Applied Statistics and Probability for Engineers, 7th edition

6-1

CHAPTER 6

Section 6.1

6.1.1 No, usually not. For example, if the sample is {2, 3} the mean is 2.5 which is not an observation in the sample.

6.1.7 Sample average:

==

351.8 43.975

8

x

6.1.8 Sample average:

=

= = =

119.56 2.173mm

9

n

i

i

x

xn

Applied Statistics and Probability for Engineers, 7th edition

6.1.9 Sample average of exercise group:

=

= = =

13454.68 287.89

12

n

i

i

x

xn

Applied Statistics and Probability for Engineers, 7th edition

6-3

6.1.10 Dot Diagram of CRT data (Data were rounded for the plot)

6.1.11 a)

=

65.86 F

x

6.1.12

a) All 52 clouds:

Applied Statistics and Probability for Engineers, 7th edition

6-4

b) Unseeded Clouds:

Sample mean:

c) Seeded Clouds:

Applied Statistics and Probability for Engineers, 7th edition

6-5

6.1.13 Dot Diagram for Unseeded clouds:

6.1.14 Sample mean:

=

= = =

13737.98 52.6476

71

n

i

i

x

xn

Applied Statistics and Probability for Engineers, 7th edition

6-6

Section 6.2

6.2.1 A back-to-back steam-and-leaf display is useful when two data sets are to be compared.

6.2.2 The median will be equal to the mean when the sample is symmetric about the mean value.

6.2.4

Stem-and-leaf of C1 N = 82

Leaf Unit = 0.10

1 83 4

3 84 33

4 85 3

7 86 777

6.2.5 Stem-and-leaf display for cycles to failure: unit = 100 1|2 represents 1200

1 0T|3

6.2.6 The data in the 42nd is 90.4 which is median.

The mode is the most frequently occurring data value. There are several data values that occur 3 times. These are:

Applied Statistics and Probability for Engineers, 7th edition

6-7

6.2.7 Sample median is at

( )

+=35

70 1

2.5th

observation, the median is 1436.5.

70

6.2.8 Sample mean:

= 65.811in.,x

inches, standard deviations = 2.106 inches, and sample median:

=66.000in.x

inches

6.2.9 Stem-and-leaf display. Strength: unit = 1.0 1|2 represents 12

1 532|9

1 533|

6.2.10 Stem-and-leaf of concentration, N = 60, Leaf Unit = 1.0, 2|9 represents 29

Note: Minitab has dropped the value to the right of the decimal to make this display.

1 2|9

Applied Statistics and Probability for Engineers, 7th edition

6-8

The data have a symmetrical bell-shaped distribution, and therefore may be normally distributed.

Sample Mean

Applied Statistics and Probability for Engineers, 7th edition

6-9

Sample Standard Deviation

==

==

60 60 2

11

3592.0 and 224,257

ii

ii

xx

6.2.11 Stem-and-leaf display. Rating: unit = 01.0 1|2 represents 12

1 8|3

Applied Statistics and Probability for Engineers, 7th edition

6-10

22/40 = 55% of the taste testers considered this particular Pinot Noir truly exceptional.

6.2.12 Stem-and-leaf diagram of NbOCl3N = 27

Leaf Unit = 100 0|4 represents 40 gram-mole/liter x 10-3

6 0|444444

7 0|5

Sample Standard Deviation

==

27 27 2

41553 and 87,792,869

ii

xx

6.2.13 Stem-and-leaf display. Height: unit = 0.10 1|2 represents 1.2

Female Students| Male Students

0|61 1

Applied Statistics and Probability for Engineers, 7th edition

6-11

Section 6.3

6.3.1

Frequency Tabulation for Exercise 6.2.5 - Cycles to Failure

--------------------------------------------------------------------------------

Lower Upper Relative Cumulative Cum. Rel.

Class Limit Limit Midpoint Frequency Frequency Frequency Frequency

--------------------------------------------------------------------------------

at or below .000 0 .0000 0 .0000

1 .000 266.667 133.333 0 .0000 0 .0000

2 266.667 533.333 400.000 1 .0143 1 .0143

6.3.2

Solution uses the n = 83 observations from the data set.

Frequency Tabulation for Exercise 6.2.4.Octane Data

--------------------------------------------------------------------------------

Lower Upper Relative Cumulative Cum. Rel.

Class Limit Limit Midpoint Frequency Frequency Frequency Frequency

--------------------------------------------------------------------------------

at or below 81.75 0 .0000 0 .0000

1 81.75 84.25 83.0 1 .0120 1 .0120

Applied Statistics and Probability for Engineers, 7th edition

6.3.3

Histogram 8 bins:

Histogram 16 bins:

6-13

6.3.4 Solutions uses the n = 83 observations from the data set.

Frequency Tabulation for Exercise 6.2.4.Octane Data

--------------------------------------------------------------------------------

Lower Upper Relative Cumulative Cum. Rel.

Class Limit Limit Midpoint Frequency FrequencyFrequencyFrequency

--------------------------------------------------------------------------------

at or below 83.000 0 .0000 0 .0000

1 83.000 84.125 83.5625 1 .0120 1 .0120

2 84.125 85.250 84.6875 2 .0241 3 .0361

3 85.250 86.375 85.8125 1 .0120 4 .0482

Histogram 8 bins:

Applied Statistics and Probability for Engineers, 7th edition

Histogram 16 Bins:

6.3.5 The histogram for the spot weld shear strength data shows that the data appear to be normally distributed (the same

shape that appears in the stem-leaf-diagram).

6.3.6

Frequency Tabulation for Exercise 6.2.8. Height Data

--------------------------------------------------------------------------------

Lower Upper Relative Cumulative Cum. Rel.

Class Limit Limit Midpoint Frequency FrequencyFrequencyFrequency

--------------------------------------------------------------------------------

at or below 60.500 0 .0000 0 .0000

1 60.500 61.500 61.000 1 .0270 1 .0270

2 61.500 62.500 62.000 2 .0541 3 .0811

6-15

6.3.7 Yes, the histogram of the wine rating data shows the same shape as the stem-and-leaf display.

6.3.8

Frequency Tabulation for Exercise 6.2.10. Concentration data

--------------------------------------------------------------------------------

Lower Upper Relative Cumulative Cum. Rel.

Class Limit Limit Midpoint Frequency FrequencyFrequencyFrequency

--------------------------------------------------------------------------------

at or below 29.000 0 .0000 0 .0000

1 29.0000 37.000 33.000 2 .0333 2 .0333

2 37.0000 45.000 41.000 6 .1000 8 .1333

6-16

Applied Statistics and Probability for Engineers, 7th edition

6.3.9

Section 6.4

6.4.1 a) Descriptive Statistics for unseeded cloud data:

Median = 44.2

Lower Quartile = Q1 = 23.7

c)

6-18

6.4.2 a) Descriptive Statistics:

Variable Mean SE Mean StDev Minimum Q1 Median Q3

Y 52.648 0.909 7.656 48.640 49.930 51.340 52.580

b)

c) The extreme outliers present in this data are 62.45 and 112.72

Descriptive data without extreme outliers:

Variable TotalCount N N* Mean SE Mean StDev Minimum Q1

Applied Statistics and Probability for Engineers, 7th edition

6-19

6.4.3 a)

Mean = 2.415

b)

Box Plot: There are no outliers in the data.

6.4.4 Descriptive Statistics

Variable N Mean Median Tr Mean StDev SE Mean

PMC 20 4.000 4.100 4.044 0.931 0.208

a) Sample Mean = 4, Sample Variance = 0.867, Sample Standard Deviation = 0.931

b)

6.4.5 Descriptive Statistics of O-ring joint temperature data

Variable N Mean Median TrMean StDev SE Mean

Temp 36 65.86 67.50 66.66 12.16 2.03

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