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Complete Solutions Manual
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ยฉ Cengage Learning. All rights reserved. No distribution allowed without express authorization.
Introduction to
Probability and Statistics
15th Edition
William Mendenhall, III
1925-2009
Robert J. Beaver
University of California, Riverside, Emeritus
Barbara M. Beaver
University of California, Riverside, Emerita
Prepared by
Barbara M. Beaver
Australia โข Brazil โข Mexico โข Singapore โข United Kingdom โข United States
ISBN-13: 978-1-337-55829-7
ISBN-10: 1-337-55829-X
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Contents
Chapter 1: Describing Data with Graphsโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆ…1
Chapter 2: Describing Data with Numerical Measuresโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆ….30
Chapter 3: Describing Bivariate Dataโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆ……….68
Chapter 4: Probabilityโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆ…………..93
Chapter 5: Discrete Probability Distributionโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆ..121
Chapter 6: The Normal Probability Distributionโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆ…156
Chapter 7: Sampling Distributionsโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆ.186
Chapter 8: Large-Sample Estimationโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆ.210
Chapter 9: Large-Sample Test of Hypothesesโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆ…240
Chapter 10: Inference from Small Samplesโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆ…271
Chapter 11: The Analysis of Varianceโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆ…324
Chapter 12: Linear Regression and Correlationโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆ.364
Chapter 13: Multiple Regression Analysisโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆ415
Chapter 14: The Analysis of Categorical Data………………………………………………………………..439
Chapter 15: Nonparametric Statisticsโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆ.469
1: Describing Data with Graphs
Section 1.1
1.1.1
The experimental unit, the individual or object on which a variable is measured, is the student.
1.1.2
The experimental unit on which the number of errors is measured is the exam.
1.1.3
The experimental unit is the patient.
1.1.4
The experimental unit is the azalea plant.
1.1.5
The experimental unit is the car.
1.1.6
โTime to assembleโ is a quantitative variable because a numerical quantity (1 hour, 1.5 hours, etc.) is
measured.
1.1.7
โNumber of studentsโ is a quantitative variable because a numerical quantity (1, 2, etc.) is measured.
1.1.8
โRating of a politicianโ is a qualitative variable since a quality (excellent, good, fair, poor) is measured.
1.1.9
โState of residenceโ is a qualitative variable since a quality (CA, MT, AL, etc.) is measured.
1.1.10
โPopulationโ is a discrete variable because it can take on only integer values.
1.1.11
โWeightโ is a continuous variable, taking on any values associated with an interval on the real line.
1.1.12
Number of claims is a discrete variable because it can take on only integer values.
1.1.13
โNumber of consumersโ is integer-valued and hence discrete.
1.1.14
โNumber of boating accidentsโ is integer-valued and hence discrete.
1.1.15
โTimeโ is a continuous variable.
1.1.16
โCost of a head of lettuceโ is a discrete variable since money can be measured only in dollars and cents.
1.1.17
โNumber of brothers and sistersโ is integer-valued and hence discrete.
1.1.18
โYield in bushelsโ is a continuous variable, taking on any values associated with an interval on the real
line.
1.1.19
The statewide database contains a record of all drivers in the state of Michigan. The data collected
represents the population of interest to the researcher.
1.1.20
The researcher is interested in the opinions of all citizens, not just the 1000 citizens that have been
interviewed. The responses of these 1000 citizens represent a sample.
1.1.21
The researcher is interested in the weight gain of all animals that might be put on this diet, not just the
twenty animals that have been observed. The responses of these twenty animals is a sample.
1.1.22
The data from the Internal Revenue Service contains the records of all wage earners in the United States.
The data collected represents the population of interest to the researcher.
1.1.23
a
The experimental unit, the item or object on which variables are measured, is the vehicle.
b
Type (qualitative); make (qualitative); carpool or not? (qualitative); one-way commute distance
(quantitative continuous); age of vehicle (quantitative continuous)
c
1.1.24
Since five variables have been measured, this is multivariate data.
a
The set of ages at death represents a population, because there have only been 38 different presidents
in the United States history.
b
The variable being measured is the continuous variable โageโ.
c
โAgeโ is a quantitative variable.
1
1.1.25
a
The population of interest consists of voter opinions (for or against the candidate) at the time of the
election for all persons voting in the election.
b
Note that when a sample is taken (at some time prior or the election), we are not actually sampling
from the population of interest. As time passes, voter opinions change. Hence, the population of voter
opinions changes with time, and the sample may not be representative of the population of interest.
1.1.26
a-b The variable โsurvival timeโ is a quantitative continuous variable.
c
The population of interest is the population of survival times for all patients having a particular type
of cancer and having undergone a particular type of radiotherapy.
d-e Note that there is a problem with sampling in this situation. If we sample from all patients having
cancer and radiotherapy, some may still be living and their survival time will not be measurable. Hence,
we cannot sample directly from the population of interest, but must arrive at some reasonable alternate
population from which to sample.
1.1.27
a
The variable โreading scoreโ is a quantitative variable, which is probably integer-valued and hence
discrete.
b
The individual on which the variable is measured is the student.
c
The population is hypothetical โ it does not exist in fact โ but consists of the reading scores for all
students who could possibly be taught by this method.
Section 1.2
1.2.1
The pie chart is constructed by partitioning the circle into five parts, according to the total contributed by
each part. Since the total number of students is 100, the total number receiving a final grade of A
represents 31 100 = 0.31 or 31% of the total. Thus, this category will be represented by a sector angle of
0.31(360) = 111.6๏ฐ . The other sector angles are shown next, along with the pie chart.
Final Grade
Frequency
Fraction of Total
Sector Angle
A
31
.31
111.6
B
36
.36
129.6
C
21
.21
75.6
D
9
.09
32.4
F
3
.03
10.8
D
9.0%
F
3.0%
A
31.0%
C
21.0%
B
36.0%
2
The bar chart represents each category as a bar with height equal to the frequency of occurrence of that
category and is shown in the figure that follows.
40
Frequency
30
20
10
0
A
B
C
D
F
Final Grade
1.2.2
Construct a statistical table to summarize the data. The pie and bar charts are shown in the figures that
follow.
Status
Frequency
Fraction of Total
Sector Angle
Freshman
32
.32
115.2
Sophomore
34
.34
122.4
Junior
17
.17
61.2
Senior
9
.09
32.4
Grad Student
8
.08
28.8
35
Grad Student
8.0%
30
Senior
9.0%
Freshman
32.0%
Frequency
25
Junior
17.0%
20
15
10
5
0
Sophomore
34.0%
1.2.3
Freshman
Sophomore
Junior
Senior
Grad Student
Status
Construct a statistical table to summarize the data. The pie and bar charts are shown in the figures that
follow.
Status
Humanities, Arts & Sciences
Natural/Agricultural Sciences
Business
Other
Frequency
43
32
17
8
3
Fraction of Total
.43
.32
.17
.08
Sector Angle
154.8
115.2
61.2
28.8
40
Frequency
other
8.0%
Business
17.0%
Humanities, Arts & Sciences
43.0%
30
20
10
0
Natural/Agricultural Sciences
32.0%
m
Hu
rt
,A
es
i ti
an
s&
es
nc
ie
Sc
N
es
nc
c ie
lS
a
r
tu
ul
ric
Ag
al/
r
u
at
s
es
sin
Bu
r
he
Ot
College
1.2.4
a
The pie chart is constructed by partitioning the circle into four parts, according to the total contributed
by each part. Since the total number of people is 50, the total number in category A represents
11 50 = 0.22 or 22% of the total. Thus, this category will be represented by a sector angle of
0.22(360) = 79.2o . The other sector angles are shown below. The pie chart is shown in the figure that
follows.
Category
Frequency
Fraction of Total
Sector Angle
A
11
.22
79.2
B
14
.28
100.8
C
20
.40
144.0
D
5
.10
36.0
D
10.0%
20
A
22.0%
Frequency
15
C
40.0%
10
5
B
28.0%
0
A
B
C
D
Category
b
The bar chart represents each category as a bar with height equal to the frequency of occurrence of
that category and is shown in the figure above.
c
Yes, the shape will change depending on the order of presentation. The order is unimportant.
d
The proportion of people in categories B, C, or D is found by summing the frequencies in those three
categories, and dividing by n = 50. That is, (14 + 20 + 5) 50 = 0.78 .
e
Since there are 14 people in category B, there are 50 โ14 = 36 who are not, and the percentage is
calculated as ( 36 50 )100 = 72% .
1.2.5
a-b Construct a statistical table to summarize the data. The pie and bar charts are shown in the figures
that follow.
4
State
CA
AZ
TX
Frequency
9
8
8
Fraction of Total
.36
.32
.32
Sector Angle
129.6
115.2
115.2
9
8
7
TX
32.0%
CA
36.0%
Frequency
6
5
4
3
2
1
0
AZ
32.0%
CA
AZ
TX
State
c
From the table or the chart, Texas produced 8 25 = 0.32 of the jeans.
d
The highest bar represents California, which produced the most pairs of jeans.
e
Since the bars and the sectors are almost equal in size, the three states produced roughly the same
number of pairs of jeans.
1.2.6-9 The bar charts represent each category as a bar with height equal to the frequency of occurrence of that
category.
Exercise 7
Exercise 6
90
70
80
60
70
50
Percent
Percent
60
50
40
30
40
30
20
20
10
10
0
Republicans
Independents
0
Democrats
18 to 34
35 to 54
55+
Age
Party ID
Exercise 8
Exercise 9
100
80
70
80
50
Percent
Percent
60
60
40
40
30
20
20
10
0
Republicans
Independents
0
Democrats
18 to 34
35 to 54
55+
Age
Party ID
1.2.10
Answers will vary.
1.2.11
a
The percentages given in the exercise only add to 94%. We should add another category called
โOtherโ, which will account for the other 6% of the responses.
b
Either type of chart is appropriate. Since the data is already presented as percentages of the whole
group, we choose to use a pie chart, shown in the figure that follows.
5
Too much arguing
5.0%
other
6.0%
Not good at it
14.0%
Other plans
40.0%
Too much work
15.0%
Too much pressure
20.0%
c-d Answers will vary.
1.2.12-14
The percentages falling in each of the four categories in 2017 are shown next (in parentheses), and the
pie chart for 2017 and bar charts for 2010 and 2017 follow.
Region
2010
2017
United States/Canada
99
183 (13.8%)
Europe
107
271 (20.4%)
Asia
64
453 (34.2%)
Rest of the World
58
419 (31.6%)
Total
328
1326 (100%)
Exercise 12 (2017)
U.S./Canada
13.8%
Rest of the World
31.6%
Europe
20.4%
Asia
34.2%
Exercise 14 (2017)
Exercise 13 (2010)
500
100
Average Daily Users (millions)
Average Daily Users (millions)
120
80
60
40
20
0
U.S./Canada
Europe
Asia
Rest of the World
Region
400
300
200
100
0
U.S./Canada
Europe
Asia
Region
6
Rest of the World
1.2.15
Users in Asia and the rest of the world have increased more rapidly than those in the U.S., Canada or
Europe over the seven-year period.
1.2.16
a
The total percentage of responses given in the table is only (40 + 34 + 19)% = 93% . Hence there are
7% of the opinions not recorded, which should go into a category called โOtherโ or โMore than a few
daysโ.
b
Yes. The bars are very close to the correct proportions.
c
Similar to previous exercises. The pie chart is shown next. The bar chart is probably more interesting
to look at.
Mor than a Few Days
7.0%
No Time
19.0%
One Day
40.0%
A Few Days
34.0%
1.2.17-18 Answers will vary from student to student. Since the graph gives a range of values for Zimbabweโs share,
we have chosen to use the 13% figure, and have used 3% in the โOtherโ category. The pie chart and bar
charts are shown next.
other
3.0%
25
Botswana
26.0%
20
Percent Share
Russia
20.0%
Canada
18.0%
Zimbabwe
13.0%
10
5
0
Angola
10.0%
South Africa
10.0%
15
Botswana
Zimbabwe
Angola
South Africa
Canada
Russia
Other
Country
1.2.19-20 The Pareto chart is shown below. The Pareto chart is more effective than the bar chart or the pie chart.
25
Percent Share
20
15
10
5
0
Botswana
Russia
Canada
Zimbabwe
Angola
South Africa
Other
Country
1.2.21
The data should be displayed with either a bar chart or a pie chart. The pie chart is shown next.
7
Green
1.0%
Beige/Brown
4.0%
Yellow/Gold
2.0%
other
1.0%
Silver
13.9%
White/White pearl
20.8%
Black/Black effect
20.8%
Red
10.9%
Gray
16.8%
Blue
8.9%
Section 1.3
1.3.1
The dotplot is shown next; the data is skewed right, with one outlier, x = 2.0.
1.0
1.2
1.4
1.6
1.8
2.0
Exercise 1
1.3.2
The dotplot is shown next; the data is relatively mound-shaped, with no outliers.
54
56
58
60
62
Exercise 2
1.3.3-5 The most obvious choice of a stem is to use the ones digit. The portion of the observation to the right of
the ones digit constitutes the leaf. Observations are classified by row according to stem and also within
each stem according to relative magnitude. The stem and leaf display is shown next.
1 6 8
2 1 2 5 5 5 7 8 8 9 9
3 1 1 4 5 5 6 6 6 7 7 7 7 8 9 9 9
leaf digit = 0.1
4 0 0 0 1 2 2 3 4 5 6 7 8 9 9 9
1 2 represents 1.2
5 1 1 6 6 7
6 12
3.
The stem and leaf display has a mound shaped distribution, with no outliers.
4.
From the stem and leaf display, the smallest observation is 1.6 (1 6).
5.
The eight and ninth largest observations are both 4.9 (4 9).
8
1.3.6
The stem is chosen as the ones digit, and the portion of the observation to the right of the ones digit is the
leaf.
3 |
2 3 4 5 5 5 6 6 7 9 9 9 9
4 |
0 0 2 2 3 3 3 4 4 5 8
leaf digit = 0.1 1 2 represents 1.2
1.3.7-8 The stems are split, with the leaf digits 0 to 4 belonging to the first part of the stem and the leaf digits 5 to 9
belonging to the second. The stem and leaf display shown below improves the presentation of the data.
3 | 2 3 4
3 | 5 5 5 6 6 7 9 9 9 9
leaf digit = 0.1 1 2 represents 1.2
4 | 0 0 2 2 3 3 3 4 4
4 | 5 8
1.3.9
The scale is drawn on the horizontal axis and the measurements are represented by dots.
0
1
2
Exercise 9
1.3.10
Since there is only one digit in each measurement, the ones digit must be the stem, and the leaf will be a
zero digit for each measurement.
0 | 0 0 0 0 0
1 | 0 0 0 0 0 0 0 0 0
2 | 0 0 0 0 0 0
1.3.11
The distribution is relatively mound-shaped, with no outliers.
1.3.12
The two plots convey the same information if the stem and leaf plot is turned 90 o and stretched to resemble
the dotplot.
1.3.13
The line chart plots โdayโ on the horizontal axis and โtimeโ on the vertical axis. The line chart shown next
reveals that learning is taking place, since the time decreases each successive day.
45
Time (seconds)
40
35
30
25
1
2
3
4
5
Day
1.3.14
The line graph is shown next. Notice the change in y as x increases. The measurements are decreasing
over time.
9
63
62
Measurement
61
60
59
58
57
56
0
2
4
6
8
10
Year
1.3.15
The dotplot is shown next.
1
2
3
4
5
6
7
Number of Cheeseburgers
1.3.16
a
The distribution is somewhat mound-shaped (as much as a small set can be); there are no outliers.
b
2 10 = 0.2
a
The test scores are graphed using a stem and leaf plot generated by Minitab.
b-c The distribution is not mound-shaped, but is rather has two peaks centered around the scores 65 and
85. This might indicate that the students are divided into two groups โ those who understand the material
and do well on exams, and those who do not have a thorough command of the material.
1.3.17
a
We choose a stem and leaf plot, using the ones and tenths place as the stem, and a zero digit as the
leaf. The Minitab printout is shown next.
10
Dotplot of Calcium
b
The data set is relatively mound-shaped, centered at 5.2.
c
The value x = 5.7 does not fall within the range of the other cell counts, and would be considered
somewhat unusual.
1.3.18
a-b The dotplot and the stem and leaf plot are drawn using Minitab.
2.68
2.70
2.72
2.74
2.76
2.78
2.80
2.82
Calcium
c
The measurements all seem to be within the same range of variability. There do not appear to be any
outliers.
1.3.19
a Stem and leaf displays may vary from student to student. The most obvious choice is to use the tens
digit as the stem and the ones digit as the leaf.
7| 8 9
8| 0 1 7
9| 0 1 2 4 4 5 6 6 6 8 8
10 | 1 7 9
11 | 2
b The display is fairly mound-shaped, with a large peak in the middle.
1.3.20
a
The sizes and volumes of the food items do increase as the number of calories increase, but not in the
correct proportion to the actual calories. The differences in calorie content are not accurately portrayed in
the graph.
b
The bar graph which accurately portrays the number of calories in the six food items is shown next.
11
900
800
Number of Calories
700
600
500
400
300
200
100
0
Hershey’s Kiss
Oreo
12 oz Coke
12 oz Beer
Pizza
Whopper
Food Item
1.3.21
a-b The bar charts for the median weekly earnings and unemployment rates for eight different levels of
education are shown next.
2000
8
Median wkly earnings
Unemployment rate
7
6
5
4
3
2
1500
1000
500
1
0
c
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Pr
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Educational Attainment
Educational Attainment
g
Hi
h
h
sc
L
ld
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l
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hi
gh
h
sc
ld
oo
l
ip
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om
c The unemployment rate drops and the median weekly earnings rise as the level of educational attainment
increases.
1.3.22
a Similar to previous exercises. The pie chart is shown next.
Judaism
0.2%
Chinese Traditional
6.8%
Sikhism
0.4%
other
1.1%
Buddhism
6.5%
Primal Indigenous & African Traditional
6.9%
Christianity
36.4%
Islam
26.0%
Hinduism
15.6%
b
The bar chart is shown next.
12
Members (millions)
2000
1500
1000
500
0
d
Bu
i
dh
sm
s ti
ri
Ch
i
an
ty
n
Hi
Pr
c
i
du
im
sm
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am
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da
Ju
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m
is
kh
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Ch
se
ne
on
iti
ad
Tr
al
h
Ot
er
Religion
The Pareto chart is a bar chart with the heights of the bars ordered from large to small. This display is more
effective than the pie chart.
Members (millions)
2000
1500
1000
500
0
ty
ni
tia
ris
h
C
Isl
am
I
al
im
Pr
n
u
nd
Hi
m
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it i
ad
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Ch
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h
dd
Bu
m
is
r
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Ot
m
his
Sik
Ju
m
is
da
Religion
1.3.23
a
The distribution is skewed to the right, with a several unusually large measurements. The five states
marked as HI are California, New Jersey, New York and Pennsylvania.
b
Three of the four states are quite large in area, which might explain the large number of hazardous
waste sites. However, New Jersey is relatively small, and other large states do not have unusually large
number of waste sites. The pattern is not clear.
1.3.24
a
The distribution is skewed to the right, with two outliers.
b
The dotplot is shown next. It conveys nearly the same information, but the stem-and-leaf plot may be
more informative.
13
1.4
2.8
4.2
5.6
7.0
8.4
9.8
Weekend Gross
1.3.25
a
Answers will vary.
b
The stem and leaf plot is constructed using the tens place as the stem and the ones place as the leaf.
Notice that the distribution is roughly mound-shaped.
c-d Three of the five youngest presidents โ Kennedy, Lincoln and Garfield โ were assassinated while in
office. This would explain the fact that their ages at death were in the lower tail of the distribution.
Section 1.4
1.4.1
The relative frequency histogram displays the relative frequency as the height of the bar over the
appropriate class interval and is shown next. The distribution is relatively mound-shaped.
.50
Relative Frequency
.40
.30
.20
.10
0
100
120
140
160
180
200
x
.
1.4.2
Since the variable of interest can only take integer values, the classes can be chosen as the values 0, 1, 2, 3,
4, 5 and 6. The table containing the classes, their corresponding frequencies and their relative frequencies
and the relative frequency histogram are shown next. The distribution is skewed to the right.
14
Number of Household Pets
0
1
2
3
4
5
6
Total
Frequency
13
19
12
4
1
0
1
50
Relative Frequency
13/50 = .26
19/50 = .38
12/50 = .24
4/50 = .08
1/50 = .02
0/50 = .00
1/50 = .02
50/50 = 1.00
.40
Relative Frequency
.30
.20
.10
0
0
1
2
3
4
5
6
Number of Pets
1.4.3-8 The proportion of measurements falling in each interval is equal to the sum of the heights of the bars over
that interval. Remember that the lower class boundary is included, but not the upper class boundary.
3. .20 + .40 + .15 = .75
4. .05 + .15 + .20 = .40
5. .05
6. .40 + .15 = .55
7. .15
8. .05 + .15 + .20 = .40
1.4.9
Answers will vary. The range of the data is 110 โ10 = 90 and we need to use seven classes. Calculate
90 / 7 = 12.86 which we choose to round up to 15. Convenient class boundaries are created, starting at 10:
10 to < 25, 25 to < 40, โฆ, 100 to < 115.
1.4.10
Answers will vary. The range of the data is 76.8 โ 25.5 = 51.3 and we need to use six classes. Calculate
51.3 / 6 = 8.55 which we choose to round up to 9. Convenient class boundaries are created, starting at 25:
25 to < 34, 34 to < 43, โฆ, 70 to < 79.
1.4.11
Answers will vary. The range of the data is 1.73 โ .31 = 1.42 and we need to use ten classes. Calculate
1.42 /10 = .142 which we choose to round up to .15. Convenient class boundaries are created, starting at
.30: .30 to < .45, .45 to < .60, โฆ, 1.65 to < 1.80.
1.4.12
Answers will vary. The range of the data is 192 โ 0 = 192 and we need to use eight classes. Calculate
192 / 8 = 24 which we choose to round up to 25. Convenient class boundaries are created, starting at 0: 0
to < 25, 25 to < 50, โฆ, 175 to < 200.
1.4.13-16 The table containing the classes, their corresponding frequencies and their relative frequencies and the
relative frequency histogram are shown next.
15
Class i
Class Boundaries
Tally
fi
Relative frequency, fi/n
1
2
3
4
5
1.6 to < 2.1
2.1 to < 2.6
2.6 to < 3.1
3.1 to < 3.6
3.6 to < 4.1
11
11111
11111
11111
11111 11111 1111
2
5
5
5
14
.04
.10
.10
.10
.28
6
7
8
9
10
4.1 to < 4.6
4.6 to < 5.1
5.1 to < 5.6
5.6 to < 6.1
6.1 to < 6.6
11111 11
11111
11
111
11
7
5
2
3
2
.14
.10
.04
.06
.04
Relative Frequency
.30
.20
.10
0
1.6
2.1
2.6
3.1
3.6
4.1
4.6
5.1
5.6
6.1
6.6
DATA
13. The distribution is roughly mound-shaped.
14. The fraction less than 5.1 is that fraction lying in classes 1-7, or ( 2 + 5 +
15. The fraction larger than 3.6 lies in classes 5-10, or (14 + 7 +
+ 7 + 5) 50 = 43 50 = 0.86 .
+ 3 + 2 ) 50 = 33 50 = 0.66 .
16. The fraction from 2.6 up to but not including 4.6 lies in classes 3-6, or
( 5 + 5 + 14 + 7 ) 50 = 31 50 = 0.62 .
1.4.17-20 Since the variable of interest can only take the values 0, 1, or 2, the classes can be chosen as the integer
values 0, 1, and 2. The table shows the classes, their corresponding frequencies and their relative
frequencies. The relative frequency histogram follows the table.
Value
0
1
2
Frequency
5
9
6
Relative Frequency
.25
.45
.30
16
0.5
Relative Frequency
0.4
0.3
0.2
0.1
0.0
0
1
2
17. Using the table above, the proportion of measurements greater than 1 is the same as the proportion of
โ2โs, or 0.30.
18. The proportion of measurements less than 2 is the same as the proportion of โ0โs and โ1โs, or
0.25 + 0.45 = .70 .
19. The probability of selecting a โ2โ in a random selection from these twenty measurements is 6 20 = .30 .
20. There are no outliers in this relatively symmetric, mound-shaped distribution.
1.4.21-23 Answers will vary. The range of the data is 94 โ 55 = 39 and we choose to use 5 classes. Calculate
39 / 5 = 7.8 which we choose to round up to 10. Convenient class boundaries are created, starting at 50 and
the table and relative frequency histogram are created.
Class Boundaries
50 to < 60
60 to < 70
70 to < 80
80 to < 90
90 to < 100
Frequency
2
6
3
6
3
Relative Frequency
.10
.30
.15
.30
.15
.30
Relative Frequency
.25
.20
.15
.10
.05
0
50
60
70
80
90
100
Scores
21. The distribution has two peaks at about 65 and 85. Depending on the way in which the student
constructs the histogram, these peaks may or may not be clearly seen.
17
22. The shape is unusual. It might indicate that the students are divided into two groups โ those who
understand the material and do well on exams, and those who do not have a thorough command of the
material.
23. The shapes are roughly the same, but this may not be the case if the student constructs the histogram
using different class boundaries.
1.4.24 a
There are a few extremely small numbers, indicating that the distribution is probably skewed to the
left.
b
The range of the data 165 โ 8 = 157 . We choose to use seven class intervals of length 25, with
subintervals 0 to < 25, 25 to < 50, 50 to < 75, and so on. The tally and relative frequency histogram are
shown next.
Class i
1
2
3
4
5
6
7
Class Boundaries
0 to < 25
25 to < 50
50 to < 75
75 to < 100
100 to < 125
125 to < 150
150 to < 175
Tally
11
111
111
11
11111 11
111
fi
2
0
3
3
2
7
3
Relative frequency, fi/n
2/20
0/20
3/20
3/20
2/20
7/20
3/20
.40
Relative Frequency
.30
.20
.10
0
0
25
50
75
100
125
150
175
Times
c
1.4.25
The distribution is indeed skewed left with two possible outliers: x = 8 and x = 11.
a
The range of the data 32.3 โ 0.2 = 32.1 . We choose to use eleven class intervals of length 3 (
32.1 11 = 2.9 , which when rounded to the next largest integer is 3). The subintervals 0.1 to < 3.1, 3.1 to <
6.1, 6.1 to < 9.1, and so on, are convenient and the tally and relative frequency histogram are shown next.
Class i
1
2
3
4
5
6
7
8
9
10
11
Class Boundaries
0.1 to < 3.1
3.1 to < 6.1
6.1 to < 9.1
9.1 to < 12.1
12.1 to < 15.1
15.1 to < 18.1
18.1 to < 21.1
21.1 to < 24.1
24.1 to < 37.1
27.1 to < 30.1
30.1 to < 33.1
Tally
11111 11111 11111
11111 1111
11111 11111
111
1111
111
11
11
1
1
18
fi
15
9
10
3
4
3
2
2
1
0
1
Relative frequency, fi/n
15/50
9/50
10/50
3/50
4/50
3/50
2/50
2/50
1/50
0/50
1/50
0.30
Relative Frequency
0.25
0.20
0.15
0.10
0.05
0
0.1
6.1
12.1
18.1
24.1
30.1
TIME
b
The data is skewed to the right, with a few unusually large measurements.
c
Looking at the data, we see that 36 patients had a disease recurrence within 10 months. Therefore, the
fraction of recurrence times less than or equal to 10 is 36 50 = 0.72 .
1.4.26
a
We use class intervals of length 5, beginning with the subinterval 30 to < 35. The tally and the relative
frequency histogram are shown next.
Class i
1
2
3
4
5
6
Class Boundaries
30 to < 35
35 to < 40
40 to < 45
45 to < 50
50 to < 55
55 to < 60
Tally
11111 11111 11
11111 11111 11111
11111 11111 11
11111 111
11
1
fi
12
15
12
8
2
1
Relative frequency, fi/n
12/50
15/50
12/50
8/50
2/50
1/50
.30
Relative Frequency
.25
.20
.15
.10
.05
0
30
35
40
45
50
55
60
Ages
b
Use the table or the relative frequency histogram. The proportion of children in the interval 35 to < 45
is (15 + 12)/50 = .54.
c
1.4.27
The proportion of children aged less than 50 months is (12 + 15 + 12 + 8)/50 = .94.
a
The data ranges from .2 to 5.2, or 5.0 units. Since the number of class intervals should be between
five and twelve, we choose to use eleven class intervals, with each class interval having length 0.50 (
5.0 11 = .45 , which, rounded to the nearest convenient fraction, is .50). We must now select interval
boundaries such that no measurement can fall on a boundary point. The subintervals .1 to < .6, .6 to < 1.1,
and so on, are convenient and a tally is constructed.
19
Class i
Class Boundaries
Tally
1
0.1 to < 0.6
11111 11111
2
0.6 to < 1.1
11111 11111 11111
3
1.1 to < 1.6
11111 11111 11111
4
1.6 to < 2.1
11111 11111
5
2.1 to < 2.6
1111
6
2.6 to < 3.1
1
7
3.1 to < 3.6
11
8
3.6 to < 4.1
1
9
4.1 to < 4.6
1
10
4.6 to < 5.1
11
5.1 to < 5.6
1
The relative frequency histogram is shown next.
fi
10
15
15
10
4
1
2
1
1
0
1
Relative frequency, fi/n
.167
.250
.250
.167
.067
.017
.033
.017
.017
.000
.017
.25
Relative Frequency
.20
.15
.10
.05
0
0.1
1.1
2.1
3.1
4.1
5.1
Times
b
The distribution is skewed to the right, with several unusually large observations.
c
For some reason, one person had to wait 5.2 minutes. Perhaps the supermarket was understaffed that
day, or there may have been an unusually large number of customers in the store.
1.4.28
a
Histograms will vary from student to student. A typical histogram generated by Minitab is shown
next.
.25
Relative Frequency
.20
.15
.10
.05
0
0.34
0.36
0.38
0.40
0.42
Batting Avg
b
1.4.29
Since 1 of the 20 players has an average above 0.400, the chance is 1 out of 20 or 1 20 = 0.05 .
a-b Answers will vary from student to student. The students should notice that the distribution is skewed
to the right with a few pennies being unusually old. A typical histogram is shown next.
20
.50
Relative Frequency
.40
.30
.20
.10
0
0
8
16
24
32
Age (Years)
1.4.30
a
Answers will vary from student to student. A typical histogram is shown next. It looks very similar to
the histogram from Exercise 1.4.29.
40
Relative Frequency
30
20
10
0
0
4
8
12
16
20
24
28
32
36
40
44
Age (Years)
b
1.4.31
There is one outlier, x = 41.
a
Answers will vary from student to student. The relative frequency histogram below was constructed
using classes of length 1.0 starting at x = 4 . The value x = 35.1 is not shown in the table but appears on
the graph shown next.
Class i
1
2
3
4
5
6
7
8
Class Boundaries
4.0 to < 5.0
5.0 to < 6.0
6.0 to < 7.0
7.0 to < 8.0
8.0 to < 9.0
9.0 to < 10.0
10.0 to < 11.0
11.0 to < 12.0
Tally
1
0
11111 1
11111 11111 11111
11111 111
11111 11111 111
11111 11
111
21
fi
1
0
6
15
8
13
7
3
Relative frequency, fi/n
1/54
0/54
6/54
15/54
8/54
13/54
7/54
3/54
.30
Relative Frequency
.25
.20
.15
.10
.05
0
5
10
15
20
25
30
35
Wind speed
b
Since Mt. Washington is a very mountainous area, it is not unusual that the average wind speed would
be very high.
c
The value x = 9.9 does not lie far from the center of the distribution (excluding x = 35.1 ). It would
not be considered unusually high.
1.4.32
a-b The data is somewhat mound-shaped, but it appears to have two local peaks โ high points from which
the frequencies drop off on either side.
c
Since these are student heights, the data can be divided into two groups โ heights of males and heights
of females. Both groups will have an approximate mound-shape, but the average female height will be
lower than the average male height. When the two groups are combined into one data set, it causes a
โmixtureโ of two mound-shaped distributions and produces the two peaks seen in the histogram.
1.4.33
a
The relative frequency histogram below was constructed using classes of length 1.0 starting at
x = 0.0 .
Class i
1
2
3
4
5
6
7
8
9
10
Class Boundaries
0.0 to < 1.0
1.0 to < 2.0
2.0 to < 3.0
3.0 to < 4.0
4.0 to < 5.0
5.0 to < 6.0
6.0 to < 7.0
7.0 to < 8.0
8.0 to < 9.0
9.0 to < 10.0
Tally
11
11
1
111
111
11111
111
11111
11111 111
11111 1
22
fi
2
2
1
3
4
5
3
5
8
6
Relative frequency, fi/n
2/39
2/39
1/39
3/39
4/39
5/39
3/39
5/39
8/39
6/39
Relative Frequency
.20
.15
.10
.05
0
0
2
4
6
8
10
Distance
a
The distribution is skewed to the left, with slightly higher frequency in the first two classes (within
two miles of UCR).
b
As the distance from UCR increases, each successive area increases in size, thus allowing for more
Starbucks stores in that region.
Reviewing What Youโve Learned
1.R.1
a
โEthnic originโ is a qualitative variable since a quality (ethnic origin) is measured.
b
โScoreโ is a quantitative variable since a numerical quantity (0-100) is measured.
c โType of establishmentโ is a qualitative variable since a category (Carlโs Jr., McDonaldโs or Burger
King) is measured.
d
1.R.2
โMercury concentrationโ is a quantitative variable since a numerical quantity is measured.
To determine whether a distribution is likely to be skewed, look for the likelihood of observing extremely
large or extremely small values of the variable of interest.
a
The price of an 8-oz can of peas is not likely to contain unusually large or small values.
b
Not likely to be skewed.
c If a package is dropped, it is likely that all the shells will be broken. Hence, a few large number of
broken shells is possible. The distribution will be skewed.
d If an animal has one tick, he is likely to have more than one. There will be some โ0โs with uninfected
rabbits, and then a larger number of large values. The distribution will not be symmetric.
1.R.3
a The length of time between arrivals at an outpatient clinic is a continuous random variable, since it can
be any of the infinite number of positive real values.
b The time required to finish an examination is a continuous random variable as was the random variable
described in part a.
1.R.4
c
Weight is continuous, taking any positive real value.
d
Body temperature is continuous, taking any real value.
e
Number of people is discrete, taking the values 0, 1, 2, โฆ
a
Number of properties is discrete, taking the values 0, 1, 2, โฆ
b
Depth is continuous, taking any non-negative real value.
c
Length of time is continuous, taking any non-negative real value.
d
Number of aircraft is discrete.
23
1.R.5
a
b
.25
Relative Frequency
.20
.15
.10
.05
0
50
100
150
200
250
300
350
400
Length
c
1.R.6
These data are skewed right.
a
The five quantitative variables are measured over time two months after the oil spill. Some sort of
comparative bar charts (side-by-side or stacked) or a line chart should be used.
b
As the time after the spill increases, the values of all five variables increase.
c-d The line chart for number of personnel and the bar chart for fishing areas closed are shown next.
35
30
20
25
Areas closed %
Number of Personnel (thousands)
25
15
10
20
15
10
5
5
0
10
20
30
40
0
50
13
Day
e
26
39
51
Day
The line chart for amount of dispersants is shown next. There appears to be a straight-line trend.
Dispersants Used (1000 gallons)
1200
1000
800
600
400
200
10
20
30
40
50
Day
1.R.7
a
The popular vote within each state should vary depending on the size of the state. Since there are
several very large states (in population) in the United States, the distribution should be skewed to the right.
b-c Histograms will vary from student to student but should resemble the histogram generated by Minitab
in the next figure. The distribution is indeed skewed to the right, with three โoutliersโ โ California, Florida
and Texas.
24
14/50
Relative Frequency
12/50
10/50
8/50
6/50
4/50
2/50
0
0
1000
2000
3000
4000
Popular vote
1.R.8
a-b Once the size of the state is removed by calculating the percentage of the popular vote, the unusually
large values in the Exercise 7 data set will disappear, and each state will be measured on an equal basis.
Student histograms should resemble the histogram shown next. Notice the relatively mound-shape and the
lack of any outliers.
10/50
Relative Frequency
8/50
6/50
4/50
2/50
0
30
40
50
60
70
Percentage
1.R.9
a-b Popular vote is skewed to the right while the percentage of popular vote is roughly mound-shaped.
While the distribution of popular vote has outliers (California, Florida and Texas), there are no outliers in
the distribution of percentage of popular vote. When the stem and leaf plots are turned 90 o, the shapes are
very similar to the histograms.
c
Once the size of the state is removed by calculating the percentage of the popular vote, the unusually
large values in the set of โpopular votesโ will disappear, and each state will be measured on an equal basis.
The data then distribute themselves in a mound-shape around the average percentage of the popular vote.
1.R.10 a
The measurements are obtained by counting the number of beats for 30 seconds, and then multiplying
by 2. Thus, the measurements should all be even numbers.
b
The stem and leaf plot is shown next.
25
c
Answers will vary. A typical histogram, generated by Minitab, is shown next.
.30
Relative Frequency
.25
.20
.15
.10
.05
0
40
50
60
70
80
90
100
110
Pulse
d
The distribution of pulse rates is mound-shaped and relatively symmetric around a central location of
75 beats per minute. There are no outliers.
1.R.11 a-b Answers will vary from student to student. A typical histogram is shown nextโthe distribution is
skewed to the right, with an extreme outlier (Texas).
.70
Relative Frequency
.60
.50
.40
.30
.20
.10
0
0
10000
20000
30000
40000
50000
60000
70000
Capacity
c
Answers will vary.
1.R.12 a-b Answers will vary. A typical histogram is shown next. Notice the gaps and the bimodal nature of the
histogram, probably due to the fact that the samples were collected at different locations.
.20
Relative Frequency
.15
.10
.05
0
10
12
14
16
18
AL
26
20
c The dotplot is shown as follows. The locations are indeed responsible for the unusual gaps and peaks
in the relative frequency histogram.
11.2
12.6
14.0
15.4
16.8
18.2
19.6
21.0
AL
Site
L
I
C
A
1.R.13 a-b The Minitab stem and leaf plot is shown next. The distribution is slightly skewed to the right.
c
Pennsylvania (58.20) has an unusually high gas tax.
1.R.14 a-b Answers will vary. The Minitab stem and leaf plot is shown next. The distribution is skewed to the
right.
1.R.15 a-b The distribution is approximately mound-shaped, with one unusual measurement, in the class with
midpoint at 100.8ยฐ. Perhaps the person whose temperature was 100.8 has some sort of illness coming on?
c
The value 98.6ยฐ is slightly to the right of center.
On Your Own
1.R.16 Answers will vary from student to student. The students should notice that the distribution is skewed to the
right with a few presidents (Truman, Cleveland, and F.D. Roosevelt) casting an unusually large number of
vetoes.
27

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