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CHAPTER 4 SECTION 2: NUMERICAL DESCRIPTIVE TECHNIQUES
MULTIPLE CHOICE
75. A sample of 20 observations has a standard deviation of 3. The sum of the squared deviations from the
sample mean is:
a. 20.
b. 23.
c. 29.
d. 171.
ANS: D
PTS: 1
REF: SECTION 4.2
76. If two data sets have the same range:
a. the distances from the smallest to largest observations in both sets will be the same.
b. the smallest and largest observations are the same in both sets.
c. both sets will have the same standard deviation.
d. both sets will have the same interquartile range.
ANS: A
PTS: 1
REF: SECTION 4.2
77. The Empirical Rule states that the approximate percentage of measurements in a data set (providing
that the data set has a bell shaped distribution) that fall within two standard deviations of their mean is
approximately:
a. 68%.
b. 75%.
c. 95%.
d. 99%.
ANS: C
PTS: 1
REF: SECTION 4.2
78. Which of the following summary measures is affected most by extreme values?
a. The median.
b. The mean.
c. The range.
d. The interquartile range.
ANS: C
PTS: 1
REF: SECTION 4.2
79. Chebysheff's Theorem states that the percentage of measurements in a data set that fall within three
standard deviations of their mean is:
a. 75%.
b. at least 75%.
c. 89%.
d. at least 89%.
ANS: D
PTS: 1
REF: SECTION 4.2
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copied, or distributed without the prior consent of the publisher.
80. Which of the following is a measure of variability?
a. The interquartile range
b. The variance
c. The coefficient of variation
d. All of these choices are true.
ANS: D
PTS: 1
REF: SECTION 4.2
81. The smaller the spread of scores around the mean:
a. the smaller the variance of the data set.
b. the smaller the standard deviation of the data set.
c. the smaller the coefficient of variation of the data set.
d. All of these choices are true.
ANS: D
PTS: 1
REF: SECTION 4.2
82. Is a standard deviation of 10 a large number indicating great variability, or is it small number
indicating little variability? To answer this question correctly, one should look carefully at the value of
the:
a. mean.
b. standard deviation.
c. coefficient of variation.
d. mean dividing by the standard deviation.
ANS: C
PTS: 1
REF: SECTION 4.2
83. Which of the following types of data has no measure of variability?
a. Interval data
b. Nominal data
c. Bimodal data
d. None of these choices.
ANS: B
PTS: 1
REF: SECTION 4.2
84. Which of the following statements is true regarding the data set 8, 8, 8, 8, and 8?
a. The range equals 0.
b. The standard deviation equals 0.
c. The coefficient of variation equals 0.
d. All of these choices are true.
ANS: D
PTS: 1
REF: SECTION 4.2
TRUE/FALSE
85. The value of the standard deviation may be either positive or negative, while the value of the variance
will always be positive.
ANS: F
PTS: 1
REF: SECTION 4.2
86. The difference between the largest and smallest observations in an ordered data set is called the range.
ANS: T
PTS: 1
REF: SECTION 4.2
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copied, or distributed without the prior consent of the publisher.
87. The standard deviation is expressed in terms of the original units of measurement but the variance is
not.
ANS: T
PTS: 1
REF: SECTION 4.2
88.  is a population parameter and s is a sample statistic.
ANS: T
PTS: 1
REF: SECTION 4.2
89. While Chebysheff's Theorem applies to any distribution, regardless of shape, the Empirical Rule
applies only to distributions that are bell shaped.
ANS: T
PTS: 1
REF: SECTION 4.2
90. The mean of fifty sales receipts is $65.75 and the standard deviation is $10.55. Using Chebysheff's
Theorem, 75% of the sales receipts were between $44.65 and $86.85.
ANS: T
PTS: 1
REF: SECTION 4.2
91. The data set 10, 20, 30 has the same variance as the data set 100, 200, 300.
ANS: F
PTS: 1
REF: SECTION 4.2
92. According to Chebysheff's Theorem, at least 93.75% of observations should fall within 4 standard
deviations of the mean.
ANS: T
PTS: 1
REF: SECTION 4.2
93. Chebysheff's Theorem states that the percentage of observations in a data set that should fall within
five standard deviations of their mean is at least 96%.
ANS: T
PTS: 1
REF: SECTION 4.2
94. The Empirical Rule states that the percentage of observations in a data set (providing that the data set
is bell shaped) that fall within one standard deviation of their mean is approximately 75%.
ANS: F
PTS: 1
REF: SECTION 4.2
95. A population with 200 elements has a variance of 20. From this information, it can be shown that the
population standard deviation is 10.
ANS: F
PTS: 1
REF: SECTION 4.2
96. If two data sets have the same range, the distances from the smallest to largest observations in both
sets will be the same.
ANS: T
PTS: 1
REF: SECTION 4.2
97. The coefficient of variation is a measure of variability.
ANS: T
PTS: 1
REF: SECTION 4.2
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copied, or distributed without the prior consent of the publisher.
98. The standard deviation is the positive square root of the variance.
ANS: T
PTS: 1
REF: SECTION 4.2
99. If two data sets have the same standard deviation, they must have the same coefficient of variation.
ANS: F
PTS: 1
REF: SECTION 4.2
100. The units for the variance are the same as the units for the original data (for example, feet, inches,
etc.).
ANS: F
PTS: 1
REF: SECTION 4.2
101. The units for the standard deviation are the same as the units for the original data (for example, feet,
inches, etc.).
ANS: T
PTS: 1
REF: SECTION 4.2
102. The variance is more meaningful and easier to interpret compared to the standard deviation.
ANS: F
PTS: 1
REF: SECTION 4.2
103. The range is considered the weakest measure of variability.
ANS: T
PTS: 1
REF: SECTION 4.2
104. Chebysheff's Theorem applies only to data sets that have a bell shaped distribution.
ANS: F
PTS: 1
REF: SECTION 4.2
105. The coefficient of variation allows us to compare two sets of data based on different measurement
units.
ANS: T
PTS: 1
REF: SECTION 4.2
106. If the observations are in the millions, a standard deviation of 10 would be considered small. If the
observations are all less than 50, a standard deviation of 10 would be considered large.
ANS: T
PTS: 1
REF: SECTION 4.2
COMPLETION
107. According to the Empirical Rule, if the data form a bell shaped normal distribution, approximately
____________________ percent of the observations will be contained within 2 standard deviations
around the mean.
ANS: 95
PTS: 1
REF: SECTION 4.2
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copied, or distributed without the prior consent of the publisher.
108. According to the Empirical Rule, if the data form a bell shaped normal distribution approximately
____________________ percent of the observations will be contained within 1 standard deviation
around the mean.
ANS: 68
PTS: 1
REF: SECTION 4.2
109. According to the Empirical Rule, if the data form a bell shaped normal distribution approximately
____________________ percent of the observations will be contained within 3 standard deviations
around the mean.
ANS: 99.7
PTS: 1
REF: SECTION 4.2
110. There are three statistics used to measure variability in a data set; the range, the
____________________ and the ____________________.
ANS:
variance; standard deviation
standard deviation; variance
PTS: 1
REF: SECTION 4.2
111. The ____________________ is the square root of the ____________________.
ANS: standard deviation; variance
PTS: 1
REF: SECTION 4.2
112. The ____________________ is the least effective of all the measures of variability.
ANS: range
PTS: 1
REF: SECTION 4.2
113. The ____________________ uses both the mean and the standard deviation to interpret standard
deviation for bell shaped histograms.
ANS: Empirical Rule
PTS: 1
REF: SECTION 4.2
114. ____________________ uses both the mean and the standard deviation to interpret standard deviation
for histograms of any shape.
ANS: Chebysheff's Theorem
PTS: 1
REF: SECTION 4.2
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copied, or distributed without the prior consent of the publisher.
115. A statistic that interprets the standard deviation relative to the size of the numbers in the data set is
called the ____________________ of ____________________.
ANS: coefficient; variation
PTS: 1
REF: SECTION 4.2
116. The range, variance, standard deviation, and coefficient of variation are to be used only on
____________________ data.
ANS: interval
PTS: 1
REF: SECTION 4.2
SHORT ANSWER
117. A basketball player has the following points for seven games: 20, 25, 32, 18, 19, 22, and 30. Compute
the following measures of variability.
a.
b.
c.
Standard deviation
Coefficient of variation
Compare the standard deviation and coefficient of variation and use them to discuss the
variability in the data.
ANS:
a.
b.
c.
s = 5.499
cv = 0.232
The standard deviation is 5.499 and the coefficient of variation is 0.232. The coefficient of
variation is smallest because the mean is larger than the standard deviation.
PTS: 1
REF: SECTION 4.2
118. The following data represent the number of children in a sample of 10 families from a certain
community: 4, 2, 1, 1, 5, 3, 0, 1, 0, and 2.
a.
b.
c.
d.
e.
Compute the range.
Compute the variance.
Compute the standard deviation.
Compute the coefficient of variation.
Explain why in this case range > variance > standard deviation > coefficient of variation.
ANS:
a.
b.
c.
d.
e.
5
2.77
1.66
0.87
The range is the difference between the largest and smallest observation, which compares
the numbers to each other; the variance is in essence "the average squared deviation from
mean", which compares the numbers to the mean. The standard deviation is less than the
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copied, or distributed without the prior consent of the publisher.
variance because it's the square root of a number larger than one. The coefficient of
variation is even smaller because the mean is larger than the standard deviation.
PTS: 1
REF: SECTION 4.2
Weights of Workers
The following data represent the weights in pounds of a sample of 25 workers: 164, 148, 137, 157,
173, 156, 177, 172, 169, 165, 145, 168, 163, 162, 174, 152, 156, 168, 154, 151, 174, 146, 134, 140,
and 171.
119. {Weights of Workers Narrative} Compute the sample variance, and sample standard deviation.
ANS:
s2 = 156.12, and s = 12.49
PTS: 1
REF: SECTION 4.2
120. {Weights of Workers Narrative} Compute the range and coefficient of variation.
ANS:
Range = 43,
cv = 12.49 / 159.04 = 0.079
PTS: 1
REF: SECTION 4.2
121. {Weights of Workers Narrative} Which is a better measure of variability in the weights of the workers,
the standard deviation or the coefficient of variation?
ANS:
Each has its own place. The standard deviation is in essence the "average distance from the mean."
The coefficient of variation takes the standard deviation into perspective by dividing the mean.
PTS: 1
REF: SECTION 4.2
122. Is it possible for the standard deviation of a data set to be larger than its variance? Explain.
ANS:
Yes. A standard deviation is larger than its corresponding variance when the variance is between 0 and
1 (exclusive).
PTS: 1
REF: SECTION 4.2
Ages of Teachers
The ages (in years) of three groups of teachers are shown below:
Group A:
Group B:
Group C:
17
30
44
22
28
39
20
35
54
18
40
21
23
25
52
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copied, or distributed without the prior consent of the publisher.
123. {Ages of Teachers Narrative} Calculate and compare the standard deviations for the three samples.
ANS:
s = 2.55 years, 5.94 years, and 13.21 years for Groups A, B, and C, respectively. Group A has the
smallest standard deviation because the teachers ages are very close, and Group C has ages which are
more spread out from the middle (the largest being 54 and the smallest being 21), so it has the largest
standard deviation.
PTS: 1
REF: SECTION 4.2
124. {Ages of Teachers Narrative} Compute and compare the ranges for the three groups.
ANS:
Range = 6, 15, and 33 (years) for Groups A, B, and C, respectively. Group C has a much wider range
mainly because the ages themselves are so much further apart than for the other two groups. Group A's
ages are closer together, giving them the smallest range.
PTS: 1
REF: SECTION 4.2
125. {Ages of Teachers Narrative} Compute and compare the coefficient of variation for the three samples.
ANS:
cv = 0.127, 0.188, and 0.315 for Groups A, B, and C, respectively. All of the groups have mean ages
that are larger than their standard deviations. Each group has a larger standard deviation than the next,
and a larger mean than the next, but the standard deviations increase faster than the means do.
PTS: 1
REF: SECTION 4.2
126. Suppose your data set contains ages (in years) and you calculate the range, variance, standard
deviation, and coefficient of variation for the data. Explain what units each of these measures is in.
ANS:
Range subtracts two years (largest minus smallest) so it is still in units of years. The variance is the
average squared deviation from the mean, which is measured in square units. In this case, years
squared. (These units have no meaning in this case.) The standard deviation is the square root of the
variance, taking the units back to years. The coefficient of variation takes the standard deviation (in
years) divided by the mean (in years) and hence has no units at all.
PTS: 1
REF: SECTION 4.2
127. The number of hours a college student spent studying during the final exam week was recorded as
follows: 7, 6, 4, 9, 8, 5, and 10. Compute the range for the data, express the number in the appropriate
unit.
ANS:
Range = 6 hours
PTS: 1
REF: SECTION 4.2
128. The number of hours a college student spent studying during the final exam week was recorded as
follows: 7, 6, 4, 9, 8, 5, and 10. Compute s2 and s for the data and express the numbers in the
appropriate unit.
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copied, or distributed without the prior consent of the publisher.
ANS:
s2 = 4.667 (hours)2
s = 2.160 hours
PTS: 1
REF: SECTION 4.2
129. The annual percentage rates of return over the past 10 years for two mutual funds are as shown below.
Which fund would you classify as having the higher level of risk?
Fund A:
Fund B:
7.4
4.1
7.1
10.8
19.7
5.1
3.9
10.9
32.4
26.5
41.7
24.0
23.2
16.9
4.0
9.4
1.9
2.6
29.3
10.1
ANS:
The variance of returns will be used as the measure of risk of an investment. Since,
and
, fund A has the higher level of risk.
PTS: 1
REF: SECTION 4.2
Ages of Employees
The following data represent the ages in years of a sample of 25 employees from a government
department: 31, 43, 56, 23, 49, 42, 33, 61, 44, 28, 48, 38, 44, 35, 40, 64, 52, 42, 47, 39, 53, 27, 36, 35,
and 20.
130. {Ages of Employees Narrative} Compute the range of the data, and express the number in the
appropriate unit.
ANS:
Range = 44 years
PTS: 1
REF: SECTION 4.2
131. {Ages of Employees Narrative} Compute the sample variance, and sample standard deviation, and
express the numbers in the appropriate units.
ANS:
s2 = 124.83 (years)2, and s = 11.17 years
PTS: 1
REF: SECTION 4.2
132. {Ages of Employees Narrative} Compute the coefficient of variation, and express the number in the
appropriate unit.
ANS:
cv = 11.17 years / 41.2 years = 0.271. Coefficient of variation has no units attached to it.
PTS: 1
REF: SECTION 4.2
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copied, or distributed without the prior consent of the publisher.
Salaries of Employees
The following data represent the salaries (in thousands of dollars) of a sample of 13 employees of a
firm: 26.5, 23.5, 29.7, 24.8, 21.1, 24.3, 20.4, 22.7, 27.2, 23.7, 24.1, 24.8, and 28.2.
133. {Salaries of Employees Narrative} Compute the variance, and standard deviation of the salaries, and
express the numbers in the appropriate units.
ANS:
s2 = 7.097 (thousand dollars, squared), and s = 2.664 thousand dollars.
PTS: 1
REF: SECTION 4.2
134. {Salaries of Employees Narrative} Compute the coefficient of variation, and express the number in the
appropriate unit.
ANS:
cv = 2.664 thousand dollars / 24.692 thousand dollars = 0.108. No units are attached to the cv.
PTS: 1
REF: SECTION 4.2
135. {Salaries of Employees Narrative} Compute the range.
ANS:
Range = 9.3 thousand dollars
PTS: 1
REF: SECTION 4.2
136. Consider the following population of measurements: 162, 152, 177, 157, 184, 176, 165, 181, 170, and
163. Label and compute the variance and standard deviation.
ANS:
2 = 101.61 and  = 10.08
PTS: 1
REF: SECTION 4.2
Egg Demand
A supermarket has determined that daily demand for egg cartons has an approximate bell shaped
distribution, with a mean of 55 cartons and a standard deviation of six cartons.
137. {Egg Demand Narrative} How often can we expect between 49 and 61 cartons to be sold in a day?
(Give a percentage.)
ANS:
Approximately 68% by the Empirical Rule.
PTS: 1
REF: SECTION 4.2
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138. {Egg Demand Narrative} What percentage of the time will the number of cartons of eggs sold be more
than 2 standard deviations from the mean?
ANS:
Approximately 5% by the Empirical Rule.
PTS: 1
REF: SECTION 4.2
139. {Egg Demand Narrative} If the supermarket begins each morning with a supply of 67 cartons of eggs,
how often will demand exceed the supply? (Give a percentage.)
ANS:
Approximately 2.5% by the Empirical Rule.
PTS: 1
REF: SECTION 4.2
140. A sample of 13 high school teachers has a mean age of 30 years and a standard deviation of 5 years.
Suppose that the sample is enlarged to 15 high school teachers, by including two additional teachers
that are each 30 years old. Will the standard deviation increase, decrease, or stay the same, and why?
ANS:
The new standard deviation will be smaller than the original one because the two new teachers' ages
don't deviate from the mean, but they add two more observations to the data set, bringing down the
average deviation from the mean. The new standard deviations actually is s = 4.629.
PTS: 1
REF: SECTION 4.2
141. The price-earnings ratios of a sample of stocks have a mean value of 13.5 and a standard deviation of
2. If the ratios have a bell shaped distribution, what can we say about the proportion of ratios that fall
between
a.
b.
c.
11.5 and 15.5?
9.5 and 17.5?
7.5 and 19.5?
ANS:
a.
b.
c.
The interval contains approximately 68% of the ratios, according to the Empirical Rule.
The interval contains approximately 95% of the ratios according to the Empirical Rule.
The interval contains approximately 99.7 of the ratios according to the Empirical Rule.
PTS: 1
REF: SECTION 4.2
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