Chapter 3 – Descriptive Statistics: Numerical Measures
d.
the mean
87. A graphical summary of data that is based on a five-number summary is a
a.
histogram
b.
stem-and-leaf display
c.
scatter diagram
d.
box plot
88. A numerical measure of linear association between two variables is the
a.
b.
c.
d.
89. Positive values of covariance indicate
a.
a positive variance of the x values
b.
a positive variance of the y values
c.
the standard deviation is positive
d.
a positive relation between the x and the y variables
90. Excel’s __________ function can be used to compute the sample covariance.
a.
MAX
b.
COVARIANCE.S
c.
VAR
d.
CORREL
91. A numerical measure of linear association between two variables is the
a.
variance
b.
z-score
c.
correlation coefficient
d.
None of the other answers are correct.
c
92. The correlation coefficient ranges between
Chapter 3 – Descriptive Statistics: Numerical Measures
a.
0 and 1
b.
−1 and +1
c.
minus infinity and plus infinity
d.
1 and 100
93. The correlation coefficient
a.
is the same as the covariance
b.
can be larger than 1
c.
cannot be larger than 1
d.
cannot be negative
c
94. Excel’s __________ function can be used to compute the sample correlation coefficient.
a.
MAX
b.
COVAR
c.
VAR
d.
CORREL
95. A mean computed in such a way that each data value is given a weight reflecting its importance is referred to as
a.
an important mean
b.
a trimmed mean
c.
a weighted mean
d.
a heavy mean
c
96. In computing descriptive statistics from grouped data,
a.
data values are treated as if they occur at the midpoint of a class
b.
the grouped data result is more accurate than the ungrouped result
c.
the grouped data computations are used only when a population is being analyzed
d.
All of these answers are correct.
a
97. When should measures of location and dispersion be computed from grouped data rather than from individual data
values?
a.
as much as possible since computations are easier
b.
only when individual data values are unavailable
c.
whenever computer packages for descriptive statistics are unavailable
d.
only when the data are from a population
Chapter 3 – Descriptive Statistics: Numerical Measures
b
1
98. An important numerical measure of the shape of a distribution is the
a.
correlation coefficient
b.
variance
c.
skewness
d.
relative location
c
1
99. If the data distribution is symmetric, the skewness is
a.
0
b.
.5
c.
1
d.
None of the other answers is correct.
a
1
100. For data skewed to the left, the skewness is
a.
between 0 and .5
b.
less than 1
c.
positive
d.
negative
d
1
101. When the data are positively skewed, the mean will usually be
a.
less than the median
b.
greater than the median
c.
less than the mode
d.
greater than the mode
b
1
102. Which of the following statements about financial growth factors using the geometric mean is untrue?
a.
The growth factor cannot be less than zero.
b.
A growth factor less than 1 indicates negative growth.
c.
A growth factor greater than 2 indicates more than 100 percent growth.
d.
A growth factor greater than 0 indicates positive growth.
d
1
103. In a box plot, data value x is considered an outlier if
a.
x < Q1 – 1.0(IQR) or x > Q3 + 1.0(IQR)
Chapter 3 – Descriptive Statistics: Numerical Measures
b.
x < Q2 – 1.0(IQR) or x > Q2 + 1.0(IQR)
c.
x < Q1 – 1.5(IQR) or x > Q3 + 1.5(IQR)
d.
x < Q2 – 1.5(IQR) or x > Q2 + 1.5(IQR)
c
104. The empirical rule states that, for data having a bell-shaped distribution, the percentage of data values being within
two standard deviations of the mean is approximately
a.
65
b.
75
c.
85
d.
95
105. The empirical rule states that, for data having a bell-shaped distribution, the percentage of data values being within
three standard deviations of the mean is
a.
approximately 85
b.
approximately 90
c.
approximately 95
d.
nearly 100
106. The textbook authors recommend treating any data value with a z-score less than ___ or greater than ___ as an
outlier.
a.
-2.5, +2.5
b.
-3.0, +3.0
c.
-3.5, +3.5
d.
-4.0, +4.0
Subjective Short Answer
107. The hourly wages of a sample of eight individuals is given below.
Individual
Hourly Wage ($)
A
27
B
25
C
20
D
10
E
12
F
14
G
17
H
19
For the above sample, determine the following measures:
Chapter 3 – Descriptive Statistics: Numerical Measures
a.
The mean.
b.
The standard deviation.
c.
The 25th percentile.
a.
18
b.
c.
13
1
108. In 1998, the average age of students at UTC was 22 with a standard deviation of 3.96. In 1999, the average age was
24 with a standard deviation of 4.08. In which year do the ages show a more dispersed distribution? Show your complete
work and support your answer.
1
109. For the following data
5
7
9
11
15
19
Compute the following measures:
a.
The mean
b.
The variance
c.
The standard deviation
d.
The coefficient of variation
e.
The 25th percentile
f.
The median
g.
The 75th percentile
a.
11
b.
c.
d.
e.
f.
10
g.
15
1
110. For the following data
20
18
17
23
22
19
21
17
23
Compute the following measures:
a.
The mean
b.
The variance
c.
The standard deviation
d.
The coefficient of variation
e.
The 25th percentile
f.
The median
g.
The 75th percentile
a.
20
c.
d.
111. A private research organization studying families in various countries reported the following data for the amount of
time 4-year old children spent alone with their fathers each day.
Country
Time with Dad (minutes)
Belgium
30
Canada
44
China
54
Finland
50
Germany
36
Nigeria
42
Sweden
46
United States
42
For the above sample, determine the following measures:
a.
The mean
b.
The standard deviation
c.
The mode
d.
The 75th percentile
a.
43
b.
7.56
c.
42
d.
48
112. The following data show the yearly salaries of football coaches at some state-supported universities.
University
Salary ($1,000)
A
53
B
44
C
68
D
47
E
62
F
59
G
53
H
94
For the above sample, determine the following measures.
a.
The mean yearly salary
b.
The standard deviation
c.
The mode
d.
The median
e.
The 70th percentile
a.
60
b.
15.8
c.
53
d.
56
e.
62
e.
18
20
g.
22
113. The amount of time that a sample of students spends watching television per day is given below.
Student
Time (minutes)
1
40
2
28
3
71
4
48
5
49
6
35
7
40
8
57
a.
Compute the mean.
b.
Compute the median.
c.
Compute the standard deviation.
d.
Compute the 75th percentile.
a.
46
b.
44
c.
13.5
d.
53
1
114. The number of hours worked per week for a sample of ten students is shown below.
Student
Hours
1
20
2
0
3
18
4
16
5
22
6
40
7
8
8
6
9
30
10
40
a.
Determine the median and explain its meaning.
b.
Compute the 70th percentile and explain its meaning.
c.
What is the mode of the above data? What does it signify?
a.
19; approximately 50% of the students work at least 19 hours
b.
26; at least 70% of the students work less than or equal to 26 hours per week
c.
40; the most frequent data element
1
115. A researcher has obtained the number of hours worked per week during the summer for a sample of fifteen students.
40
25
35
30
20
40
30
20
40
10
30
20
10
5
20
Using this data set, compute the
a.
median
b.
mean
c.
mode
d.
40th percentile
Chapter 3 – Descriptive Statistics: Numerical Measures
e.
range
f.
sample variance
g.
standard deviation
a.
25
b.
25
c.
20
d.
20
e.
35
f.
128.571
g.
11.339
POINTS:
1
116. A sample of twelve families was taken. Each family was asked how many times per week they dine in restaurants.
Their responses are given below.
2
1
0
2
0
2
1
2
0
2
1
2
Using this data set, compute the
a.
mode
b.
median
c.
mean
d.
range
e.
interquartile range
f.
variance
g.
standard deviation
h.
coefficient of variation
a.
b.
1.5
c.
1.25
d.
e.
1.5
f.
0.75
g.
0.866
h.
69.28%
POINTS:
1
117. A sample of 9 mothers was taken. The mothers were asked the age of their oldest child. You are given their
responses below.
3
12
4
7
14
6
2
9
11
a.
Compute the mean.
b.
Compute the variance.
c.
Compute the standard deviation.
d.
Compute the coefficient of variation.
e.
Determine the 25th percentile.
f.
Determine the median
g.
Determine the 75th percentile.
h.
Determine the range.
b.
17.78
c.
4.22
d.
55.8
f.
7.0
g.
11
h.
12
POINTS:
1
118. A sample of 11 individuals shows the following monthly incomes.
Individual
Income ($)
1
1,500
2
2,000
3
2,500
4
4,000
5
4,000
6
2,500
7
2,000
8
4,000
9
3,500
10
3,000
11
43,000
a.
What would be a representative measure of central location for the above data? Explain.
b.
Determine the mode.
c.
Determine the median.
d.
Determine the 60th percentile.
e.
Drop the income of individual number 11 and compute the standard deviation for the first 10
individuals.
a.
Median, because the income of individual 11 is unusually high.
b.
4000
c.
3000
d.
3500
e.
936.90
POINTS:
1
119. Suppose annual salaries for sales associates from a particular store have a mean of $32,500 and a standard deviation
of $2,500.
a.
Calculate and interpret the z-score for a sales associate who makes $36,000.
b.
Use Chebyshev’s theorem to calculate the percentage of sales associates with salaries between
$26,250 and $38,750.
c.
Suppose that the distribution of annual salaries for sales associates at this store is bell-shaped.
Use the empirical rule to calculate the percentage of sales associates with salaries between
$27,500 and $37,500.
d.
Use the empirical rule to determine the percentage of sales associates with salaries less than
$27,500.
e.
Still suppose that the distribution of annual salaries for sales associates at this store is bell-
shaped. A sales associate makes $42,000. Should this salary be considered an outlier? Explain.
a.
1.4. This sales associate’s annual salary is 1.4 standard deviations higher than the mean annual
b.
84%
c.
95%
d.
2.5%
e.
e.
4.0
3.8.
POINTS:
1
120. Provide a five-number summary for the follow data.
115
191
153
194
236
184
216
185
183
202
POINTS:
1
121. The following observations are given for two variables.
y
x
5
2
8
12
18
3
20
6
22
11
30
19
10
18
7
9
a.
Compute and interpret the sample covariance for the above data.
b.
Compute and interpret the sample correlation coefficient.
b.
0.345. There is a positive relationship between x and y. The relationship is not very strong.
POINTS:
1
122. The following data represent the daily demand (y in thousands of units) and the unit price (x in dollars) for a product.
Daily Demand (y)
Unit Price (x)
47
1
39
3
35
5
44
3
34
6
20
8
15
16
30
6
a.
Compute and interpret the sample covariance for the above data.
b.
Compute and interpret the sample correlation coefficient.
b.
-0.922. There is a strong negative relationship between daily demand and unit price.
POINTS:
1
123. Compute the weighted mean for the following data.
xi
Weight (wi)
9
10
8
12
5
4
3
5
2
3
6.676
1
124. Compute the weighted mean for the following data.
xi
Weight (wi)
19
12
17
30
14
28
13
10
18
10
16
1
125. Paul, a freshman at a local college just completed 15 credit hours. His grade report is presented below.
Course
Credit Hours
Grades
Calculus
5
C
Biology
4
A
English
3
D
Music
2
B
P.E.
1
A
The local university uses a 4 point grading system, i.e., A = 4, B = 3, C = 2, D = 1, F = 0. Compute Paul’s semester grade
point average.
2.6
1
126. Consider the data in the following frequency distribution. Assume the data represent a population.
Class
Frequency
2 − 6
2
7 − 11
3
12 − 16
4
17 − 21
1
For the above data, compute the following.
a.
The mean
b.
The variance
c.
The standard deviation
a.
11
b.
21
c.
4.58
1
127. The following frequency distribution shows the ACT scores of a sample of students:
Score
Frequency
14 − 18
2
19 − 23
5
24 − 28
12
29 − 33
1
For the above data, compute the following.
a.
The mean
b.
The standard deviation
a.
24
b.
3.77
1
128. The following is a frequency distribution of grades for a statistics examination.
Examination Grade
Frequency
40 − 49
3
50 − 59
5
60 − 69
11
70 − 79
22
80 − 89
15
90 − 99
6
Treating these data as a sample, compute the following:
a.
The mean
b.
The standard deviation
c.
The variance
d.
The coefficient of variation
a.
74.016
b.
12.601
c.
158.778
d.
17.02%
1
129. The starting salaries of a sample of college students are given below.
Starting Salary ($1000s)
Frequency
10 − 14
2
15 − 19
3
20 − 24
5
25 − 29
7
30 − 34
2
35 − 39
1
a.
Compute the mean.
b.
Compute the variance.
c.
Compute the standard deviation.
d.
Compute the coefficient of variation.
a.
23.7
b.
42.83
130. The following frequency distribution shows the time (in minutes) that a sample of students uses the computer
terminals per day.
Time (minutes)
Frequency
20 − 39
2
40 − 59
4
60 − 79
6
80 − 99
4
100 − 119
2
a.
Compute the mean.
b.
Compute the variance.
c.
Compute the standard deviation.
d.
Compute the coefficient of variation.
a.
69.5
b.
564.54
c.
23.76
d.
34.19%
1
131. A sample of charge accounts at a local drug store revealed the following frequency distribution of unpaid balances.
Unpaid Balance ($)
Frequency
10 − 29
5
30 − 49
10
50 − 69
6
70 − 89
9
90 − 109
20
a.
Determine the mean unpaid balance.
b.
Determine the standard deviation.
c.
Compute the coefficient of variation.
a.
71.1
b.
28.88
c.
40.62%
1
132. The following is a frequency distribution for the ages of a sample of employees at a local company.
Age
Frequency
30 − 39
2
40 − 49
3
50 − 59
7
60 − 69
5
70 − 79
1
a.
Determine the average age for the sample.
b.
Compute the variance.
c.
Compute the standard deviation.
c.
6.54
d.
27.59
1
Chapter 3 – Descriptive Statistics: Numerical Measures
d.
Compute the coefficient of variation.
54.5
b.
117.65
10.85
d.
19.91%
133. Del Michaels had a successful morning, or so he thinks, selling 1300 surplus notebook computers over the telephone
to three commercial customers. The three customers were not equally skillful at negotiating a low unit price. Customer A
bought 600 computers for $1252 each, B bought 300 units at $1310 each, and C bought 400 at $1375 each.
a. What is the average unit price at which Del sold the 1300 computers?
b. Del’s manager told Del he expected him to sell, by the end of the day, a total of 2500 surplus computers at an average
price of $1312 each. What is the average unit price at which Del must sell the remaining 1200 computers?
134. Missy Walters owns a mail-order business specializing in baby clothes. She is considering offering her customers a
discount on shipping charges based on the dollar-amount of the mail order. Before Missy decides the discount policy, she
needs a better understanding of the dollar-amount distribution of the mail orders she receives. Missy had an assistant
randomly select 50 recent orders and record the value, to the nearest dollar, of each order as shown below.
136
281
226
123
178
445
231
389
196
175
211
162
212
241
182
290
434
167
246
338
194
242
368
258
323
196
183
209
198
212
277
348
173
409
264
237
490
222
472
248
231
154
166
214
311
141
159
362
189
260
a. Determine the mean, median, and mode for this data set.
b. Determine the 80th percentile.
c. Determine the first quartile.
d. Determine the range and interquartile range.
e. Determine the sample variance, sample standard deviation, and coefficient of variation.
f. Determine the z-scores for the minimum and maximum values in the data set.
135. Ron Butler, a custom home builder, is looking over the expenses he incurred for a house he just completed
constructing. For the purpose of pricing future construction projects, he would like to know the average wage ($/hour) he
paid the workers he employed. Listed below are the categories of worker he employed, along with their respective wage
and total hours worked. What is the average wage ($/hour) he paid the workers?
Worker
Wage ($/hr)
Total Hours
Carpenter
21.60
520
Electrician
28.72
230
Laborer
11.80
410
Chapter 3 – Descriptive Statistics: Numerical Measures
Painter
19.75
270
Plumber
24.16
160
$20.05