# 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

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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

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

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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