Table Of ContentCHAPTER 1
Section 1.1
1.
a. Houston Chronicle, Des Moines Register, Chicago Tribune, Washington Post
b. Capital One, Campbell Soup, Merrill Lynch, Pulitzer
c. Bill Jasper, Kay Reinke, Helen Ford, David Menendez
d. 1.78, 2.44, 3.50, 3.04
2.
a. 29.1 yd., 28.3 yd., 24.7 yd., 31.0 yd.
b. 432, 196, 184, 321
c. 2.1, 4.0, 3.2, 6.3
d. 0.07 g, 1.58 g, 7.1 g, 27.2 g
3.
a. In a sample of 100 phones, what are the chances that more than 20 need service while under warranty?
What are the chances than none need service while still under warranty?
b. What proportion of all phones of this brand and model will need service within the warranty period?
4. Concrete populations include: all living U.S. Citizens, all mutual funds marketed in the U.S., all books
published in 2019. For the population of all mutual funds, possible questions include
• Probability: In a sample of 5 mutual funds, what is the chance that all 5 have rates of return which
exceeded 10% last year?
• Statistical inference: If previous year rates-of-return for 5 mutual funds were 9.6, 14.5, 8.3, 9.9 and
10.2, can we conclude that the average rate for all funds was below 10%?
Hypothetical populations include: all grade point averages for University of California undergraduates
during the next academic year, page lengths for all books published during the next calendar year, batting
averages for all major league players during the next baseball season. For the page length population,
possible questions include
• Probability: In a sample of 10 books to be published next year, how likely is it that the average number
of pages for the 10 is between 200 and 250?
• Statistical inference: If the sample average number of pages for 10 books is 227, can we be highly
confident that the average for all books is between 200 and 245?
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Chapter 1: Overview and Descriptive Statistics
5.
a. Two variables (at least) were recorded: skin color and hourly wages.
b. Skin color is categorical (with four categories), while hourly wages is quantitative (units: $/hr).
6.
a. Two variables were recorded: brand and percent of polyunsaturated fat.
b. Brand is categorical (with as many categories as there were brands in the sample), while
polyunsaturated fat is quantitative (units: percent).
c. Based on the sample data, can we conclude that the mean percent of polyunsaturated fat in all
margarine containers exceeds 20%? Can we conclude that the population mean polyunsaturated fat
percent differs by brand? What proportion of all margarine containers have a polyunsaturated fat
percent between 15% and 25%?
d. Since this only describes observations in the sample and does not draw a conclusion about all Parkay
margarine containers, this is an example of descriptive statistics.
7. a. categorical b. quantitative c. categorical d. categorical e. categorical
8. a. categorical b. quantitative c. quantitative d. categorical e. quantitative f. quantitative
9.
a. No, the relevant conceptual population is all scores of all students who participate in the SI in
conjunction with this particular statistics course.
b. The advantage to randomly assigning students to the two groups is that the two groups should then be
fairly comparable before the study. If the two groups perform differently in the class, we can
reasonably attribute this to the treatments (SI and control). If it were left to students to choose, stronger
or more dedicated students might gravitate toward SI, confounding the results.
c. If all students were put in the treatment group there would be no results with which to compare the
treatments.
10. One could take a simple random sample of students from all students in the California State University
system and ask each student in the sample to report the distance from their hometown to campus.
Alternatively, the sample could be generated by taking a stratified random sample by taking a simple
random sample from each of the 23 campuses and again asking each student in the sample to report the
distance from their hometown to campus. Certain problems might arise with self-reporting of distances,
such as recording error or poor recall. This study is enumerative because there exists a finite, identifiable
population of objects from which to sample.
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Chapter 1: Overview and Descriptive Statistics
11. One could generate a simple random sample of all single-family homes in the city or a stratified random
sample by taking a simple random sample from each of the 10 district neighborhoods. From each of the
homes in the sample the necessary variables would be collected. This would be an enumerative study
because there exists a finite, identifiable population of objects from which to sample.
12.
a. Number observations equal 2 x 2 x 2 = 8
b. This could be called a conceptual population because the data would be collected on an existing
process. There is no sampling frame.
13.
a. There could be several explanations for the variability of the measurements. Among them could be
measuring error, (due to mechanical or technical changes across measurements), recording error,
differences in weather conditions at time of measurements, etc.
b. This could be called a conceptual because there is no sampling frame.
Section 1.2
14.
a. Truncate each observation at the decimal point and use two rows for each tens place. So, the first row
will represent values 30-34, the next row 35-39, then 40-44, and so on.
3L 444 Stem: tens digit
3H 66 Leaf: ones digit
4L 123
4H 5777889
5L 223
5H 6
b. The distribution shows some slight asymmetry, extending further toward lower values (from a “center”
of about 47) than toward higher values. We might describe this as a (somewhat) negatively-skewed
distribution.
c. No, none of the data values in the stem-and-leaf display appear to be outliers.
d. Ten of the 19 times exceed 45 minutes, for a proportion of 10/19 ≈ .526.
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Chapter 1: Overview and Descriptive Statistics
15.
6L 034
6H 667899
7L 00122244
7H Stem=Tens
8L 001111122344 Leaf=Ones
8H 5557899
9L 03
9H 58
This display brings out the gap in the data: There are no scores in the high 70s.
16. The accompanying stem-and-leaf display divides each tens place into five groups: 0-1, 2-3, 4-5, 6-7, 8-9.
The graph suggests that noise levels followed a positively-skewed distribution, ranging from about 55 to 83
dBA. There are a few gaps in the distribution, and the three 83.0 dBA observations might be high outliers.
5 5555555
5 66666666777777777 Stem: Tens digit
5 888999 Leaf: Ones digit
6
6 22333333
6 4445555555
6 7777
6 88889
7 00111
7 3333
7 4444
7
7 9999
8
8 333
17.
a. The stem-and-leaf display appears at the top of the next page.
b. Arguably, a representative crack depth might be around 9-10 μm.
c. This is somewhat subjective, but the display appears quite spread out.
d. No, the distribution is certainly not symmetric. Rather, crack depths appear to be strongly positively
skewed.
e. Yes: All of the values 66.5, 76.1, and 81.1 μm appear to be high outliers. (Using an outlier convention
described later in the chapter, even the values in the 50s would be considered outliers!)
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Chapter 1: Overview and Descriptive Statistics
0 123333333444444
0 55555667788888999
1 0000001111224
1 5789 Stem: Tens digit
2 0112 Leaf: Ones digit
2 6
3 334
3 7
4 2
4 68
5 012
5
6
6 6
7
7 6
8 1
18.
a. The accompanying display uses two rows for each ones digit.
2 22233344
2 5555666666777778888888888888888999999
3 000000000001111111111122222222222333344444444444
3 5555666777777888888899999999
4 000001111123333333334444 Stem: Ones digit
4 555556678999 Leaf: Tenths digit
5 000122334
5 556666789
6 0
6 7
7 1
7 5558
8 1
8 68
9 24
9 6
b. A typical median multiple is around 3.4.
c. The median multiple distribution for these 188 cities is positively skewed, with no obvious gaps or
outliers.
d. A quick count finds 28 of these 188 cities have a median multiple above 5.0, so the proportion of these
cities classified as “Severely Unaffordable” is 28/188 ≈ .149.
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Chapter 1: Overview and Descriptive Statistics
19.
American French
8 1
755543211000 9 00234566
9432 10 2356
6630 11 1369
850 12 223558
8 13 7
14
15 8 Stem: Tens digit
2 16 Leaf: Ones digit
The American distribution is positively skewed, but the French distribution is fairly symmetric. Almost
half of the American movies are in the 90s, but the French movies are more spread out.
20. Drivers in very large urban areas generally experience more extra time due to traffic than do drivers in large
urban areas. In very large cities, drivers typically experience 58-60 extra hours per traveler per year due to
traffic, compared to roughly 39 hours in large cities. One very large urban area has an especially high
value: 93 extra hours per year due to traffic. (Not surprisingly, that’s Los Angeles, CA.)
Large Very Large
874320 1
963 2 Stem: Tens unit
9853300 3 8 Leaf: Ones unit
6630 4 99
553210 5 1178
6 0379
7 2
8
9 3
21.
a.
Value Freq. Rel. Freq.
( = Freq. / 60)
0 7 .117
1 12 .200
2 13 .217
3 14 .233
4 6 .100
5 3 .050
6 3 .050
7 1 .017
8 1 .017
Note: Relative frequencies add to 1.001, not 1, due to rounding.
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Chapter 1: Overview and Descriptive Statistics
b. The number of batches with at most 5 nonconforming items is 7+12+13+14+6+3 = 55, which is a
proportion of 55/60 = .917. The proportion of batches with (strictly) fewer than 5 nonconforming
items is 52/60 = .867.
Notice that these proportions could also have been computed by using the relative frequencies: e.g.,
proportion of batches with 5 or fewer nonconforming items = 1 – (.05+.017+.017) = .916; proportion
of batches with fewer than 5 nonconforming items = 1 – (.05+.05+.017+.017) = .866.
c. The center of the histogram is somewhere around 2 or 3 and it shows that there is some positive
skewness in the data. The histogram also shows that there is a lot of spread/variation in this data.
14
12
10
Frequency 86
4
2
0
0 2 4 6 8
Number of nonconforming transducers per batch
22.
a. The most interesting feature of the histogram is the heavy positive skewness of the data, corresponding
to the fact that the vast majority of authors in the sample published exactly 1 paper.
700
600
500
Frequency430000
200
100
0
1 2 3 4 5 6 7 8 9 10 11
Number of papers
b. There were 988 authors in the sample. From the frequency distribution (or from the histogram), the
number of authors who published at least 5 papers is 35 + 30 + 23 + 17 + 15 + 10 + 5 = 135, so the
proportion who published 5 or more papers is 135/988 = .137, or 13.7%. Similarly, the proportion who
published more than 5 papers is 100/988 = .101 = 10.1%.
c. No. Strictly speaking, the class described by “10+” has no upper boundary, so it is impossible to draw
a rectangle above it having finite area (i.e., frequency). We would not know where to end the rectangle.
d. Yes The category 10-11 has finite width of 2 (10, 11), so the cumulated frequency of 15 can be plotted
as a rectangle of height 15/2 = 7.5 over this interval.
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Chapter 1: Overview and Descriptive Statistics
23.
a. 589/1570 = .375.
b. 1 – (589 + 190 + 176 + 157 + 115)/1570 = .218.
c. (115 + 89 + 57 + 55 + 33 = 31)/1570 = .242.
d. The herd size distribution in the accompanying histogram is extremely positively skewed.
0.4
0.3
ncy
ue
q
e
e fr0.2
Relativ
0.1
0.0
0 5 10 15 20 25 30
Herd size
24.
a. The following stem-and-leaf display was constructed:
0 123334555599
1 00122234688 stem: thousands
2 1112344477 leaf: hundreds
3 0113338
4 37
5 23778
A typical data value is somewhere in the low 2000s. The display is almost unimodal (the stem at 5
would be considered a mode, the stem at 0 another) and has a positive skew.
b. A histogram of this data, using classes boundaries of 0, 1000, 2000, …, 6000 is shown below. The
proportion of subdivisions with total length less than 2000 is (12+11)/47 = .489, or 48.9%. Between
2000 and 4000, the proportion is (10+7)/47 = .362, or 36.2%. The histogram shows the same general
shape as depicted by the stem-and-leaf in part a.
Frequency
10
5
0
0 1000 2000 3000 4000 5000 6000
length
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Chapter 1: Overview and Descriptive Statistics
25.
a. A histogram of the y data appears below. From this histogram, the number of subdivisions having no
cul-de-sacs (i.e., y = 0) is 17/47 = .362, or 36.2%. The proportion having at least one cul-de-sac (y ≥ 1)
is (47 – 17)/47 = 30/47 = .638, or 63.8%. Note that subtracting the number of cul-de-sacs with y = 0
from the total, 47, is an easy way to find the number of subdivisions with y ≥ 1.
Frequency
20
10
0
0 1 2 3 4 5
y
b. A histogram of the z data appears below. From this histogram, the number of subdivisions with at
most 5 intersections (i.e., z ≤ 5) is 42/47 = .894, or 89.4%. The proportion having fewer than 5
intersections (z < 5) is 39/47 = .830, or 83.0%.
Frequency
10
5
0
0 1 2 3 4 5 6 7 8
z
26. A very large percentage of the data values are greater than 0, which indicates that most, but not all, runners
do slow down at the end of the race. The histogram is also positively skewed, which means that some
runners slow down a lot compared to the others. A typical value for this data would be in the neighborhood
of 200 seconds. The proportion of the runners who ran the last 5 km faster than they did the first 5 km is
very small, about 1% or so.
27.
a. The distribution of these by-state values is slightly positively skewed with one extremely high outlier
(Washington DC, 54.6%) and two other potential outliers (Massachusetts, 40.5% and West Virginia,
19.2%). The “typical” state percentage appears to be between 25 and 30 percent.
20 25 30 35 40 45 50 55
Percent of State with 4-Year Degree
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Chapter 1: Overview and Descriptive Statistics
b. No: Since the population sizes of the 50 states + DC are not equal, the mean of these percentages
would not equal the overall percentage. (If we knew all 51 population sizes, we could take the
appropriate weighted average, effectively re-constructing the total count of people with 4-year degrees
and dividing by the total population size.)
28.
a. A value of 0 means the spare tire was inflated to exactly the pressure specified by the manufacturer. A
negative value indicates a spare tire that was over-inflated; e.g., –6 means the spare tire’s pressure was
6 psi more than the manufacturer’s specification.
b. The underinflation distribution for these 95 spare tires is somewhat negatively skewed. A small
number of tires were overinflated (negative values in the graph). Many tires were either slightly
underinflated (0-10 psi) or massively underinflated (50-60 psi). The latter group corresponds to spare
tires that were completely flat but have a spec of around 60 psi.
20
15
Percent10
5
0
-20 0 20 40 60
Underinflation of spare tire (psi)
c. Because the intervals do not have equal width, we must us a density scale. For each interval, the
density is defined as (frequency) / (length of interval).
Histogram of Underinflation
0.020
0.015
Density0.010
0.005
0.000
-20 0 10 20 30 40 60
Underinflation
d. Taking “within ± 10 psi” to include 10 psi, 24 of the 95 tires fit this description, for a proportion of
24/95 ≈ .253.
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