Ch. 5-7 Practice Problems Name: ___________________
Instructions: Please complete the following applied statistical exercises to the best of your ability after reading Chapters 5, 6, and 7. Please type your responses and bold them, CAPITALIZE THEM, or highlight them in yellow. This exercise is worth up to 20 points. You get full points if this exercise is fully complete. However, you will lose points if you leave questions blank.
1. A technician works on the mailing machines of companies in the Phoenix, Arizona area. Depending on the type of malfunction, the service call can take 1, 2, 3, or 4 hours. The different types of malfunctions occur at about the same time frequency.
|
x |
f(x) |
|
1 |
|
|
2 |
|
|
3 |
|
|
4 |
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Total |
1.00 |
a. Develop a probability distribution for a duration of a service call.
b. Draw a graph of the probability distribution.
c. Show that your probability distribution satisfied the two conditions required for a discrete probability function.
d. What is the probability a service call will take three hours?
2. The probability distribution for damage claims paid by the Newton Automobile Insurance Company on collision insurance follows.
|
Payment ($) x
|
Probability f(x)
|
Xf(x)
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|
0 |
.85 |
|
|
500 |
.04 |
|
|
1000 |
.04 |
|
|
3000 |
.03 |
|
|
5000 |
.02 |
|
|
8000 |
.01 |
|
|
10000 |
.01 |
|
|
|
E(x) =μ = Σxf(x) =
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|
a. Use the expected value of collision payment to determine the collision insurance premium that would enable the company to break even.
b. The insurance company charges an annual rate of $250 for the collision coverage. What is the expected value of the collision policy for a policyholder? (Hint: The expected payments from the company minus the cost of coverage.) Why does the policyholder decide to purchase a collision policy with this expected value?
3. This following probability distributions of job satisfaction scores for a survey sample of information systems (IS) senior executives and (IS) middle managers range from 1 (very dissatisfied) to 5 (very satisfied).
|
Job Satisfaction Score (x) |
IS Senior Executives f(x) – probability |
x f (x)
|
IS Middle Managers f(x) – probability |
x f (x) |
|
1 |
.05 |
|
.04 |
|
|
2 |
.09 |
|
.10 |
|
|
3 |
.03 |
|
.12 |
|
|
4 |
.42 |
|
.46 |
|
|
5 |
.41 |
|
.28 |
|
|
|
Sx f (x)= |
|
Sx f (x)= |
|
a. What is the expected value of the job satisfaction score for senior executives? ___
b. What is the expected value of the job satisfaction score for middle managers? ___
|
Job Satisfaction Score (x) |
(x – μ) Subtract the expected value of executives |
(x – μ)² Square the value |
f(x)
|
(x – μ)²f(x)
|
|
1 |
|
|
.05 |
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2 |
|
|
.09 |
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3 |
|
|
.03 |
|
|
4 |
|
|
.42 |
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|
5 |
|
|
.41 |
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|
|
|
|
σ² =
|
|
c. Compute the variance of job satisfaction scores for executives. _____
d. Compute the standard deviation of job satisfaction scores for executives. _____
|
Job Satisfaction Score (x) |
(x – μ) Subtract the expected value of middle managers |
(x – μ)² Square the value |
f(x)
|
(x – μ)²f(x)
|
|
1 |
|
|
.04 |
|
|
2 |
|
|
.10 |
|
|
3 |
|
|
.12 |
|
|
4 |
|
|
.46 |
|
|
5 |
|
|
.28 |
|
|
|
|
|
σ² =
|
|
e. Compute the variance of job satisfaction scores for middle managers. _____
f. Compute the standard deviation of job satisfaction for the middle managers. _____
g. Compare the overall job satisfaction of senior executives and middle managers given your interpretations of your findings.
4. Delta Airlines quotes a flight time of 2 hours for its flights from Cincinnati to Tampa. Suppose we believe that actual flight times are uniformly distributed between 2 hours and 2 hours, 20 minutes.
a. Describe or show the graph of the probability density function, f(x), for a flight time in interval minutes (e.g., 100, 110, 120, etc.). You may use Excel or a similar software to obtain this image.
Example: What is the probability that the flight will be no more than 4 minutes late?
P(x £ 124) = Height (1/b-a)X Width (b-a) = (1/20) (124 – 120) = 0.20
b. What is the probability that the flight will be no more than 5 minutes?
P(x < __) = Height (1/b-a)X Width (b-a) =
c. What is the probability that the flight will be no more than 10 minutes late?
P(x < __) = Height (1/b-a)X Width (b-a) =
d. What is the expected value flight time?
e. What is the variance?
f. What is the standard deviation?
5. A sample of 50 Fortune 500 companies (Fortune, April 14, 2003) showed 5 were based in New York, 6 in California, 2 in Minnesota, and 1 in Wisconsin.
a. Develop a sample estimate of the proportion of Fortune 500 companies based in New York.
b. Develop a sample estimate of the number of Fortune 500 companies based in Minnesota.
