65. 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.
66. 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.
67. 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.
68. 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.
69. 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:
70. There are three statistics used to measure variability in a data set; the range, the
____________________, and the ____________________.
71. The ____________________ is the square root of the ____________________.
72. The ____________________ is the least effective of all the measures of variability.
73. The ____________________ uses both the mean and the standard deviation to interpret standard
deviation for bell shaped histograms.
74. ____________________ uses both the mean and the standard deviation to interpret standard deviation
for histograms of any shape.
75. A statistic that interprets the standard deviation relative to the size of the numbers in the data set is
called the ____________________ of ____________________.
76. The range, variance, standard deviation, and coefficient of variation are to be used only on
____________________ data.
77. 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.
Standard deviation
b.
Coefficient of variation
c.
Compare the standard deviation and coefficient of variation and use them to discuss the
variability in the data.
a.
b.
cv = 0.232
c.
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.
78. 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.
Compute the range.
b.
Compute the variance.
c.
Compute the standard deviation.
d.
Compute the coefficient of variation.
e.
Explain why in this case range > variance > standard deviation > coefficient of variation.
ANS:
5
b.
2.77
1.66
d.
0.87
Weights of Teachers
The following data represent the weights in pounds of a sample of 25 teachers: 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.
79. {Weights of Teachers Narrative} Compute the sample variance and sample standard deviation.
80. {Weights of Teachers Narrative} Compute the range and coefficient of variation.
81. {Weights of Teachers Narrative} Which is a better measure of variability in the weights of the
teachers, the standard deviation or the coefficient of variation?
82. Is it possible for the standard deviation of a data set to be larger than its variance? Explain.
Ages of Workers
The ages (in years) of three groups of workers are shown below:
Group A:
17
22
20
18
23
Group B:
30
28
35
40
25
Group C:
44
39
54
21
52
83. {Ages of Workers Narrative} Calculate and compare the standard deviations for the three samples.
84. {Ages of Workers Narrative} Compute and compare the ranges for the three groups.
85. {Ages of Workers Narrative} Compute and compare the coefficient of variation for the three samples.
86. 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.
87. The number of hours a college student spent studying during the final exam week was recorded as
follows: 6, 5, 2, 8, 7, 4, and 9. Compute the range for the data, express the number in the appropriate
unit.
88. 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.
89. 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:
7.1
−7.4
19.7
−3.9
41.7
23.2
4.0
1.9
29.3
Fund B:
10.8
−4.1
5.1
10.9
24.0
16.9
9.4
−2.6
10.1
Ages of Volunteers
The following data represent the ages in years of a sample of 25 volunteers from a charitable
organization: 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.
90. {Ages of Volunteers Narrative} Compute the range of the data and express the number in the
appropriate unit.
91. {Ages of Volunteers Narrative} Compute the sample variance and sample standard deviation, and
express the numbers in the appropriate units.
92. {Ages of Volunteers Narrative} Compute the coefficient of variation and express the number in the
appropriate unit.
Salaries of Office Workers
The following data represent the salaries (in thousands of dollars) of a sample of 13 office workers 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.
93. {Salaries of Office Workers Narrative} Compute the variance and standard deviation of the salaries,
and express the numbers in the appropriate units.
94. {Salaries of Office Workers Narrative} Compute the coefficient of variation and express the number in
the appropriate unit.
95. {Salaries of Office Workers Narrative} Compute the range.
96. 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.
Milk Demand
A supermarket has determined that daily demand for milk containers has an approximate bell shaped
distribution, with a mean of 55 containers and a standard deviation of six containers.
97. {Milk Demand Narrative} How often can we expect between 49 and 61 containers to be sold in a day?
(Give a percentage.)
98. {Milk Demand Narrative} What percentage of the time will the number of containers of milk sold be
more than 2 standard deviations from the mean?
99. {Milk Demand Narrative} If the supermarket begins each morning with a supply of 67 containers of
milk, how often will demand exceed the supply? (Give a percentage.)
ANS:
100. A sample of 13 college professors has a mean age of 30 years and a standard deviation of 5 years.
Suppose that the sample is enlarged to 15 college professors, by including two additional professors
that are each 30 years old. Will the standard deviation increase, decrease, or stay the same, and why?
101. 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.
11.5 and 15.5?
b.
9.5 and 17.5?
c.
7.5 and 19.5?
a.
The interval contains approximately 68% of the ratios, according to the Empirical Rule.
b.
The interval contains approximately 95% of the ratios according to the Empirical Rule.
c.
The interval contains approximately 99.7% of the ratios according to the Empirical Rule.
102. The distance between the 25th percentile and the median is always the same as the distance between
the median and the 75th percentile.
103. The interquartile range will always exceed that of the range.
104. The interquartile range is found by taking the difference between the 1st and 3rd quartiles and dividing
that value by 2.
105. Quartiles divide the observations in a data set into four parts with the same amount of data in each
part.
106. The length of the box in a box plot portrays the interquartile range.
107. Expressed in percentiles, the interquartile range is the difference between the 25th and 75th
percentiles.
108. 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.
109. The first and second quartiles of a data set can never be equal.
110. The value for Q3 can never be smaller than the value for Q1.
111. In symmetric data, the value for Q2 is always halfway between Q1 and Q3.
112. Percentiles can be converted into quintiles and deciles, where quintiles divide the data into fifths, and
deciles divide the data into tenths.
113. The 5-number summary consists of the smallest observation, the first quartile, the median, the third
quartile, and the largest observation.
114. In a negatively skewed distribution, the distance from the smallest observation to Q1 exceeds the
distance from Q3 to the largest observation.
115. A box plot is a graphical representation of the 5-number summary.
116. 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.
117. The interquartile range is an interval of numbers starting at Q1 and ending at Q3.
118. The line drawn within the box of a box plot always represents the mean.
119. The line drawn within the box of a box plot always represents the median.
120. The interquartile range is a measure of variability in a set of data.
121. Expressed in percentiles, the fifth decile is the 50th percentile or the median.
122. 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
123. The length of the box in the box plot portrays the:
a.
median.
b.
interquartile range.
c.
range.
d.
third quartile.
124. 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 Q1 to Q2 is larger than the distance from Q2 to Q3.
c.
The distance from Q1 to Q2 is half the distance from Q2 to Q3.
d.
The distance from Q1 to Q3 is twice the distance from the Q1 to Q2.
125. 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.