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Chapter 2: Q60E (page 103)

Calculate the range, variance, and standard deviation for the following samples:

a.39, 42, 40, 37, 41

b.100, 4, 7, 96, 80, 3, 1, 10, 2

c.100, 4, 7, 30, 80, 30, 42, 2

Short Answer

Expert verified
  1. Range = 5, Variance = 3.79, Standard Deviation = 1.95
  2. Range = 99, Variance = 1949.25, Standard Deviation = 44.15
  3. Range = 98, Variance = 1307.839, Standard Deviation = 36.16

Step by step solution

01

Calculating the range, variance and standard deviation for a

39, 42, 40, 37, 41

Range = 42 鈥� 37 = 5

x

2

39

-0.8

0.64

42

2.2

4.84

40

0.2

0.4

37

-2.8

7.84

41

1.2

1.44

Sum

0

15.16

Mean(x炉)=sumofxn=1995=39.8

Variance=鈭�(x-x炉)2n-1=15.165-1=15.164=3.79

StandardDeviation=Variance=3.79=1.95

Therefore,the variance and standard deviation are 3.79 and 1.95 respectively.

02

Finding the range, variance and standard deviation for b

100, 4, 7, 96, 80, 3, 1, 10, 2

Range = 100 鈥� 1 = 99

x

2

100

66.33

4400.11

4

-29.67

880.11

7

-26.67

711.11

96

62.33

3885.44

80

46.33

2146.778

3

-30.67

940.44

1

-32.67

1067.11

10

-23.67

560.11

2

-31.67

1002.778

Sum

0

15594

Mean(x炉)=sumofxn=3019=33.66

Variance=鈭�(x-x炉)2n-1=155949-1=155948=1949.25

StandardDeviation=Variance=1949.25=44.15

Therefore, Variance = 1949.25 and Standard Deviation = 44.15

03

Computing the range, variance and standard deviation for c

100, 4, 7, 30, 80, 30, 42, 2

Range = 100 鈥� 2 = 98

x

2

100

63.125

3984.766

4

-32.875

1080.766

7

-29.875

892.5156

30

-6.875

47.26563

80

43.125

1859.766

30

-6.875

47.26563

42

5.125

26.26563

2

-34.875

1216.266

Sum

0

9154.875

Mean(x炉)=sumofxn=2958=36.875

Variance=鈭�(x-x炉)2n-1=9154.8758-1=9154.8757=1307.839

StandardDeviation=Variance=1307.839=36.16

Hence, thevariance is 1307.839 and standard deviation is 36.16.

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Most popular questions from this chapter

Corporate sustainability of CPA firms.Refer to the Business and Society(March 2011) study on the sustainability behaviors of CPA corporations, Exercise 2.23 ( p. 83). Recall that level of support for corporate sustainability (measured on a quantitative scale ranging from 0 to 160 points) was obtained for each of 992 senior managers at CPA firms. Numerical measures of central tendency for level of support are shown in the accompanying Minitab printout.

Descriptive Statistics: Support

Variable

Support

N

Mean

Minimum

Median

Maximum

Mode

N for mode

992

67.755

0.000

68.000

155.000

64

20

a.Locate the mean on the printout. Comment on the accuracy of the statement: 鈥淥n average, the level of support for corporate sustainability for the 992 senior managers was 67.76 points.鈥�

b.Locate the median on the printout. Comment on the accuracy of the statement: 鈥淗alf of the 992 senior managers reported a level of support for corporate sustainability below 68 points.鈥�

c.Locate the mode on the printout. Comment on the accuracy of the statement: 鈥淢ost of the 992 senior managers reported a level of support for corporate sustainability below 64 points.鈥�

d.Based on the values of the measures of central tendency, make a statement about the type of skewness (if any) that exists in the distribution of 992 support levels. Check your answer by examining the histogram shown in Exercise 2.23.

Voltage sags and swells.Refer to the Electrical Engineering(Vol. 95, 2013) study of power quality (measured by鈥渟ags鈥� and 鈥渟wells鈥�) in Turkish transformers, Exercise 2.96(p. 116). For a sample of 103 transformers built for heavyindustry, the mean and standard deviation of the numberof sags per week were 353 and 30, respectively; also, themean and standard deviation of the number of swells perweek were 184 and 25, respectively. Consider a transformerthat has 400 sags and 100 swells in a week.

a.Would you consider 400 sags per week unusual, statistically? Explain.

b.Would you consider 100 swells per week unusual, statistically? Explain.

For each of the following data sets, compute xbar, s2, and s. If appropriate, specify the units in which your answers are expressed.

a.4, 6, 6, 5, 6, 7

b.- \(1, \)4, - \(3, \)0, - \(3, - \)6

c.3/5 %, 4/5 %, 2/5 %, 1/5 %, 1/16 %

d.Calculate the range of each data set in parts a鈥揷.

Symmetric or skewed?Would you expect the data sets described below to possess relative frequency distributions that are symmetric, skewed to the right, or skewed to the left? Explain.

a.The salaries of all persons employed by a large university

b.The grades on an easy test

c.The grades on a difficult test

d.The amounts of time students in your class studied last week

e.The ages of automobiles on a used-car lot

f.The amounts of time spent by students on a difficult examination (maximum time is 50 minutes)

Question: Permeability of sandstone during weathering.Refer to the Geographical Analysis(Vol. 42, 2010) study of the decay properties of sandstone when exposed to the weather, Exercise 2.47 (p. 96). Recall that slices of sandstone blocks were tested for permeability under three conditions: no exposure to any type of weathering (A), repeatedly sprayed with a 10% salt solution (B), and soaked in a 10% salt solution and dried (C). Measures of variation for the permeability measurements (mV) of each sandstone group are displayed in the accompanying Minitab printout.

Descriptive Statistics: PermA, PermB, PermC

Variable

N

StDev.

Variance

Minimum

Maximum

Range

PermA

100

14.48

209.53

55.20

122.40

67.20

PermB

100

21.97

482.75

50.40

150.00

99.60

PermC

100

20.05

401.94

52.20

129.00

76.80

a.Find the range of the permeability measurements for Group A sandstone slices. Verify its value using the minimum and maximum values shown on the printout.

b.Find the standard deviation of the permeability measurements for Group A sandstone slices. Verify its value using the variance shown on the printout.

c.Which condition (A, B, or C) has the more variable permeability data?
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