Chapter 03 When should measures of location and dispersion be computed

Document Type
Test Prep
Book Title
Essentials of Modern Business Statistics 4th (Fourth) Edition By Williams 4th Edition
Authors
J.K
94. 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
Exhibit 3-4
The following is the frequency distribution for the speeds of a sample of automobiles traveling on
an interstate highway.
Speed (MPH)
Frequency
50 - 54
2
55 - 59
4
60 - 64
5
65 - 69
10
70 - 74
9
75 - 79
5
35
95. Refer to Exhibit 3-4. The mean is
a. 35
b. 670
c. 10
d. 67
96 Refer to Exhibit 3-4. The variance is
a. 6.969
b. 7.071
c. 48.570
d. 50.000
97. Refer to Exhibit 3-4. The standard deviation is
a. 6.969
b. 7.071
c. 48.570
d. 50.000
98. An important numerical measure of the shape of a distribution is the
a. correlation coefficient
b. variance
c. skewness
d. relative location
99. If the data distribution is symmetric, the skewness is
a. 0
b. .5
c. 1
d. None of the other answers is correct.
100. For data skewed to the left, the skewness is
a. between 0 and .5
b. less than 1
c. positive
d. negative
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
PROBLEMS
1. 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:
a. The mean.
b. The standard deviation.
c. The 25th percentile.
2. 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.
3. For the following data
5
7
9
11
15
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
4. 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
5. 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
6. The following data show the yearly salaries of football coaches at some state-
supported universities.
University
Salary ($1,000)
A
53
B
44
C
68
D
47
E
62
F
59
G
53
H
94
For the above sample, determine the following measures.
a. The mean yearly salary
b. The standard deviation
c. The mode
d. The median
e. The 70th percentile
7. The amount of time that a sample of students spends watching television per day is given
below.
Student
Time (minutes)
1
40
2
28
3
71
4
48
5
49
6
35
7
40
8
57
a. Compute the mean.
b. Compute the median.
c. Compute the standard deviation.
d. Compute the 75th percentile.
8. The number of hours worked per week for a sample of ten students is shown below.
Student
Hours
1
20
2
0
3
18
4
16
5
22
6
40
7
8
8
6
9
30
10
40
a. Determine the median and explain its meaning.
b. Compute the 70th percentile and explain its meaning.
c. What is the mode of the above data? What does it signify?
9. A researcher has obtained the number of hours worked per week during the summer for a
sample of fifteen students.
40
25
35
30
20
40
30
20
40
10
30
20
10
5
20
Using this data set, compute the
a. median
b. mean
c. mode
d. 40th percentile
e. range
f. sample variance
g. standard deviation
10. A sample of twelve families was taken. Each family was asked how many times per
week they dine in restaurants. Their responses are given below.
2
1
0
2
0
2
1
2
0
2
1
2
Using this data set, compute the
a. mode
b. median
c. mean
d. range
e. interquartile range
f. variance
g. standard deviation
h. coefficient of variation
h. 69.28%
11. A sample of 9 mothers was taken. The mothers were asked the age of their oldest child.
You are given their responses below.
3
12
4
7
14
6
2
9
11
a. Compute the mean.
b. Compute the variance.
c. Compute the standard deviation.
d. Compute the coefficient of variation.
e. Determine the 25th percentile.
f. Determine the median
g. Determine the 75th percentile.
h. Determine the range.
12. A sample of 11 individuals shows the following monthly incomes.
Individual
Income ($)
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.
13. 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.
14. Provide a five-number summary for the follow data.
115
191
153
194
236
184
216
185
183
202
15. 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.
16. 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.
17. Compute the weighted mean for the following data.
xi
Weight (wi)
9
10
8
12
5
4
3
5
2
3
18. Compute the weighted mean for the following data.
Xi
Weight (wi)
19
12
17
30
14
28
13
10
18
10
19. 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.
20. 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
21. 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
22. 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
23. The starting salaries of a sample of college students are given below.
Starting Salary ($1000s)
Frequency
10 14
2
15 19
3
20 24
5
25 29
7
30 34
2
35 39
1
a. Compute the mean.
b. Compute the variance.
c. Compute the standard deviation.
d. Compute the coefficient of variation.
24. The following frequency distribution shows the time (in minutes) that a sample of
students uses the computer terminals per day.
Time (minutes)
Frequency
20 39
2
40 59
4
60 79
6
80 99
4
100 119
2
a. Compute the mean.
b. Compute the variance.
c. Compute the standard deviation.
d. Compute the coefficient of variation.
25. A sample of charge accounts at a local drug store revealed the following frequency
distribution of unpaid balances.
Unpaid Balance ($)
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.
26. 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.
d. Compute the coefficient of variation.
27. 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 $1312 each. What is the average
unit price at which Del must sell the remaining 1200 computers?
28. Missy Walters owns a mail-order business specializing in baby clothes. She is
considering offering her customers a discount on shipping charges based on the dollar-
amount of the mail order. Before Missy decides the discount policy, she needs a better
understanding of the dollar-amount distribution of the mail orders she receives. Missy
had an assistant randomly select 50 recent orders and record the value, to the nearest
dollar, of each order as shown below.
136
281
226
123
178
445
231
389
196
175
211
162
212
241
182
290
434
167
246
338
194
242
368
258
323
196
183
209
198
212
277
348
173
409
264
237
490
222
472
248
231
154
166
214
311
141
159
362
189
260
a. Determine the mean, median, and mode for this data set.
b. Determine the 80th percentile.
c. Determine the first quartile.
d. Determine the range and interquartile range.
e. Determine the sample variance, sample standard deviation, and coefficient of
variation.
f. Determine the z-scores for the minimum and maximum values in the data set.
29. Ron Butler, a custom home builder, is looking over the expenses he incurred for a house
he just completed constructing. For the purpose of pricing future construction projects,
he would like to know the average wage ($/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
Painter
19.75
270
Plumber
24.16
160

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