Decision theory, Correlation, Regression, Quality Control

1. You are a corporate inspector for an organization with 10 manufacturing plants, and you are interested in determining how well each plant audit score correlates with injury experience. The individual plant audit scores and their injury frequency rates for the year, along with their respective rankings, are tabulated in the following table.
Plant audit score (X) rank(x) injury rate (Y) rank(v)
A 50 10 10.00 10
B 65 8 8.50 8
C 82 5 5.00 4
D 85 4 7.00 6
E 73 7 8.00 7
F 75 6 5.50 5
G 92 3 4.00 3
H 95 2 3.00 2
I 55 9 9.00 9
J 98 1 2.00 1

a. What is the Spearman rank coefficient of correlation between rankings of audit scores and injury rates?
i. +0.96
ii. +0.90
iii. -0.90
iv. -0.96

b. Assume that you have calculated that the coefficient of correlation is significant at the 1 % level. This means that:
I the items measured by the audits account for 1 % of the factors affecting injury rates.
II The coefficient of correlation has a 99% probability of resulting. from chance.
III The coefficient of correlation is only 1 % correct.
IV The coefficient of correlation has a 1% probability of resulting from chance.

2. A quality-control technician with a bath soap manufacturer periodically draws samples of size 50 from the assembly line and examines each bar of soap for defects. The following table shows the number of defective bars of soap in each of the last 25 samples drawn. Determine whether or not the process appears to be in control.

Sample Number of Sample Number of
Number defective bars number defective bars
1 2 13 4
2 2 14 4
3 0 15 4
4 3 16 0
5 2 17 2
6 10 18 0
7 4 19 10
8 10 20 4
9 0 21 2
10 4 22 1
11 14 23 0
12 10 24 3
25 5
3. The manufacturer of a lawn fertilizer must decide whether or not to introduce their product in a new market area. They know that demand may be strong, weak, or nonexistent. If they decide not to enter the new market area, no additional profit will be realized. If they decide to enter the new market area and demand is strong, the profit will be $100,000.00. If demand is weak, the profit will be $5,000.00, and if there is no demand, the manufacturers will realize a loss of $20,000.00.

a. Construct a payoff table for this situation.
b. If the decision maker uses the maxi-min criterion to reach a decision, which act will be chosen? Why?
c. Construct an opportunity loss table for the manufacturers.
d. If the decision-maker uses the mini-max criterion to reach a decision, which act will be chosen? Why?
e. If the decision-maker uses the maxi-max criterion to reach a decision, which act will be chosen? Why?

4. Efficiency experts with a manufacturing firm wish to study the relationship between the time required for an employee to complete a certain task and the light intensity and noise level at the workstation. Their data on 20 employees are presented in the following table.
Time Required Light Noise
To complete task intensity level
(Y) (Xl) (X2)
58 5 58
90 8 70
60 7 65
45 4 54
125 11 83
100 9 72
115 10 76
52 6 59
85 8 68
135 10 60
125 11 81
40 5 59
95 9 71
70 6 62
120 10 75
86 7 64
80 7 66
65 6 61
95 8 66
25 5 57

a. Find the multiple regression equation describing the relationship among these variables.
b. Compute R2.
c. Let Xl = 10 and X2 = 60 and find the predicted value of Y.

5. A spokesperson for a real estate firm states that housing units in a certain community contain a good mix of two ethnic groups. A community representative says that the two ethnic groups tend to cluster together rather than mix. A random sample of two blocks in the community showed the following sequence of homes of the two ethnic groups A and B.
AAABABBAAAAABBABBAAA
Can one conclude from these data that the homes of the two ethnic groups are not randomly mixed in the community? Show the 7 steps of hypothesis testing.

6. The following data reported by a manufacturing company show, for a sample of twelve months, selected from monthly data kept over a period of several years, the monthly demand for a product and the work-hours expended in the production of the item for the given month.
Sample Month Work-hours Demand

1 142 70
2 79 40
3 118 60
4 180 90
5 60 30
6 158 80
7 145 70
8 95 50
9 140 70
10 84 40
11 185 95
12 75 45
a. Find the slope and y-intercept, using work-hours as the dependent variable.
b. Compute the coefficient of determination and the correlation coefficient.

7. The following table shows the number of persons covered by private health insurance for hospital and surgical benefits for 1990 through 2002.

Year #of
Persons
1990 121
1991 126
1992 130
1993 134
1994 139
Year
#of Persons
Year
#of Persons
1995 1996 1997 1998 1999
145 148 151 156 161
2000 2001 2002
167 174 175
a. Plot the original data.
b. Determine the trend equation and graph the trend line
c. Compute Y c for each year
d. Using the above data, compute the exponential smoothing, using a smoothing coefficient of 0.3.

8. A manufacturing process fills boxes of cereal. The nominal net weight per box is 16 ounces. We wish to establish control limits for the filling process by drawing a random sample of 5 boxes each hour during 24 hours of operation. The deviations (above and below the mean) of each box sample is as shown:
Deviations in hundredths Deviations in hundredths
Of an ounce from 16 oz. Of an ounce from 16 oz.
Sample XI X2 X3 X4 X5 Sample XI X2 X3 X4 X5
1 9 -60 32 -15 99 13 -45 -23 90 -39 55
2 -54 -19 -7 -10 7 14 68 -56 -80 -37 -80
3 42 -47 52 50 -24 15 -59 -15 28 11 10
4 -1 55 89 45 -63 16 -48 86 50 -75 72
5 80 48 -64 -27 67 17 12 8 -51 -47 -74
6 6 -52 -37 -89 80 18 35 -18 -46 16 76
7 -6 49 15 -34 -20 19 -91 95 -72 -24 -82
8 -26 54 7 -20 31 20 -89 -75 -10 -94 4
9 -57 -96 -57 -24 3 21 -49 63 25 38 -31
10 -79 -80 5 5 69 22 33 2 22 -96 -23
11 52 5 -32 89 -30 23 10 17 47 14 93
12 -80 17 -52 -42 66 24 55 -66 94 -55 42
a. Construct an X -bar control chart from these data.
b. Construct an R-chart from these data.
c. Perform a "runs-test" on this data.
d. Is the process in control?

9. There are 10,000 students at a college: 2700 are freshmen, 2300 are sophomores, 3000 are juniors, and 2000 are seniors. Recently a new president was appointed. Two thousand students attended the reception party for the president. The attendance breakdown is shown below in the following table.
Freshman Sophomores Juniors Seniors
Attended Yes 300 700 650 350
Reception No 2400 1600 2350 1650
Total 2700 2300 3000 2000
Test the null hypothesis that the proportion of freshmen, sophomores, juniors, and seniors that attended the reception is the same. (Use a 5% level of significance)

10. As part of a factory's quality-control operation, a technician periodically draws a sample of 20 finished items from the assembly line and inspects each for defects. Fifty consecutive samples yielded the following results:
Number of Defects (Xi) Items in sample of 20 Number of Samples in which (Xi) defectives Occurred

0 17
1 13
2 9
3 5
4 3
5 2
6 1
7 or more 0

Test the Goodness of Fit of these data to a Poisson distribution. Let a = 0.05