Standard Deviation
I. Choose the best answer for each multiple choice. Please use CAPITAL letters to indicate your answer and write neatly. (20 points)
1. ____ 3.____ 5. ____ 7.____ 9. ____
2. ____ 4.____ 6.____ 8.____ 10.____
1. Larger values of the standard deviation result in a normal curve that is
A. shifted to the right
B. shifted to the left
C. narrower and more peaked
D. wider and flatter
2. The options from which a decision maker chooses a course of action are
A. called the decision alternatives.
B. under the control of the decision maker.
C. not the same as the states of nature.
D. All of the above alternatives are true.
3. A numerical description of the outcome of an experiment is
A. a normal variable.
B. a discrete variable.
C. a random variable.
D. an experimental variable.
4. Experiments with repeated independent trials will be described by the binomial distribution if
A. the trials are continuous.
B. each trial result influences the next.
C. the time between trials is constant.
D. each trial has exactly two outcomes whose probabilities do not change.
5. If x is normally distributed with mean 12 and standard deviation 2, then P(x 9) is
A. P(z 9/10).
B. P(z -3/2)
C. P(z 2/3)
D. P(z -3/4)
6. For a maximization problem, the pessimistic approach is often referred to as the
A. minimax approach
B. maximin approach
C. maximax approach
D. minimin approach
7. A standard normal distribution is a normal distribution with
A. a mean of 1 and a standard deviation of 0
B. a mean of 0 and a standard deviation of 1
C. any mean and a standard deviation of 1
D. any mean and any standard deviation
8. Which of the following are continuous random variables?
I. the weight of an elephant
II. the time to answer a questionnaire
III. the number of floors in a skyscraper
IV. the square feet of countertop in a kitchen
A. I and II only
B. III and IV only
C. I, II and IV
D. I, II, III, and IV
9. Which of the following can convert a normal random variable x to a standard normal
random variable?
A. We use each value of x plus its mean and then divided by the standard deviation
B. We use each value of x subtract its mean and then divided by the standard deviation
C. We use each value of x divided by the standard deviation and then plus its mean
D. We use each value of x divided by the standard deviation and then subtract its mean
10. Z is a standard normal random variable. The P(z > 2.23) equals
A. 0.0129
B. 0.9821
C. 0.5
D. 0.0174
II. Problem Solving
1. A calculus instructor uses computer aided instruction and allows students to take the midterm exam as many times as needed until a passing grade is obtained. Following is a record of the number of students in a class of 50 who took the test each number of times.
| Students | Number of Tests |
| 22 | 1 |
| 15 | 2 |
| 8 | 3 |
| 5 | 4 |
a. Find the expected value of the number of tests taken. (10 points)
b. Compute the variance and the standard deviation of the number of tests taken.
(10 points)
2. Scores on an endurance test for cardiac patients are normally distributed with mean of 185 and standard deviation of 30.
a. What is the probability a patient will score above 192? (10 points)
b. What score does a patient at the 80th percentile receive? (10 points)
3. You are a marketing manager for a food products company, considering the introduction of a new brand of organic salad dressings. You need to develop a marketing plan for the salad dressings in which you must decide whether you will have a gradual introduction of the salad dressings (with only a few different salad dressings introduced to the market) or a concentrated introduction of the salad dressings (in which a full line of salad dressings will be introduced to the market). You estimate that if there is a low demand for the salad dressings, your first year’s profit will be $1 million for a gradual introduction and million (a loss of $5 million) for a concentrated introduction. If there is high demand, you estimate that your first year’s profit will be $4 million for a gradual introduction and $10 million for a concentrated introduction. The payoff table for the organic salad dressings marketing is given as follows:
| Low Demand | High Demand | |
| Gradual | 1 | 4 |
| Concentrated | -5 | 10 |
a. If nothing is known about the probabilities of the chance outcomes, what is the recommended decision using the pessimistic, and minimax regret approaches? (10 points)
b. Suppose you believe that the probability of demand being low is 0.75. Use the expected monetary value approach to determine an optimal decision. (Provide the expected monetary value for each decision alternative.) (10 points)
c. Given the information in part b), what is the EVPI? (10 points)
d. Use graphical sensitivity analysis to determine the range of demand probabilities for which each of the decision alternatives has the largest expected value. (10 points)