 standard deviation intervals. How closely
do the three percentages compare to the percentages claimed in the
empirical rule?
b. Repeat part a. Did you get results similar to those in part a? Explain.
c. Consider repeating part a several more times. Are the results similar
each time? If so, in what way?
d. What do you conclude about the truth of the empirical rule?
2.158 Chebyshev’s theorem states that “at least
” of the data of a
distribution will lie within k standard deviations of the mean.
a. Use the computer commands to randomly generate a sample of 100
data from a uniform (nonnormal) distribution that has a low value of 1
and a high value of 10. Construct a histogram using class boundaries of
0 to 11 in increments of 1 (see the commands). Calculate the mean and
the standard deviation using the commands found; then inspect the
histogram to determine the percentage of the data that fell within each
of the 1, 2, 3, and 4 standard deviation intervals. How closely do these
percentages compare to the percentages claimed in Chebyshev’s
theorem and in the empirical rule?
b. Repeat part a. Did you get results similar to those in part a? Explain.
c. Consider repeating part a several more times. Are the results similar
each time? If so, in what way are they similar?
d. What do you conclude about the truth of Chebyshev’s theorem and the
empirical rule?
Book Title: eTextbook: Elementary Statistics
3.1. Bivariate Data
Section 3.1 Exercises
Section 3.1 Exercises
3.1 [EX03-001] Refer to “Weighing Your Fish with a Ruler” to answer the
following questions:
a. Is there a relationship (pattern) between the two variables length of a
rainbow trout and weight of a rainbow trout? Explain why or why not.
SHOW ANSWER
b. Do you think it is reasonable (or possible) to predict the weight of a
rainbow trout based on the length of the rainbow trout? Explain why
or why not.
SHOW ANSWER
3.2
a. Is there a relationship between a person’s height and shoe size as he or
she grows from an infant to age 16? As one variable gets larger, does
the other also get larger? Explain your answers.
b. Is there a relationship between height and shoe size for people who
are older than 16 years of age? Do taller people wear larger shoes?
Explain your answers.
3.3 [EX03-003] In a national survey of 500 business and 500 leisure
travelers, each was asked where he or she would most like “more space.”
a. Express the table as percentages of the total.
SHOW ANSWER
b. Express the table as percentages of the row totals. Why might one
prefer the table to be expressed this way?
SHOW ANSWER
c. Express the table as percentages of the column totals. Why might one
prefer the table to be expressed this way?
SHOW ANSWER
3.4 The “In the eye of the beholder” graphic shows two circle graphs, each
with four sections. This same information could be represented in the form
of a 2 × 4 contingency table of two qualitative variables.
a. Identify the population and name the two variables.
b. Construct the contingency table using entries of percentages based on
row totals.
131
Figure for Exercise 3.4 In the Eye of the Beholder
Source: Energizer online survey of 1,051 married adults, ages 44–62.
3.5 “The perfect age” graphic shows the results from 9 × 2 acontingency
table for one qualitative and one quantitative variable.
“The Perfect Age”
Source: Data from Cindy Hall and Genevieve Lynn, USA TODAY; IRC Research for Walt
Disney. © 1998 USA TODAY, reprinted by permission.
a. Identify the population and name the qualitative and quantitative
variables.
SHOW ANSWER
b. Construct a bar graph showing the two distributions side by side.
SHOW ANSWER
c. Does there seem to be a big difference between the genders on this
subject?
SHOW ANSWER
3.6 [EX03-006] The National Highway System Designation Act of 1995
allows states to set their own highway speed limits. Most of the states have
raised the limits. The November 2008 maximum speed limits on interstate
highways (rural) for cars and trucks by each state are given in the
following table (in miles per hour):
State
Cars
Trucks
Alabama
70
70
Alaska
65
65
Arizona
75
75
Arkansas
70
65
California
70
55
Colorado
75
75
Connecticut
65
65
Delaware
65
65
Florida
70
70
Georgia
70
70
Hawaii
60
60
Idaho
75
65
Illinois
65
55
Indiana
70
65
Iowa
70
70
Kansas
70
70
Kentucky
65
65
State
Cars
Trucks
Louisiana
70
70
Maine
65
65
Maryland
65
65
Massachusetts
65
65
Michigan
70
60
Minnesota
70
70
Mississippi
70
70
Missouri
70
70
Montana
75
65
Nebraska
75
75
Nevada
75
75
New Hampshire
65
65
New Jersey
65
65
New Mexico
75
75
New York
65
65
North Carolina
70
70
North Dakota
75
75
State
Cars
Trucks
Ohio
65
55
Oklahoma
75
75
Oregon
65
55
Pennsylvania
65
65
Rhode Island
65
65
South Carolina
70
70
South Dakota
75
75
Tennessee
70
70
Texas
75
65
Utah
75
75
Virginia
65
65
Vermont
65
65
Washington
70
60
West Virginia
70
70
Wisconsin
65
65
Wyoming
75
75
Source: American Trucking Association
a. Build a cross-tabulation of the two variables vehicle type and
maximum speed limit on interstate highways. Express the results in
frequencies, showing marginal totals.
b. Express the contingency table you derived in part a in percentages
based on the grand total.
c. Draw a bar graph showing the results from part b.
d. Express the contingency table you derived in part a in percentages
based on the marginal total for speed limit.
e. Draw a bar graph showing the results from part d.
If you are using a computer or a calculator, try the cross-tabulation
table commands.
3.7 [EX03-007] A statewide survey was conducted to investigate the
relationship between viewers’ preferences for ABC, CBS, NBC, PBS, or FOX
for news information and their political party affiliation. The results are
shown in tabular form:
a. How many viewers were surveyed?
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SHOW ANSWER
b. Why are these bivariate data? Name the two variables. What type of
variable is each one?
SHOW ANSWER
c. How many viewers preferred to watch CBS?
SHOW ANSWER
d. What percentage of the survey was Republican?
SHOW ANSWER
e. What percentage of the Democrats preferred ABC?
SHOW ANSWER
f. What percentage of the viewers were Republican and preferred PBS?
SHOW ANSWER
3.8 [EX03-008] Consider the accompanying contingency table, which
presents the results of an advertising survey about the use of credit by
Martan Oil Company customers.
a. How many customers were surveyed?
b. Why are these bivariate data? What type of variable is each one?
c. How many customers preferred to use an oil-company card?
d. How many customers made 20 or more purchases last year?
e. How many customers preferred to use an oil-company card and made
between five and nine purchases last year?
f. What does the 80 in the fourth cell in the second row mean?
3.9 [EX03-009] The June 2009 unemployment rates for eastern and western
U.S. states were as follows:
Source: U.S. Bureau of Labor Statistics
Display these rates as two dotplots using the same scale; compare means
and medians.
SHOW ANSWER
3.10 [EX03-010] What effect does the minimum amount have on the
interest rate being offered on 3-month certificates of deposit (CDs)? The
following are advertised rates of return, y, for a minimum deposit of $500,
$1000, $2500, $5000, or $10,000, x. (Note that x is in $100 and y is annual
percentage rate of return.)
Source: Bankrate.com, July 28, 2009
a. Prepare a dotplot of the five sets of data using a common scale.
b. Prepare a 5-number summary and a boxplot of the five sets of data.
Use the same scale for the boxplots.
c. Describe any differences you see among the three sets of data.
If you are using a computer or calculator for Exercise 3.10, try the
commands on Side-by-Side Boxplots and Dotplots.
3.11 [EX03-011] Can a woman’s height be predicted from her mother’s
height? The heights of some mother–daughter pairs are listed; x is the
mother’s height and y is the daughter’s height.
x
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y
63
65
65
65
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64
63
62
63
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x
64
63
64
64
63
67
61
65
64
65
y
64
64
65
65
62
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62
63
66
66
a. Draw two dotplots using the same scale and showing the two sets of
data side by side.
SHOW ANSWER
b. What can you conclude from seeing the two sets of heights as separate
sets in part a? Explain.
SHOW ANSWER
c. Draw a scatter diagram of these data as ordered pairs.
SHOW ANSWER
d. What can you conclude from seeing the data presented as ordered
pairs? Explain.
SHOW ANSWER
3.12 [EX03-012] The following tables list the ages, heights (in inches), and
weights (in pounds) of the players on the 2009 roster for the National
Hockey League teams Boston Bruins and Edmonton Oilers.
Boston Bruins
Edmonton Oilers
Age
Height
Weight
Age
Height
Weight
31
72
193
22
70
180
24
72
186
22
69
178
23
71
176
19
70
191
32
70
195
24
71
183
22
71
194
24
71
190
32
71
209
24
72
190
35
74
186
24
73
195
34
73
175
23
71
200
21
76
220
30
73
202
30
72
195
24
76
217
25
77
215
28
78
265
25
72
192
33
74
220
21
72
189
26
76
243
27
72
195
23
71
180
24
75
188
25
72
191
22
75
196
32
73
203
Boston Bruins
Edmonton Oilers
Age
Height
Weight
Age
Height
Weight
41
70
195
23
75
217
29
72
192
22
72
196
32
73
209
26
75
210
22
75
185
25
73
195
25
74
225
21
74
223
32
81
261
23
75
204
26
73
211
33
76
227
30
70
189
36
73
200
24
70
187
34
76
225
30
72
220
32
70
188
23
73
185
25
76
189
25
74
218
36
73
208
26
72
200
34
72
207
22
74
171
25
73
190
Boston Bruins
Edmonton Oilers
Age
Height
Weight
28
74
200
35
71
182
Age
Height
Weight
Source: Source: http://sports.espn.go.com
133
a. Compare each of the three variables—height, weight, and age—using
either a dotplot or a histogram (use the same scale).
b. Based on what you see in the graphs in part a, can you detect a
substantial difference between the two teams in regard to these three
variables? Explain.
c. Explain why the data, as used in part a, are not bivariate data.
3.13 Consider the two variables of a person’s height and weight. Which
variable, height or weight, would you use as the input variable when
studying their relationship? Explain why.
SHOW ANSWER
3.14 Draw a coordinate axis and plot the points (0, 6), (3, 5), (3, 2), and (5, 0)
to form a scatter diagram. Describe the pattern that the data show in this
display.
3.15 Does studying for an exam pay off?
a.
Draw a scatter diagram of the number of hours studied, x, compared
with the exam grade received, y.
b.
Explain what you can conclude based on the pattern of data shown
on the scatter diagram drawn in part a. (Retain these solutions to use
in Exercise 3.55.)
SHOW ANSWER
3.16 Refer to Figure 3.8 in “Americans Love Their Automobiles” (Applied
Example 3.4) to answer the following questions:
a. Name the two variables used.
b. Does the scatter diagram suggest a relationship between the two
variables? Explain.
c. What conclusion, if any, can you draw from the appearance of the
scatter diagram?
3.17 Growth charts are commonly used by a child’s pediatrician to monitor
a child’s growth. Consider the growth chart that follows.
a. What are the two variables shown in the graph?
SHOW ANSWER
b. What information does the ordered pair (3, 87) represent?
SHOW ANSWER
c. Describe how the pediatrician might use this chart and what types of
conclusions might be based on the information displayed by it.
SHOW ANSWER
3.18 [EX03-012]
a. Draw a scatter diagram showing height, x, and weight, y, for the Boston
Bruins hockey team, using the data in Exercise 3.12.
b. Draw a scatter diagram showing height, x, and weight, y, for the
Edmonton Oilers hockey team using the data in Exercise 3.12.
c. Explain why the data, as used in parts a and b, are bivariate data.
FYI
If you are using a computer or calculator, try the commands on
Scatter Diagram.
3.19 [EX03-019] The accompanying data show the number of hours, x,
studied for an exam and the grade received, y (y is measured in tens; that
is, y = 8 means that the grade, rounded to the nearest 10 points, is 80).
Draw the scatter diagram. (Retain this solution to use in Exercise 3.37.)
x
2
3
3
4
4
5
5
6
6
6
7
7
y
5
5
7
5
7
7
8
6
9
8
7
9
SHOW ANSWER
3.20 [EX03-020] An experimental psychologist asserts that the older a child
is, the fewer irrelevant answers he or she will give during a controlled
experiment. To investigate this claim, the following data were collected.
Draw a scatter diagram. (Retain this solution to use in Exercise 3.38.)
134
3.21 [EX03-021] A sample of 15 upper-class students who commute to
classes was selected at registration. They were asked to estimate the
distance (x) and the time (y) required to commute each day to class (see the
following table).
a. Do you expect to find a linear relationship between the two variables
commute distance and commute time? If so, explain what relationship
you expect.
SHOW ANSWER
b. Construct a scatter diagram depicting these data.
SHOW ANSWER
c. Does the scatter diagram in part b reinforce what you expected in part
a?
SHOW ANSWER
3.22 [EX03-022] Refer to the 2009 4-wheel-drive, 6-cylinder SUVs chart in
Applied Example 3.4 and the two variables gas tank capacity, x, and the
cost to fill it, y.
a. If you were to draw scatter diagrams of these two variables, on the
same graph but separate, for the SUVs that use regular and premium
gasoline, do you think the two sets of data would be distinguishable?
Explain what you anticipate seeing.
b. Construct a scatter diagram of tank capacity, x, and fill-up cost, y, for
the SUVs using regular gasoline.
c. Construct a scatter diagram of tank capacity, x, and fill-up cost, y, for
the SUVs using premium gasoline on the scatter diagram for part b.
d. Are the two sets of data distinguishable?
e. How does your answer in part a compare to your answer in part d?
Explain any difference.
3.23 [EX03-023] Baseball stadiums vary in age, style and size, and many
other ways. Fans might think of the size of a stadium in terms of the
number of seats, while players might measure the size of a stadium in
terms of the distance from home plate to the centerfield fence.
Seats
CF
38,805
420
41,118
400
56,000
400
45,030
400
34,077
400
40,793
400
56,144
408
50,516
400
40,615
400
48,190
406
36,331
434
43,405
405
48,911
400
50,449
415
50,091
400
43,772
404
49,033
407
Seats
CF
47,447
405
40,120
422
41,503
404
40,950
435
38,496
400
41,900
400
42,271
404
43,647
401
42,600
396
46,200
400
41,222
403
52,355
408
45,000
408
CF = distance from home plate to centerfield fence Source: Source:
http://mlb.mlb.com
Is there a relationship between these two measurements of the “size” of
the 30 Major League Baseball stadiums?
a. What do you think you will find? Bigger fields have more seats?
Smaller fields have more seats? No relationship between field size and
number of seats? A strong relationship between field size and number
of seats? Explain.
SHOW ANSWER
b. Construct a scatter diagram.
SHOW ANSWER
c. Describe what the scatter diagram tells you, including a reaction to
your answer in part a.
SHOW ANSWER
3.24 [EX03-024] Most adult Americans drive. But do you have any idea how
many licensed drivers there are in each U.S. state? The table here lists the
number of male and female drivers licensed in each of 15 randomly
selected U.S. states during 2007.
Source: Federal Highway Admin., U.S. Dept. of Transportation
a. Do you expect to find a linear (straight-line) relationship between
number of male and number of female licensed drivers per state? How
strong do you anticipate this relationship to be? Describe.
b. Construct a scatter diagram using x for the number of male drivers
and y for the number of female drivers.
c. Compare the scatter diagram to your expectations in part a. How did
you do? Explain.
d. Are there data points that look like they are separate from the pattern
created by the rest of ordered pairs? If they were removed from the
dataset, would the results change? What caused these point(s) to be
135
separate from the others, but yet still be part of the extended pattern?
Explain.
e. Use the dataset for all 51 states to construct a scatter diagram.
Compare the pattern of the sample of 15 to the pattern shown by all 51.
Describe in detail.
f. Did the sample provide enough information for you to understand the
relationship between the two variables in this situation? Explain.
3.25 [EX03-025] Ronald Fisher, an English statistician (1890–1962),
collected measurements for a sample of 150 irises. Of concern were five
variables: species, petal width (PW), petal length (PL), sepal width (SW),
and sepal length (SL) (all in mm). Sepals are the outermost leaves that
encase the flower before it opens. The goal of Fisher’s experiment was to
produce a simple function that could be used to classify flowers correctly.
A random sample of his complete dataset is given in the accompanying
table.
Type
PW
PL
SW
SL
0
2
15
35
52
2
18
48
32
59
1
19
51
27
58
0
3
13
35
50
0
3
15
38
51
2
12
44
26
55
1
20
64
38
79
2
15
49
31
69
2
15
45
29
60
2
12
39
27
58
1
22
56
28
64
1
13
52
30
67
0
2
14
29
44
2
16
51
27
60
0
5
17
33
51
1
24
51
28
58
1
19
50
25
63
Type
PW
PL
SW
SL
0
1
15
31
49
1
23
59
32
68
2
13
44
23
63
2
15
42
30
59
1
25
57
33
67
1
21
57
33
67
0
2
15
37
54
1
18
49
27
63
1
17
45
25
49
1
24
56
34
63
0
2
14
36
50
2
10
50
22
60
0
2
12
32
50
a. Construct a scatter diagram of petal length, x, and petal width, y. Use
different symbols to represent the three species.
SHOW ANSWER
b. Construct a scatter diagram of sepal length, x, and sepal width, y. Use
different symbols to represent the three species.
SHOW ANSWER
c. Explain what the scatter diagrams in parts a and b portray.
Let’s see how well a random sample represents the data from which it
was selected.
SHOW ANSWER
d. Repeat parts a and b using the dataset containing all 150 of Fisher’s
data on [EX03-025]
SHOW ANSWER
e. Aside from the fact that the scatter diagrams in parts a and b have
fewer data, comment on the similarities and differences between the
distributions shown for 150 data and for the 30 randomly selected
data.
3.26 [EX03-026] Total solar eclipses actually take place nearly as often as
total lunar eclipses, but the former are visible over a much narrower path.
Both the path width and the duration vary substantially from one eclipse
to the next. The table below shows the duration (in seconds) and path
width (in miles) of 44 total solar eclipses measured in the past and those
projected to the year 2010:
Date
Duration (s)
Width (mi)
1950
73
83
1952
189
85
1954
155
95
1955
427
157
1956
284
266
1958
310
129
1959
181
75
1961
165
160
1962
248
91
1963
99
63
1965
315
123
1966
117
52
1968
39
64
1970
207
95
1972
155
109
1973
423
159
1974
308
214
Date
Duration (s)
Width (mi)
1976
286
123
1977
157
61
1979
169
185
1980
248
92
1981
122
67
1983
310
123
1984
119
53
1985
118
430
1986
1
1
1987
7
3
1988
216
104
1990
152
125
1991
413
160
1992
320
182
1994
263
117
1995
129
48
1997
170
221
Date
Duration (s)
Width (mi)
1998
248
94
1999
142
69
2001
296
125
2002
124
54
2003
117
338
2005
42
17
2006
247
114
2008
147
144
2009
399
160
2010
320
160
Source: Source:The World Almanac and Book of Facts 1998.
a. Draw a scatter diagram showing duration, y, and path width, x, for the
total solar eclipses.
b. How would you describe this diagram?
c. The durations and path widths for the years 2006–2009 were
projections. The recorded values were:
Year
Path Width
Duration
2006
65 miles
247 sec
2008
147 miles
147 sec
2009
160 miles
399 sec
Compare the recorded values to the projections. Comment on accuracy.
136
Book Title: eTextbook: Elementary Statistics
3.2. Linear Correlation
Section 3.2 Exercises
Section 3.2 Exercises
3.27 Skillbuilder Applet Exercise provides scatter diagrams for various
correlation coefficients.
a. Starting at r = 0 , move the slider to the right until r = 1 . Explain what
is happening to the corresponding scatter diagrams.
SHOW ANSWER
b. Starting at r = 0 , move the slider to the left until r = −1 . Explain what
is happening to the corresponding scatter diagrams.
SHOW ANSWER
3.28 How would you interpret the findings of a correlation study that
reported a linear correlation coefficient of −1.34?
3.29 How would you interpret the findings of a correlation study that
reported a linear correlation coefficient of +0.37?
SHOW ANSWER
3.30 Explain why it makes sense for a set of data to have a correlation
coefficient of zero when the data show a very definite pattern, as in Figure
3.11.
3.31 Does studying for an exam pay off? The number of hours studied, x, is
compared with the exam grade received, y:
x
2
5
1
4
2
y
80
80
70
90
60
a. Complete the preliminary calculations: extensions, five sums, SS(x),
SS(y), and SS(xy).
SHOW ANSWER
b. Find r.
SHOW ANSWER
3.32 [EX03-032] Cell phones and iPods are necessities for the current
generation. Does the use of one indicate the use of the other? Seven junior
high students who own both a cell phone and an iPod were randomly
selected, resulting in the following data:
a. Complete the preliminary calculations: extensions, five sums, and
SS(x), SS(y), and SS(xy).
b. Find r.
3.33 [EX03-033] Many organizations offer “special” magazine rates to their
members. The American Federation of Teachers is no different, and here
are a few of the rates they offer their members.
Source: AFT, Feb. 2009
a. a Construct a scatter diagram with “your price” as the dependent
variable, y, and “usual rate” as the independent variable, x.
SHOW ANSWER
b. Find:
SS(x)
SHOW ANSWER
c. SS(y)
SHOW ANSWER
d. SS(xy)
SHOW ANSWER
e. Pearson’s product moment, r
SHOW ANSWER
3.34 [EX03-034] A random sample of 10 seventh-grade students gave the
following data on x = number of minutes watching television on an
average weeknight versus of minutes spent on homework on an average
weeknight.
143
a. Construct a scatter diagram with “homework minutes” as the
dependent variable, y, and “television minutes” as the independent
variable, x.
Find:
b. SS(x)
c. SS(y)
d. SS(xy)
e. Pearson’s product moment, r
3.35 Manatees swim near the surface of the water. They often run into
trouble with the many powerboats in Florida. Consider the graph that
follows.
a. What two groups of subjects are being compared?
SHOW ANSWER
b. What two variables are being used to make the comparison?
SHOW ANSWER
c. What conclusion can one make based on this scatterplot?
SHOW ANSWER
d. What might you do if you were a wildlife official in Florida?
3.36 Estimate the correlation coefficient for each of the following:
3.37
a. Use the scatter diagram you drew in Exercise 3.19 to estimate r for the
sample data on the number of hours studied and the exam grade.
SHOW ANSWER
b. Calculate r.
SHOW ANSWER
3.38
a. Use the scatter diagram you drew in Exercise 3.20 to estimate r
