Multiple choices: 1. Decision alternatives A) should be identified before decision criteria are established. B) are limited to quantitative solutions C) are evaluated as a part of the problem definition stage. D) are best generated by brain-storming. 2. When the value of the output cannot be determined even if the value of the controllable input is known, the model is A) analog. B) digital. C) stochastic. D) deterministic. 3. A snack food manufacturer buys corn for tortilla chips from two cooperatives, one in Iowa and one in Illinois. The price per unit of the Iowa corn is $6.00 and the price per unit of the Illinois corn is $5.50. a. Define variables that would tell how many units to purchase from each source. b. Develop an objective function that would minimize the total cost. c. The manufacturer needs at least 12000 units of corn. The Iowa cooperative can supply up to 8000 units, and the Illinois cooperative can supply at least 6000 units. Develop constraints for these conditions. 4. The relationship d = 5000 - 25p describes what happens to demand (d) as price (p) varies. Here, price can vary between $10 and $50. a. How many units can be sold at the $10 price? How many can be sold at the $50 price? b. Model the expression for total revenue. c. Consider prices of $20, $30, and $40. Which price alternative will maximize total revenue? What are the values for demand and revenue at this price? 5. There is a fixed cost of $50,000 to start a production process. Once the process has begun, the variable cost per unit is $25. The revenue per unit is projected to be $45. a. Write an expression for total cost. b. Write an expression for total revenue c. Write an expression for total profit. d. Find the break-even point. 6. A manufacturer makes two products, doors and windows. Each must be processed through two work areas. Work area #1 has 60 hours of available production time. Work area #2 has 48 hours of available production time. Manufacturing of a door requires 4 hours in work area #1 and 2 hours in work area #2. Manufacturing of a window requires 2 hours in work area #1 and 4 hours in work area #2. Profit is $8 per door and $6 per window. a. Define decision variables that will tell how many units to build (doors and windows) b. Develop an objective function that will maximize profits. c. Develop production constraints for work area #1 and #2. 7. To establish a driver education school, organizers must decide how many cars, instructors, and students to have. Costs are estimated as follows. Annual fixed costs to operate the school are $30,000. The annual cost per car is $3000. The cost per instructor is $11,000 and one instructor is needed for each car. Tuition for each student is $350. Let x be the number of cars and y be the number of students. a. Write an expression for total cost. b. Write an expression for total revenue. c. Write an expression for total profit. d. The school offers the course eight times each year. Each time the course is offered, there are two sessions. If they decide to operate five cars, and if four students can be assigned to each car, will they break even? 8. In the set of all past due accounts, let the event A mean the account is between 31 and 60 days past due and the event B mean the account is that of a new customer. The complement of A is A) all new customers. B) all accounts fewer than 31 or more than 60 days past due. C) all accounts from new customers and all accounts that are from 31 to 60 days past due. D) all new customers whose accounts are between 31 and 60 days past due. 9. If P(A∩B) = 0 A) A and B are independent events. B) P(A) + P(B) = 1 C) A and B are mutually exclusive events. D) either P(A) = 0 or P(B) = 0. 10. If P(A|B) = .4, then A) P(B|A) = .6 B) P(A)*P(B) = .4 C) P(A) / P(B) = .4 D) None of the alternatives is correct. 11. A package of candy contains 12 brown, 5 red, and 8 green candies You grab three pieces from the package. Give the sample space of colors you could get. Order is not important. 12. There are two more assignments in a class before its end, and if you get an A on at least one of them, you will get an A for the semester. Your subjective assessment of your performance is Event Probability A on paper and A on exam .25 A on paper only .10 A on exam only .30 A on neither .35 a. What is the probability of getting an A on the paper? b. What is the probability of getting an A on the exam? c. What is the probability of getting an A in the course? d. Are the grades on the assignments independent? 13. A mail order company tracks the number of returns it receives each day. Information for the last 50 days shows Number of returns Number of days 0-99 6 100-199 20 200-299 15 300 or more 9 a. How many sample points are there? b. List and assign probabilities to sample points. c. What procedure was used to assign these probabilities? 14. Super Cola sales breakdown as 80% regular soda and 20% diet soda. While 60% of the regular soda is purchased by men, only 30% of the diet soda is purchased by men. If a woman purchases Super Cola, what is the probability that it is a diet soda? 15. It is estimated that 3% of the athletes competing in a large tournament are users of an illegal drug to enhance performance. The test for this drug is 90% accurate. What is the probability that an athlete who tests positive is actually a user? 16. A statement that matches the values of a random variable with the probabilities of those values is A) the expected value. B) the variation of the random variable. C) an experiment. D) a probability distribution. 17. In order to measure the dispersion of a random variable, look at its A) standard deviation. B) mean. C) expected value. D) average. 18. The uniform distribution defined over the interval from 25 to 40 has the probability density function A) f(x) = 1/40 for all x B) f(x) = 5/8 for 25 ≤ x ≤ 40 and f(x) = 0 elsewhere C) f(x) = 1/25 for 0 ≤ x ≤ 25 and f(x) = 1/40 for 26 ≤ x ≤ 40 D) f(x) = 1/15 for 25 ≤ x ≤ 40 and f(x) = 0 elsewhere 19. 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) 20. The high school GPAs of applicants for admission to a college program are recorded and relative frequencies are calculated for the categories. GPA f(x) x < 2.0 .08 2.0 ≤ x < 2.5 .12 2.5 ≤ x < 3.0 .35 3.0 ≤ x < 3.5 .30 3.5 ≤ x a. Complete the table to make this a valid probability distribution. b. What is the probability an applicant's GPA will be below 3.0? c. What is the probability an applicant's GPA will be 2.5 or above? 21. Students Number of Tests 10 1 7 2 2 3 1 4 a. Use the relative frequency approach to construct a probability distribution and show that it satisfies the required condition. b. Find the expected value of the number of tests taken. c. Compute the variance. d. Compute the standard deviation. 22. A video rental store has two video cameras available for customers to rent. Historically, demand for cameras has followed this distribution. The revenue per rental is $40. If a customer wants a camera and none is available, the store gives a $15 coupon for tape rental. Demand Relative Frequency Revenue Cost 0 .35 0 0 1 .30 40 0 2 .20 80 0 3 .10 80 15 4 .05 80 30 a. What is the expected demand? b. What is the expected revenue? c. What is the expected cost? d. What is the expected profit? 23. The weight of a .5 cubic yard bag of landscape mulch in uniformly distributed over the interval from 38.5 to 41.5 pounds. a. Give a mathematical expression for the probability density function. b. What is the probability that a bag will weigh more than 40 pounds? c. What is the probability that a bag will weigh less than 39 pounds? d. What is the probability that a bag will weigh between 39 and 40 pounds? 24. Time series methods A) discover a pattern in historical data and project it into the future. B) include cause-effect relationships. C) are useful when historical information is not available. D) All of the alternatives are true. 25. Gradual shifting of a time series over a long period of time is called A) periodicity. B) cycle. C) regression. D) trend. 26. To select a value for α for exponential smoothing A) use a small α when the series varies substantially. B) use a large α when the series has little random variability. C) use any value between 0 and 1 D) All of the alternatives are true. 27. Causal models A) should avoid the use of regression analysis. B) attempt to explain a time series' behavior. C) do not use time series data. D) All of the alternatives are true. 28. The number of cans of soft drinks sold in a machine each week is recorded below. Develop forecasts using a three period moving average. 338, 219, 278, 265, 314, 323, 299, 259, 287, 302 29. The number of girls who attend a summer basketball camp has been recorded for the seven years the camp has been offered. Use exponential smoothing with a smoothing constant of .8 to forecast attendance for the eighth year. 47, 68, 65, 92, 98, 121, 146 30. A trend line for the attendance at a restaurant's Sunday brunch is given by Number = 264 + .72(t) How many guests would you expect in week 20?