Download chapter 4 section 1: numerical descriptive techniques

CHOSUN UNIVERSITY SEOK-GANG,PARK
CHAPTER 4 SECTION 1: NUMERICAL DESCRIPTIVE TECHNIQUES
MULTIPLE CHOICE
1. Which of the following statistics is a measure of central location?
a. The mean
b. The median
c. The mode
d. All of these choices are true.
ANS: D
PTS: 1
REF: SECTION 4.1
2. Which measure of central location is meaningful when the data are ordinal?
a. The mean
b. The median
c. The mode
d. All of these choices are meaningful for ordinal data.
ANS: C
PTS: 1
REF: SECTION 4.1
3. Which of the following statements about the mean is not always correct?
a. The sum of the deviations from the mean is zero.
b. Half of the observations are on either side of the mean.
c. The mean is a measure of the central location.
d. The value of the mean times the number of observations equals the sum of all
observations.
ANS: B
PTS: 1
REF: SECTION 4.1
4. Which of the following statements is true for the following observations: 7, 5, 6, 4, 7, 8, and 12?
a. The mean, median and mode are all equal.
b. Only the mean and median are equal.
c. Only the mean and mode are equal
d. Only the median and mode are equal.
ANS: A
PTS: 1
REF: SECTION 4.1
5. In a histogram, the proportion of the total area which must be to the left of the median is:
a. exactly 0.50.
b. less than 0.50 if the distribution is negatively skewed.
c. more than 0.50 if the distribution is positively skewed.
d. unknown.
ANS: A
PTS: 1
REF: SECTION 4.1
6. Which measure of central location can be used for both interval and nominal variables?
a. The mean
b. The median
c. The mode
d. All of these choices are true.
ANS: C
PTS: 1
REF: SECTION 4.1
7. Which of these measures of central location is not sensitive to extreme values?
CHOSUN UNIVERSITY SEOK-GANG,PARK
a.
b.
c.
d.
The mean
The median
The mode
All of these choices are true.
ANS: B
PTS: 1
REF: SECTION 4.1
8. In a positively skewed distribution:
a. the median equals the mean.
b. the median is less than the mean.
c. the median is larger than the mean.
d. the mean can be larger or smaller than the median.
ANS: B
PTS: 1
REF: SECTION 4.1
9. Which of the following statements about the median is not true?
a. It is more affected by extreme values than the mean.
b. It is a measure of central location.
c. It is equal to Q2.
d. It is equal to the mode in a bell shaped distribution.
ANS: A
PTS: 1
REF: SECTION 4.1
10. Which of the following summary measures is sensitive to extreme values?
a. The median
b. The interquartile range
c. The mean
d. The first quartile
ANS: C
PTS: 1
REF: SECTION 4.1
11. In a perfectly symmetric bell shaped "normal" distribution:
a. the mean equals the median.
b. the median equals the mode.
c. the mean equals the mode.
d. All of these choices are true.
ANS: D
PTS: 1
REF: SECTION 4.1
12. Which of the following statements is true?
a. When the distribution is positively skewed, mean > median > mode.
b. When the distribution is negatively skewed, mean < median < mode.
c. When the distribution is symmetric and unimodal, mean = median = mode.
d. When the distribution is symmetric and bimodal, mean = median = mode.
ANS: C
PTS: 1
REF: SECTION 4.1
13. In a histogram, the proportion of the total area which must be to the right of the mean is:
a. less than 0.50 if the distribution is negatively skewed.
b. exactly 0.50.
c. more than 0.50 if the distribution is positively skewed.
d. exactly 0.50 if the distribution is symmetric and unimodal.
ANS: D
PTS: 1
REF: SECTION 4.1
14. The average score for a class of 30 students was 75. The 20 male students in the class averaged 70.
The 10 female students in the class averaged:
CHOSUN UNIVERSITY SEOK-GANG,PARK
a.
b.
c.
d.
75.
85.
65.
70.
ANS: B
PTS: 1
REF: SECTION 4.1
TRUE/FALSE
15. The mean is affected by extreme values but the median is not.
ANS: T
PTS: 1
REF: SECTION 4.1
16. The mean is a measure of variability.
ANS: F
PTS: 1
REF: SECTION 4.1
17. In a histogram, the proportion of the total area which must be to the left of the median is more than
0.50 if the distribution is positively skewed.
ANS: F
PTS: 1
REF: SECTION 4.1
18. A data sample has a mean of 107, a median of 122, and a mode of 134. The distribution of the data is
positively skewed.
ANS: F
PTS: 1
19.  is a population parameter and
ANS: T
PTS: 1
REF: SECTION 4.1
is a sample statistic.
REF: SECTION 4.1
20. In a bell shaped distribution, there is no difference in the values of the mean, median, and mode.
ANS: T
PTS: 1
REF: SECTION 4.1
21. Lily has been keeping track of what she spends to eat out. The last week's expenditures for meals eaten
out were $5.69, $5.95, $6.19, $10.91, $7.49, $14.53, and $7.66. The mean amount Lily spends on
meals is $8.35.
ANS: T
PTS: 1
REF: SECTION 4.1
22. In a negatively skewed distribution, the mean is smaller than the median and the median is smaller
than the mode.
ANS: T
PTS: 1
REF: SECTION 4.1
23. The median of a set of data is more representative than the mean when the mean is larger than most of
the observations.
ANS: T
PTS: 1
REF: SECTION 4.1
24. The value of the mean times the number of observations equals the sum of the observations.
ANS: T
PTS: 1
REF: SECTION 4.1
CHOSUN UNIVERSITY SEOK-GANG,PARK
25. In a histogram, the proportion of the total area which must be to the left of the median is less than 0.50
if the distribution is negatively skewed.
ANS: F
PTS: 1
REF: SECTION 4.1
26. In a histogram, the proportion of the total area which must be to the right of the mean is exactly 0.50 if
the distribution is symmetric and unimodal.
ANS: T
PTS: 1
REF: SECTION 4.1
27. Suppose a sample of size 50 has a sample mean of 20. In this case, the sum of all observations in the
sample is 1,000.
ANS: T
PTS: 1
REF: SECTION 4.1
28. The median of an ordered data set with 30 items would be the average of the 15th and the 16th
observations.
ANS: T
PTS: 1
REF: SECTION 4.1
29. If the median, median and mode are all equal, the histogram must be symmetric and bell shaped.
ANS: F
PTS: 1
REF: SECTION 4.1
COMPLETION
30. Another word for the mean of a data set is the ____________________.
ANS: average
PTS: 1
REF: SECTION 4.1
31. The size of a sample is denoted by the letter ____________________, and the size of a population is
denoted by the letter ____________________.
ANS: n; N
PTS: 1
REF: SECTION 4.1
32. The sample mean is denoted by ____________________ and the population mean is denoted by
____________________.
ANS:
; x
PTS: 1
REF: SECTION 4.1
33. There are three measures of central location; the mean, the ____________________ and the
____________________.
ANS:
median; mode
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mode; median
PTS: 1
REF: SECTION 4.1
34. The ____________________ is calculated by finding the middle of the data set, when the data are
ordered from smallest to largest.
ANS: median
PTS: 1
REF: SECTION 4.1
35. The ____________________ is the least desirable of all the measures of central location.
ANS: mode
PTS: 1
REF: SECTION 4.1
36. The ____________________ is not as sensitive to extreme values as the ____________________.
ANS:
median; mean
median; average
PTS: 1
REF: SECTION 4.1
37. The ____________________ mean is used whenever we wish to find the "average" growth rate, or
rate of change, in a variable over time.
ANS: geometric
PTS: 1
REF: SECTION 4.1
38. The ____________________ mean of n returns (or growth rates) is the appropriate mean to calculate
if you wish to estimate the mean rate of return (or growth rate) for any single period in the future.
ANS: arithmetic
PTS: 1
REF: SECTION 4.1
39. If a data set contains an even number of observations, the median is found by taking the
____________________ of these two numbers.
ANS:
average
arithmetic mean
PTS: 1
REF: SECTION 4.1
40. If a data set is composed of 5 different numbers, there are ____________________ modes.
ANS:
0
no
zero
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PTS: 1
REF: SECTION 4.1
SHORT ANSWER
Monthly Rent
Monthly rent data in dollars for a sample of 10 one-bedroom apartments in a small town in Iowa are as
follows: 220, 216, 220, 205, 210, 240, 195, 235, 204, and 250.
41. {Monthly Rent Narrative} Compute the sample monthly average rent.
ANS:
= $219.50
PTS: 1
REF: SECTION 4.1
42. {Monthly Rent Narrative} Compute the sample median.
ANS:
$218
PTS: 1
REF: SECTION 4.1
43. {Monthly Rent Narrative} What is the mode?
ANS:
$220
PTS: 1
REF: SECTION 4.1
Pets Survey
A sample of 25 families were asked how many pets they owned. Their responses are summarized in
the following table.
Number of Pets
Number of Families
0
3
1
10
2
5
3
4
4
2
5
1
44. {Pets Survey Narrative} Determine the mean, the median, and the mode of the number of pets owned
per family.
ANS:
= [(0  3) + (1  10) + (2  5) + (3  4) + (4  2) + (5  1)]/25 = 1.80 pets, median = 1 pet, mode = 1
pet.
PTS: 1
REF: SECTION 4.1
45. {Pets Survey Narrative} Explain what the mean, median, and mode tell you about this particular data
set.
ANS:
CHOSUN UNIVERSITY SEOK-GANG,PARK
The "average" number of pets owned was 1.80 pets. (This represents the overall average, rather than
the number of pets for the average family.) Half the families own at most one pet, and the other half
own at least one pet. The most frequent number of pets owned was one pet.
PTS: 1
REF: SECTION 4.1
46. How do the mean, median and mode compare to each other when the distribution is:
a.
b.
c.
symmetric?
negatively skewed?
positively skewed?
ANS:
a.
b.
c.
mean = median = mode
mean < median < mode
mean > median > mode
PTS: 1
REF: SECTION 4.1
47. A basketball player has the following points for seven games: 20, 25, 32, 18, 19, 22, and 30. Compute
the following measures of central location:
a.
b.
c.
mean
median
mode
ANS:
a.
b.
c.
= 23.714
median = 22.0
There is no mode
PTS: 1
REF: SECTION 4.1
Children
The following data represent the number of children in a sample of 10 families from Chicago: 4, 2, 1,
1, 5, 3, 0, 1, 0, and 2.
48. {Children Narrative} Compute the mean number of children.
ANS:
= 1.90
PTS: 1
REF: SECTION 4.1
49. {Children Narrative} Compute the median number of children.
ANS:
Median = 1.5
CHOSUN UNIVERSITY SEOK-GANG,PARK
PTS: 1
REF: SECTION 4.1
50. {Children Narrative} Is the distribution of the number of children symmetric or skewed? Why?
ANS:
The distribution is positively skewed because the mean is larger than the median.
PTS: 1
REF: SECTION 4.1
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.
51. {Weights of Workers Narrative} Construct a stem and leaf display for the weights.
ANS:
Stem
13
14
15
16
17
Leaf
47
0568
124667
2345889
123447
PTS: 1
REF: SECTION 4.1
52. {Weights of Workers Narrative} Find the median weight.
ANS:
Median = 162 pounds
PTS: 1
REF: SECTION 4.1
53. {Weights of Workers Narrative} Find the mean weight.
ANS:
PTS: 1
REF: SECTION 4.1
54. {Weights of Workers Narrative} Is the distribution of the weights of workers symmetric or skewed?
Why?
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ANS:
The distribution is negatively skewed because the mean is smaller than the median, and the stem and
leaf display also shows this negative skewness.
PTS: 1
REF: SECTION 4.1
55. 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 for the data and the value in an appropriate unit.
ANS:
= 7 hours
PTS: 1
REF: SECTION 4.1
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.
56. {Ages of Employees Narrative} Construct a stem and leaf display for the ages.
ANS:
Stem
2
3
4
5
6
Leaf
0378
1355689
022344789
236
14
PTS: 1
REF: SECTION 4.1
57. {Ages of Employees Narrative} Find the median age.
ANS:
Median = 42 years
PTS: 1
REF: SECTION 4.1
58. {Ages of Employees Narrative} Compute the sample mean age.
ANS:
= 41.2 years
PTS: 1
REF: SECTION 4.1
59. {Ages of Employees Narrative} Find the modal age.
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ANS:
Modes are 35, 42, and 44
PTS: 1
REF: SECTION 4.1
60. {Ages of Employees Narrative} Compare the mean and median ages for these employees and use them
to discuss the shape of the distribution.
ANS:
The mean and median are 41.2 years, 42 years, respectively. They are very close to each other, which
tells us the distribution of ages is approximately symmetric.
PTS: 1
REF: SECTION 4.1
Salaries of Employees
The following data represent the yearly 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.
61. {Salaries of Employees Narrative} Compute the mean salary.
ANS:
= 24.692 thousand dollars
PTS: 1
REF: SECTION 4.1
62. {Salaries of Employees Narrative} Compute the median salary.
ANS:
median = 24.3 thousand dollars
PTS: 1
REF: SECTION 4.1
63. {Salaries of Employees Narrative} Compare the mean salary with the median salary and use them to
describe the shape of the distribution.
ANS:
The mean is $24,692 thousand dollars, and the median is $24,300 thousand dollars. The mean is
slightly higher than the median. This tells us that the data are slightly positively skewed, but close to
symmetric.
PTS: 1
REF: SECTION 4.1
64. A sample of 12 construction workers has a mean age of 25 years. Suppose that the sample is enlarged
to 14 construction workers, by including two additional workers having common age of 25 each. Find
the mean of the sample of 14 workers.
ANS:
= 25 years
PTS: 1
REF: SECTION 4.1
CHOSUN UNIVERSITY SEOK-GANG,PARK
65. The mean of a sample of 15 measurements is 35.6 feet. Suppose that the sample is enlarged to 16
measurements, by including one additional measurement having a value of 42 feet. Find the mean of
the sample of the 16 measurements.
ANS:
= 36 feet.
PTS: 1
REF: SECTION 4.1
66. An investment you made in the years 2000-2003 has the following rates of return shown below:
Year
Rate of Return
2000
.50
2001
.30
2002
.10
2003
.15
Compute the geometric mean.
ANS:
PTS: 1
REF: SECTION 4.1
2-Year Investment
Suppose you make a 2-year investment of $5,000 and it grows by 100% to $10,000 during the first
year. During the second year, however, the investment suffers a 50% loss, from $10,000 back to
$5,000.
67. {2-Year Investment Narrative} Calculate the arithmetic mean.
ANS:
PTS: 1
REF: SECTION 4.1
68. {2-Year Investment Narrative} Calculate the geometric mean.
ANS:
PTS: 1
REF: SECTION 4.1
69. {2-Year Investment Narrative} Compare the values of the arithmetic and geometric means.
ANS:
Because there was no change in the value of the investment from the beginning to the end of the 2-year
period, the arithmetic mean (aka the "average" compounded rate of return) is 0%, and this is the value
of the geometric mean. This result is somewhat misleading.
PTS: 1
REF: SECTION 4.1
Ages of Senior Citizens
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A sociologist recently conducted a survey of citizens over 65 years of age whose net worth is too high
to qualify for Medicaid and who have no private health insurance. The ages of 20 uninsured senior
citizens were as follows: 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 78, 79, 80, 81, 86, 87, 91, 92, 94, and
97.
70. {Ages of Senior Citizens Narrative} Calculate the mean age of the uninsured senior citizens
ANS:
= 78.15 years.
PTS: 1
REF: SECTION 4.1
71. {Ages of Senior Citizens Narrative} Calculate the median age of the uninsured senior citizens.
ANS:
76.5 years.
PTS: 1
REF: SECTION 4.1
72. {Ages of Senior Citizens Narrative} Explain why there is no mode for this data set.
ANS:
There is no mode because every age is different.
PTS: 1
REF: SECTION 4.1
73. Suppose that a firm's sales were $2,500,000 four years ago, and sales have grown annually by 25%,
15%, 5%, and 10% since that time. What was the geometric mean growth rate in sales over the past
four years?
ANS:
If Rg is the geometric mean, then
(1 + Rg)4 = (1 + 0.25)(1 + 0.15)(1  0.05)(1 + 0.10) = 1.5022  Rg = 0.1071 or 10.71%
PTS: 1
REF: SECTION 4.1
74. Suppose that a firm's sales were $3,750,000 five years ago and are $5,250,000 today. What was the
geometric mean growth rate in sales over the past five years?
ANS:
If Rg is the geometric mean, then
3,750,000 (1 + Rg)5 = 5,250,000  Rg = 0.0696 or 6.96%
PTS: 1
REF: SECTION 4.1
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.
CHOSUN UNIVERSITY SEOK-GANG,PARK
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
CHOSUN UNIVERSITY SEOK-GANG,PARK
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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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
98. The standard deviation is the positive square root of the variance.
CHOSUN UNIVERSITY SEOK-GANG,PARK
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
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.
CHOSUN UNIVERSITY SEOK-GANG,PARK
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
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
CHOSUN UNIVERSITY SEOK-GANG,PARK
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
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
CHOSUN UNIVERSITY SEOK-GANG,PARK
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
123. {Ages of Teachers Narrative} Calculate and compare the standard deviations for the three samples.
ANS:
CHOSUN UNIVERSITY SEOK-GANG,PARK
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.
ANS:
s2 = 4.667 (hours)2
s = 2.160 hours
CHOSUN UNIVERSITY SEOK-GANG,PARK
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:
7.1
10.8
7.4
4.1
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
PTS: 1
, fund A has the higher level of risk.
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
CHOSUN UNIVERSITY SEOK-GANG,PARK
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
CHOSUN UNIVERSITY SEOK-GANG,PARK
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
MULTIPLE CHOICE
142. When extreme values are present in a set of data, which of the following descriptive summary
measures are most appropriate?
a. Coefficient of variation and range
b. Mean and standard deviation
c. Interquartile range and median
d. Variance and interquartile range
ANS: C
PTS: 1
REF: SECTION 4.3
CHOSUN UNIVERSITY SEOK-GANG,PARK
143. The length of the box in the box plot portrays the:
a. median.
b. interquartile range.
c. range.
d. third quartile.
ANS: B
PTS: 1
REF: SECTION 4.3
144. In a negatively skewed distribution, which of the following is the correct statement?
a. The distance from Q1 to Q2 is smaller than the distance from Q2 to Q3
b. The distance from the smallest observation to Q1 is larger than the distance from Q3 to the
largest observation
c. The distance from the smallest observation to Q2 is smaller than the distance from Q2 to
the largest observation
d. The distance from Q1 to Q3 is twice the distance from the Q1 to Q2
ANS: B
PTS: 1
REF: SECTION 4.3
145. In a perfectly symmetric distribution, which of the following statements is false?
a. The distance from Q1 to Q2 equals to the distance from Q2 to Q3
b. The distance from the smallest observation to Q1 is the same as the distance from Q3 to the
largest observation
c. The distance from the smallest observation to Q2 is the same as the distance from Q2 to the
largest observation
d. The distance from Q1 to Q3 is half of the distance from the smallest to the largest
observation
ANS: D
PTS: 1
REF: SECTION 4.3
146. Which of the following summary measures cannot be easily approximated from a box plot?
a. The range
b. The interquartile range
c. The second quartile
d. The standard deviation
ANS: D
PTS: 1
REF: SECTION 4.3
CHOSUN UNIVERSITY SEOK-GANG,PARK
147. The interquartile range is the difference between the:
a. largest and smallest numbers in the data set.
b. 25th percentile and the 75th percentile.
c. median and the mean.
d. None of these choices.
ANS: B
PTS: 1
REF: SECTION 4.3
148. In a positively skewed distribution, which of the following is the correct statement?
a. The distance from Q1 to Q2 is larger than the distance from Q2 to Q3
b. The distance from Q1 to Q2 is smaller than the distance from Q2 to Q3
c. The distance from Q1 to Q2 is twice the distance from Q2 to Q3
d. The distance from Q1 to Q2 is half the distance from Q2 to Q3.
ANS: B
PTS: 1
REF: SECTION 4.3
149. Which measures of central location and variability are considered to be resistant to extreme values?
a. The mean and standard deviation.
b. The mode and variance.
c. The median and interquartile range.
d. None of these choices.
ANS: C
PTS: 1
REF: SECTION 4.3
150. Which of the following measures of variability is not sensitive to extreme values?
a. The range
b. The standard deviation
c. The interquartile range
d. The coefficient of variation
ANS: C
PTS: 1
REF: SECTION 4.3
151. Which of the following statements is true?
a. The lower or first quartile is labeled Q1 and is equal to the 25th percentile.
b. The second quartile is labeled Q2 and is equal to the median.
c. The upper or third quartile is labeled Q3 and is equal to the 75th percentile.
d. All of these choices are true.
ANS: D
PTS: 1
REF: SECTION 4.3
TRUE/FALSE
152. The distance between the 25th percentile and the median is always the same as the distance between the
median and the 75th percentile.
ANS: F
PTS: 1
REF: SECTION 4.3
153. The interquartile range will always exceed that of the range.
ANS: F
PTS: 1
REF: SECTION 4.3
CHOSUN UNIVERSITY SEOK-GANG,PARK
154. The interquartile range is found by taking the difference between the 1st and 3rd quartiles and dividing
that value by 2.
ANS: F
PTS: 1
REF: SECTION 4.3
155. Quartiles divide the observations in a data set into four parts with the same amount of data in each
part.
ANS: T
PTS: 1
REF: SECTION 4.3
156. The length of the box in a box plot portrays the interquartile range.
ANS: T
PTS: 1
REF: SECTION 4.3
157. Expressed in percentiles, the interquartile range is the difference between the 25th and 75th percentiles.
ANS: T
PTS: 1
REF: SECTION 4.3
158. If the distribution of a data set were perfectly symmetric, the distance from Q1 to the median would
always equal the distance from Q3 to the median in a box plot.
ANS: T
PTS: 1
REF: SECTION 4.3
159. The first and second quartiles of a data set can never be equal.
ANS: F
PTS: 1
REF: SECTION 4.3
160. The value for Q3 can never be smaller than the value for Q1.
ANS: T
PTS: 1
REF: SECTION 4.3
161. In symmetric data, the value for Q2 is always halfway between Q1 and Q3.
ANS: T
PTS: 1
REF: SECTION 4.3
162. Percentiles can be converted into quintiles and deciles, where quintiles divide the data into fifths and
deciles divide the data into tenths.
ANS: T
PTS: 1
REF: SECTION 4.3
163. The 5-number summary consists of the smallest observation, the first quartile, the median, the third
quartile, and the largest observation.
ANS: T
PTS: 1
REF: SECTION 4.3
164. In a negatively skewed distribution, the distance from the smallest observation to Q1 exceeds the
distance from Q3 to the largest observation.
ANS: T
PTS: 1
REF: SECTION 4.3
165. A box plot is a graphical representation of the 5-number summary.
ANS: T
PTS: 1
REF: SECTION 4.3
CHOSUN UNIVERSITY SEOK-GANG,PARK
166. In a positively skewed distribution, the percentage of data between the smallest observation and Q1 is
less than the percentage of data between Q3 and the largest observation.
ANS: F
PTS: 1
REF: SECTION 4.3
167. The interquartile range is an interval of numbers starting at Q1 and ending at Q3.
ANS: F
PTS: 1
REF: SECTION 4.3
168. The line drawn within the box of a box plot always represents the mean.
ANS: F
PTS: 1
REF: SECTION 4.3
169. The line drawn within the box of a box plot always represents the median.
ANS: T
PTS: 1
REF: SECTION 4.3
170. The interquartile range is a measure of variability in a set of data.
ANS: T
PTS: 1
REF: SECTION 4.3
171. Expressed in percentiles, the fifth decile is the 50th percentile or the median.
ANS: T
PTS: 1
REF: SECTION 4.3
COMPLETION
172. If the first and second quartiles are closer to each other than are the second and third quartiles, the
shape of the histogram based on the quartiles is ____________________.
ANS: positively skewed
PTS: 1
REF: SECTION 4.3
173. If the first and second quartiles are farther apart than the second and third quartiles, the shape of the
histogram based on the quartiles is ____________________.
ANS: negatively skewed
PTS: 1
REF: SECTION 4.3
174. ____________________ are extremely large or extremely small observations.
ANS: Outliers
PTS: 1
REF: SECTION 4.3
175. The middle line inside the box in a box plot represents the ____________________.
ANS: median
PTS: 1
REF: SECTION 4.3
176. The horizontal lines extending to the left and to the right of the box in a box plot are called
____________________.
CHOSUN UNIVERSITY SEOK-GANG,PARK
ANS: whiskers
PTS: 1
REF: SECTION 4.3
177. Any points that lie outside the whiskers on a box plot are called ____________________.
ANS: outliers
PTS: 1
REF: SECTION 4.3
178. The ____________________ measures the spread between the middle 50% of the observations.
ANS: interquartile range
PTS: 1
REF: SECTION 4.3
179. The 10th ____________________ is the value for which 10% of the observations are less than that
value.
ANS: percentile
PTS: 1
REF: SECTION 4.3
180. A percentile is a measure of ____________________ standing.
ANS: relative
PTS: 1
REF: SECTION 4.3
181. Q2 is another name for the ____________________.
ANS: median
PTS: 1
REF: SECTION 4.3
SHORT ANSWER
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.
182. {Weights of Workers Narrative} Determine the location and value of the lower quartile of the weights.
ANS:
Location = L25 = 6.5, and value of Q1 = 148 + 0.50(151  148) = 149.5 pounds.
PTS: 1
REF: SECTION 4.3
183. {Weights of Workers Narrative} Determine the location and value of the second quartile of the
weights.
CHOSUN UNIVERSITY SEOK-GANG,PARK
ANS:
Location = L50 = 13, and value of Q2 is 162 pounds.
PTS: 1
REF: SECTION 4.3
184. {Weights of Workers Narrative} Determine the location and value of the upper quartile of the weights.
ANS:
Location = L75 = 19.5, and value of Q3 = 169 + 0.50(171  169) = 170 pounds.
PTS: 1
REF: SECTION 4.3
185. {Weights of Workers Narrative} Describe the shape of the distribution of weights based on the
quartiles' values.
ANS:
Since Q2  Q1 > Q3  Q2, the distribution of weights are negatively skewed.
PTS: 1
REF: SECTION 4.3
186. {Weights of Workers Narrative} Determine the location and value of the 60th percentile of the
weights.
ANS:
Location = L60 = 15.6, value of the 60th percentile = 164 + 0.60(165  164) = 164.6 pounds.
PTS: 1
REF: SECTION 4.3
187. {Weights of Workers Narrative} Construct a frequency distribution for the data, using five class
intervals, and the value 130 as the lower limit of the first class.
ANS:
Class Limits
[130, 140)
[140, 150)
[150, 160)
[160, 170)
[170, 180)
Total
PTS: 1
Frequency
2
4
6
7
6
25
REF: SECTION 4.3
188. {Weights of Workers Narrative} Construct a relative frequency histogram for the data, using five class
intervals and the value 130 as the lower limit of the first class.
CHOSUN UNIVERSITY SEOK-GANG,PARK
ANS:
PTS: 1
REF: SECTION 4.3
189. {Weights of Workers Narrative} What does the histogram tell you about the distribution of the weights
of workers?
ANS:
The histogram indicates negative skewness.
PTS: 1
REF: SECTION 4.3
190. {Weights of Workers Narrative}
a.
b.
c.
Construct a box plot for the weights.
Are there any extreme values?
What does the box plot tell you about the distribution of the data?
ANS:
a.
b.
c.
There are no extreme values.
The box plot indicates negative skewness because the left side of the box is longer than
the right side of the box.
PTS: 1
REF: SECTION 4.3
191. {Weights of Workers Narrative} Calculate the 3rd and 7th deciles of the data.
ANS:
Location = L30 = 7.8, and value of 3rd decile = 151 + 0.80(152  151) = 151.80
Location = L70 =18.2, and value of 7th decile = 168 + 0.20(169  168) = 168.20
CHOSUN UNIVERSITY SEOK-GANG,PARK
PTS: 1
REF: SECTION 4.3
Hours of TV Watching
Suppose that the following data provide the hours of television viewing per week for a sample of 15
high school students in Grand Rapids, Michigan: 5, 11, 25, 19, 18, 20, 27, 13, 8, 10, 15, 19, 18, 9, and
12.
192. {Hours of TV Watching Narrative} Determine the location and value of the first quartile.
ANS:
Location = L25 = 4, and value of Q1 = 10 hours.
PTS: 1
REF: SECTION 4.3
193. {Hours of TV Watching Narrative} Determine the location and value of the second quartile.
ANS:
Location = L50 = 8, and value of Q2 = 15 hours.
PTS: 1
REF: SECTION 4.3
194. {Hours of TV Watching Narrative} Determine the location and value of the third quartile.
ANS:
Location = L75 = 12, and value of Q3 = 19 hours.
PTS: 1
REF: SECTION 4.3
195. {Hours of TV Watching Narrative} Calculate and interpret the interquartile range.
ANS:
IQR = Q3  Q1 = 9 hours. There is a 9 hour range between the middle 50% of the TV viewing hours.
PTS: 1
REF: SECTION 4.3
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.
196. {Ages of Employees Narrative} Find the lower quartile of the ages.
ANS:
Location = L25 = 6.5, and value of Q1 = 33 + 0.50(35  33) = 34 years
PTS: 1
REF: SECTION 4.3
197. {Ages of Employees Narrative} Find the upper quartile of the ages.
CHOSUN UNIVERSITY SEOK-GANG,PARK
ANS:
Location = L75 =19.5, and value of Q3 = 48 + 0.50(49  48) = 48.50 years
PTS: 1
REF: SECTION 4.3
198. {Ages of Employees Narrative} Compute the interquartile range of the data, and interpret its meaning.
ANS:
IQR = Q3  Q1 = 48.5  34 = 14.5 years. There is a 14.5 year gap between the middle 50% of the ages
of employers from that government department.
PTS: 1
REF: SECTION 4.3
199. {Ages of Employees Narrative}
a.
b.
Construct a box plot for the ages and identify any extreme values.
What does the box plot tell you about the distribution of the data?
ANS:
a.
The box plot lies below. There are no extreme values.
b.
The box plot indicates approximate symmetry (perhaps a small amount of negative s
skewness).
PTS: 1
REF: SECTION 4.3
200. {Ages of Employees Narrative} Construct a relative frequency distribution for the data, using five
class intervals and the value 20 as the lower limit of the first class.
ANS:
Class Limits
[20, 30)
[30, 40)
[40, 50)
[50, 60)
[60, 70)
Total
Frequency
0.16
0.28
0.36
0.12
0.08
1.00
PTS: 1
REF: SECTION 4.3
201. {Ages of Employees Narrative}
a.
Construct a relative frequency histogram for the data.
CHOSUN UNIVERSITY SEOK-GANG,PARK
b.
What does the histogram tell you about the distribution of the data?
ANS:
a.
b.
The histogram is close to symmetric.
PTS: 1
REF: SECTION 4.3
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.
202. {Salaries of Employees Narrative} Compute the lower quartile for the salaries.
ANS:
Location = L25 = 3.5, and value of Q1 = 22.7 + 0.50(23.5  22.7) = 23.1 = $23,100.
PTS: 1
REF: SECTION 4.3
203. {Salaries of Employees Narrative} Compute median.
ANS:
Location = L50 = 7, and value of Q2 is the median, which is $24,300.
PTS: 1
REF: SECTION 4.3
204. {Salaries of Employees Narrative} Compute the upper quartile.
ANS:
Location = L75 = 10.5, and value of Q3 = 26.5 + 0.50(27.2  26.5) = 26.85 = $26,850.
PTS: 1
REF: SECTION 4.3
205. {Salaries of Employees Narrative}
a.
Describe the shape of distribution of salaries of employees of the firm based on the values
of the quartiles.
CHOSUN UNIVERSITY SEOK-GANG,PARK
b.
Give a possible reason for the shape of this data set, in terms of salaries.
ANS:
a.
b.
Since Q2  Q1 = 1.2 < Q3  Q2 = 2.55, then the distribution of salaries must be positively
skewed.
Most employees make a 'typical' salary, but some also make higher salaries. The higher
the salary goes, the fewer the number of people that make that salary.
PTS: 1
REF: SECTION 4.3
206. {Salaries of Employees Narrative} Compute and interpret the 90th percentile.
ANS:
Location = L90 = 12.6, and value of the 90th percentile = 27.9 + .60(28.2  27.9) = 28.08. This means
90% of the employees have a salary less than $28,080.
PTS: 1
REF: SECTION 4.3
Test Scores
Suppose that an analysis of a set of test scores reveals that: Q1 = 45, Q2 = 85, Q3 = 105.
207. {Test Scores Narrative} What do these statistics tell you about the shape of the distribution of test
scores?
ANS:
The fact that Q2  Q1 = 40 is greater that Q3  Q2 = 20 indicates that test scores are negatively skewed.
PTS: 1
REF: SECTION 4.3
208. {Test Scores Narrative} What can you say about the relative position of each of the observations 34,
84, and 104, within this data set?
ANS:
Since 34 is less than Q1, it is among the lowest 25% of the observations. The score of 84 is a just bit
smaller than the middle value, which is Q2 = 85. Since Q3 = 105, the score of 104 is larger than about
75% of the observations.
PTS: 1
REF: SECTION 4.3
209. {Test Scores Narrative} Calculate the interquartile range. What does this tell you about the data?
ANS:
IQR = Q3  Q1 = 60. This means that the middle 50% of the test scores have a range of 60 points.
PTS: 1
REF: SECTION 4.3
Ages of Senior Citizens
CHOSUN UNIVERSITY SEOK-GANG,PARK
A sociologist recently conducted a survey of citizens over 65 years of age whose net worth is too high
to qualify for Medicaid and who have no private health insurance. The ages of 20 uninsured senior
citizens were as follows: 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 78, 79, 80, 81, 86, 87, 91, 92, 94, and
97.
210. {Ages of Senior Citizens Narrative} Calculate the first quartile of the ages of the uninsured senior
citizens.
ANS:
Location = 5.25, and value of Q1 = 69 + .25(70  69) = 69.25 years.
PTS: 1
REF: SECTION 4.3
211. {Ages of Senior Citizens Narrative} Calculate the third quartile of the ages of the uninsured senior
citizens.
ANS:
Location = 15.75, and value of Q3 = 86 + .75(87  86) = 86.75 years.
PTS: 1
REF: SECTION 4.3
212. {Ages of Senior Citizens Narrative} Identify the interquartile range of the ages of the uninsured senior
citizens. What does this value tell you about the data?
ANS:
IQR = Q3  Q1 = 17.5 years. This means that the middle 50% of senior citizens whose net worth is too
high to qualify for Medicaid and who have no private health insurance have ages ranging within 17.5
years of each other.
PTS: 1
REF: SECTION 4.3
213. {Ages of Senior Citizens Narrative} What does the value of the first quartile tell you?
ANS:
One fourth of the senior citizens sampled are below 69.25 years of age.
PTS: 1
REF: SECTION 4.3
214. {Ages of Senior Citizens Narrative} What does the value of the third quartile tell you?
ANS:
Three fourths of the senior citizens are under 86.75 years of age. (Or, one forth of them are over 86.75
years old!)
PTS: 1
REF: SECTION 4.3
CHOSUN UNIVERSITY SEOK-GANG,PARK
215. {Ages of Senior Citizens Narrative} Describe the shape of the distribution of ages by comparing the
quartiles.
ANS:
The minimum age is 65; the value of Q1 is 69 + .25(70  69) = 69.25 years old; the median is (75 + 78)
/ 2 = 76.5 years old; Q3 = 86 + .75(87  86) = 86.75 years old; and the maximum age is 97. Since the
distances between each of these statistics increases in order, the ages are positively skewed (meaning
many at a 'typical' age and a few that are older than the rest).
PTS: 1
REF: SECTION 4.3
MULTIPLE CHOICE
216. Assuming a linear relationship between X and Y, if the coefficient of correlation (r) equals 0.80, this
means that:
a. there is very weak correlation.
b. the slope b1 is = 0.80.
c. the value of X is always greater than the value of Y.
d. None of these choices are true.
ANS: B
PTS: 1
REF: SECTION 4.4
217. The slope b1 of the least squares line
represents the:
a. predicted value of Y when X = 0.
b. estimated average change in Y per unit change in X.
c. predicted value of Y.
d. variation around the regression line.
ANS: B
PTS: 1
REF: SECTION 4.4
218. Generally speaking, if two variables are unrelated (as one increases, the other shows no pattern), the
covariance will be:
a. a large positive number.
b. a large negative number.
c. a positive or negative number close to zero.
d. None of these choices.
ANS: C
PTS: 1
REF: SECTION 4.4
219. If the correlation coefficient r = 1.00, then all the observations must fall exactly on:
a. a straight line with a slope that equals 1.00.
b. a straight line with a negative slope.
c. a straight line with a positive slope.
d. a horizontal straight line with a zero slope.
ANS: C
PTS: 1
REF: SECTION 4.4
220. The Y-intercept, b0, of the least squares line
represents the:
a. estimated average value of Y when X = 0.
b. estimated average change in Y per unit change in X.
c. predicted value of Y.
d. variation around the sample regression line.
ANS: A
PTS: 1
REF: SECTION 4.4
CHOSUN UNIVERSITY SEOK-GANG,PARK
221. A perfect straight line sloping downward would produce a correlation coefficient equal to:
a. +1.0
b. 1.0
c. 0.0
d. Cannot tell from the information given.
ANS: B
PTS: 1
REF: SECTION 4.4
CHOSUN UNIVERSITY SEOK-GANG,PARK
222. The least squares method minimizes which of the following sum of squares?
a.
b.
c.
d. All of these choices are true
ANS: B
PTS: 1
REF: SECTION 4.4
223. Which of the following is a property of r, the coefficient of correlation?
a. r always lies between 0 and 1.
b. r has no units.
c. If you switch the values of X and Y, the sign of r changes.
d. All of these choices are true.
ANS: B
PTS: 1
REF: SECTION 4.4
224. Which of the following is a property of the slope, b1?
a. The slope equals one if X and Y have the same variance.
b. The slope has the same sign as r, the coefficient of correlation.
c. The slope equals one if r equals one.
d. All of these choices are true.
ANS: B
PTS: 1
REF: SECTION 4.4
225. Which of the following are measures of the linear relationship between two variables?
a. The covariance
b. The coefficient of correlation
c. The variance
d. Both a and b
ANS: D
PTS: 1
REF: SECTION 4.4
226. The strength of the linear relationship between two interval variables can be measured by the:
a. coefficient of variation.
b. coefficient of correlation.
c. slope of the regression line.
d. Y-intercept.
ANS: B
PTS: 1
REF: SECTION 4.4
227. The denominator in the calculation of the sample covariance, cov (x,y), is:
a. n  2
b. n  1
c. n
d. 2n  1
ANS: B
228. If cov(x, y) = 20,
PTS: 1
and
REF: SECTION 4.4
then the sample coefficient of correlation r is:
CHOSUN UNIVERSITY SEOK-GANG,PARK
a.
b.
c.
d.
1.400
0.026
0.710
None of these choices.
ANS: C
PTS: 1
REF: SECTION 4.4
TRUE/FALSE
229. If the covariance of x and y is 26.16, and the standard deviation of x is 32.7, then the slope of the least
squares line is b1 =.80.
ANS: F
PTS: 1
230. If
,
squares line is b0 = 276.08.
ANS: T
PTS: 1
REF: SECTION 4.4
, n = 12, and the slope equals 0.5, then the y-intercept of the least
REF: SECTION 4.4
231. If the coefficient of correlation r = 0, there can be no linear relationship between the dependent
variable y and the independent variable x.
ANS: T
PTS: 1
REF: SECTION 4.4
232. If the coefficient of correlation r = 0, there can be no relationship whatsoever between the dependent
variable y and the independent variable x.
ANS: F
PTS: 1
REF: SECTION 4.4
233. If the coefficient of correlation r = .81, the standard deviations of x and y are 20 and 25, respectively,
then cov(x, y) must be 405.0.
ANS: T
PTS: 1
REF: SECTION 4.4
234. The advantage that the coefficient of correlation has over the covariance is that the former has set
lower and upper limits.
ANS: T
PTS: 1
REF: SECTION 4.4
235. If the standard deviations of x and y are 12.5 and 10.8, respectively, and the covariance is 118.8, then
the coefficient of correlation r is 0.88.
ANS: T
PTS: 1
REF: SECTION 4.4
236. Generally speaking, if two variables are unrelated, the covariance will be a number close to zero.
ANS: T
PTS: 1
REF: SECTION 4.4
237. If the standard deviation of x is 18, the covariance of x and y is 120, the coefficient r = 0.90, then the
standard deviation of y is 54.87.
ANS: F
PTS: 1
REF: SECTION 4.4
CHOSUN UNIVERSITY SEOK-GANG,PARK
238. Three measures of the linear relationship between x and y are the coefficient of correlation, the
coefficient of determination, and the coefficient of variation.
ANS: F
PTS: 1
REF: SECTION 4.4
239. The coefficient of correlation r is a number that indicates the direction and the strength of the linear
relationship between the dependent variable y and the independent variable x.
ANS: T
PTS: 1
REF: SECTION 4.4
240. The variance is a measure of the linear relationship between two variables
ANS: F
PTS: 1
REF: SECTION 4.4
241. A perfect straight line sloping upward would produce a correlation coefficient value of 1.0.
ANS: T
PTS: 1
REF: SECTION 4.4
242. When the standard deviation is expressed as a percentage of the mean, the result is the coefficient of
correlation.
ANS: F
PTS: 1
REF: SECTION 4.4
243. If the coefficient of correlation r = 1, then the best-fit linear equation will actually include all of the
observations.
ANS: T
PTS: 1
REF: SECTION 4.4
COMPLETION
244. The ____________________ of the correlation indicates the direction of a linear relationship.
ANS: sign
PTS: 1
REF: SECTION 4.4
245. The magnitude of the correlation measures the ____________________ of a linear relationship.
ANS: strength
PTS: 1
REF: SECTION 4.4
CHOSUN UNIVERSITY SEOK-GANG,PARK
246. The ____________________ of a linear relationship is hard to interpret from the covariance, but it is
easy to interpret from the correlation.
ANS:
magnitude
strength
PTS: 1
REF: SECTION 4.4
247. The method used to find the best fitting line through the observations is called the
____________________ method.
ANS: least squares
PTS: 1
REF: SECTION 4.4
248. In the equation of the least squares line,
____________________.
, b0 is the ____________________ and b1 is the
ANS: y-intercept; slope
PTS: 1
REF: SECTION 4.4
249. The ____________________ measures the margin of relative change in the dependent variable.
ANS: slope
PTS: 1
REF: SECTION 4.4
250. The y-intercept of the least squares line is the point on the line where ________________ = 0.
ANS: x
PTS: 1
REF: SECTION 4.4
251. When two variables x and y are linearly related, it does not necessarily mean that x
____________________ y.
ANS: causes
PTS: 1
REF: SECTION 4.4
252. The coefficient of determination is the percentage of variation in the ____________________ variable
that is explained by the variation in the ____________________ variable.
ANS:
dependent; independent
y; x
PTS: 1
REF: SECTION 4.4
CHOSUN UNIVERSITY SEOK-GANG,PARK
SHORT ANSWER
253. Given the following sample data:
x
y
a.
b.
c.
d.
420
2.80
610
3.60
625
3.75
500
3.00
400
2.50
450
2.70
550
3.50
650
3.90
480
2.95
565
3.30
Calculate the covariance and the correlation coefficient.
Comment on the relationship between x and y.
Determine the least squares line.
Draw the scatter diagram and plot the least squares line.
ANS:
a.
b.
c.
There is a very strong positive linear relationship between X and Y.
The least squares line is
.
d.
PTS: 1
REF: SECTION 4.4
Experience and Salary
A sample of eight observations of variables x (years of experience) and y (salary in $1,000s) is shown
below:
x
y
5
20
3
23
7
15
9
11
2
27
4
21
6
17
254. {Experience and Salary Narrative}
a.
b.
Calculate and interpret the covariance between x and y.
Give a possible reason that the covariance is negative.
ANS:
8
14
CHOSUN UNIVERSITY SEOK-GANG,PARK
a.
b.
The sign indicates a negative relationship. That is, as years of experience increases, salary
tends to decrease. The magnitude of the covariance cannot be interpreted.
The reason for the negative relationship may be that salaries become compressed over
time and are not adjusted upwards for changing market values.
PTS: 1
REF: SECTION 4.4
255. {Experience and Salary Narrative}
a.
b.
Calculate the coefficient of correlation, and comment on the relationship between x and y.
Give a possible reason that the correlation is negative.
ANS:
a.
b.
. There is a very strong (almost perfect) negative
linear relationship between X and Y. That is, as years of experience increases, salary tends
to decrease.
The reason for the negative relationship may be that salaries become compressed over
time and are not adjusted upwards for changing market values.
PTS: 1
REF: SECTION 4.4
256. {Experience and Salary Narrative} Determine the least squares line, and use it to estimate the value of
y for x = 6.
ANS:
.
When x = 6 years of experience,
PTS: 1
or $17,441
REF: SECTION 4.4
257. {Experience and Salary Narrative} Draw the scatter diagram and plot the least squares line.
ANS:
PTS: 1
REF: SECTION 4.4
258. How is the value of the correlation coefficient r affected in each of the following cases?
CHOSUN UNIVERSITY SEOK-GANG,PARK
a.
b.
c.
Each x value and y is multiplied by 4.
Each x value is switched with the corresponding y value.
Each x value is increased by 2.
ANS:
a.
b.
c.
The correlation coefficient r does not change.
The correlation coefficient r does not change.
The correlation coefficient r does not change.
PTS: 1
REF: SECTION 4.4
259. Consider the following sample:
x
y
a.
b.
9
19
6
14
7
16
5
12
8
15
Calculate the covariance and the coefficient of correlation for the sample.
What do these statistics tell you about the relationship between x and y?
ANS:
a.
b.
The covariance tells us that the two variables are positively related. The coefficient of
correlation tells us that there is a strong positive linear relationship between X and Y.
PTS: 1
REF: SECTION 4.4
MULTIPLE CHOICE
260. Which of the following characteristics does a histogram help you identify?
a. The shape of distribution.
b. The approximate center of a distribution.
c. The amount of spread in a distribution.
d. All of these choices are true.
ANS: D
PTS: 1
REF: SECTION 4.7-4.8
261. The shape of a distribution helps answer which question about the data?
a. Where is the approximate center of the distribution?
b. Are the observations close to one another, or are they widely dispersed?
c. Is the distribution symmetric?
d. All of these choices are true.
ANS: D
PTS: 1
REF: SECTION 4.7-4.8
262. Which of the following is correct about the shape of a distribution?
CHOSUN UNIVERSITY SEOK-GANG,PARK
a.
b.
c.
d.
The shape can show you how many modes there are.
The shape can help you determine the approximate center of the distribution.
The shape can help you determine whether the data are close or spread out.
All of these choices are true.
ANS: D
PTS: 1
REF: SECTION 4.7-4.8
263. A scatter diagram reveals a strong positive linear relationship between oil and gasoline prices. Which
of the following numerical techniques will not give us more detailed information about this
relationship?
a. Coefficient of determination
b. Coefficient of correlation
c. Coefficient of variation.
d. All of these choices help us describe this relationship.
ANS: C
PTS: 1
REF: SECTION 4.7-4.8
TRUE/FALSE
264. Statisticians typically apply graphical techniques as a first step in a data analysis because we first need
to know the shape of the distribution.
ANS: T
PTS: 1
REF: SECTION 4.7-4.8
265. Graphical and numerical techniques, such as histograms and least squares lines provide identical
information.
ANS: F
PTS: 1
REF: SECTION 4.7-4.8
CHOSUN UNIVERSITY SEOK-GANG,PARK
266. The coefficient of correlation and the least squares line both describe the relationship between two
interval variables.
ANS: T
PTS: 1
REF: SECTION 4.7-4.8
267. The precision provided by the numerical techniques (mean, median, and standard deviation) provides
more useful information than graphical techniques (histograms and box plots) alone.
ANS: T
PTS: 1
REF: SECTION 4.7-4.8
COMPLETION
268. Scatter diagrams, covariance, and the coefficient of correlation are useful techniques for detecting
relationships between two ____________________ variables.
ANS: interval
PTS: 1
REF: SECTION 4.7-4.8
269. Statisticians usually apply graphical techniques as a first step in analyzing data because we first need
to know the ____________________ of the distribution.
ANS: shape
PTS: 1
REF: SECTION 4.7-4.8
270. Box plots and medians work well to describe ____________________ data, while histograms and
means work well to describe ____________________ data.
ANS: skewed; symmetric
PTS: 1
REF: SECTION 4.7-4.8
271. ____________________ techniques give you the big picture of a distribution, and
____________________ techniques give you more precise details.
ANS: Graphical; numerical
PTS: 1
REF: SECTION 4.7-4.8
272. We can frequently make several inferences about the nature of the data from the
____________________ of its histogram.
ANS: shape
PTS: 1
REF: SECTION 4.7-4.8
SHORT ANSWER
273. What statistics and graphs can you use to answer the following question: Where is the approximate
center of the distribution?
ANS:
CHOSUN UNIVERSITY SEOK-GANG,PARK
The median is a better measure of center for skewed data, and the mean works well for symmetric
data. If the data is skewed, a box plot can show you where the median is. If the data is symmetric, a
histogram can give you an idea where the mean is.
PTS: 1
REF: SECTION 4.7-4.8
274. What statistics and graphs can you use to answer the following question: Are the observations close to
one another, or are they widely dispersed?
ANS:
The standard deviation can tell you whether the observations are close to or spread out from the
middle. The interquartile range tells how far apart them middle 50% of the data are, and works
especially well for skewed data. A histogram and a box plot can show you how close together/spread
out the data appear visually.
PTS: 1
REF: SECTION 4.7-4.8
275. What statistics and graphs can you use to answer the following questions: Is the distribution unimodal,
bimodal, or multimodal? If there is more than one mode, where are the peaks, and where are the
valleys?
ANS:
A histogram can show you the number of modes of a distribution, as well as where they are located. A
box plot cannot tell you where any of the modes of a distribution are, or even how many might exist. It
just shows you distances between certain milestones in the distribution, such as the five-number
summary. This is a drawback of a box plot.
PTS: 1
REF: SECTION 4.7-4.8
276. What statistics and graphs can you use to answer the following question: Is the distribution symmetric?
If not, is it skewed? If symmetric, is it bell shaped?
ANS:
A histogram can show you whether a distribution is symmetric, positively skewed, negatively skewed,
or has a different shape. If the data is symmetric, a histogram can tell you whether the data is bell
shaped, bimodal, or uniform, etc. A box plot can also show you whether a distribution is symmetric,
negatively skewed or positively skewed. However, it can't tell you what the exact shape is, if the data
are found to be symmetric. For example, a box plot cannot tell you whether a symmetric data set is
unimodal or bimodal. This is a drawback of a box plot.
PTS: 1
REF: SECTION 4.7-4.8
277. What statistics and graphs can you use to look for a relationship between two interval variables?
ANS:
Scatter diagrams, covariance, the coefficient of correlation, and the coefficient of determination are
useful techniques for detecting relationships between two interval variables.
PTS: 1
REF: SECTION 4.7-4.8