MM305 Project Cha. 2 to 4

Practice for Chapters 2, 3, 4 Chapter 2 Practice Questions Also, Be able to do a Binomial (not on practice test, but on test) see HW solutions chapter 2-32, 2-33, 2-35 Also, Be familiar with Bayes’ Theorem (not on practice test) see HW solutions chapter 2-23 (can be done with Bayes’ Theorem) and 2-28 (using the complement) Also, Be familiar with the exponential distribution (not on practice test) see HW solutions chapter 2-45 2.148 Fast Service Store has maintained daily sales records on the various size "Cool Drink" sales. Assuming that past performance is a good indicator of future sales, what is the probability of a customer purchasing a $0.50 "Cool Drink?" “Cool Drink” Price Number Sold $0.25 75 $0.35 120 $0.50 125 $0.75 50 Total 400 ANSWER: 125/400 = 0.3125 2.151 A market research study is being conducted to determine if a product modification will be received well by the public. A total of 1,000 consumers are questioned regarding this product. The table below provides information regarding this sample. Positive Reaction Neutral Reaction Negative Reaction Male 240 60 100 Female 260 220 120 (a) What is the probability that a randomly selected male would find this change favorable (positive)? (b) What is the probability that a randomly selected person would be a female who had a negative reaction? (c) If it is known that a person had a positive reaction to the study, what is the probability that the person is female? ANSWER: (a) 240/400 = 0.60 (b) 120/1000 = 0.120 (c) 260/500 = 0.520 2.152 In a production run of 200 units, there are exactly 10 defective items and 190 good items (200 total). (a) What is the probability that a randomly selected item is defective? (b) If two items are sampled without replacement, what is the probability that both are good? (c) If two items are randomly sampled without replacement, what is the probability that the first is good but the second is defective? ANSWER: (a) 10/200 = 0.05 (b) (190/200)(189/199) = 0.902 (c) (190/200)(10/199) = 0.048 2.154 Last semester, the grade distribution in a quantitative methods course had the following distribution: 10 percent A, 25 percent B, 35 percent C, 10 percent D, and 15 percent W (withdrew). (a) If this grade distribution does not change this semester, what is the probability that a randomly selected student will make at least a D? (b) If this grade distribution does not change this semester, what is the probability that a randomly selected student will fail the course? (c) If this grade distribution does not change this semester, what is the probability that a randomly selected student who finished the course (did not withdraw) made a grade of D or better? ANSWER: (a) 80 percent (b) 5 percent (c) 0.80/0.85 = 0.94 2.157 A southwestern tourist city has records indicating that the average daily temperature in the summer is 82 degrees F, which is normally distributed with a standard deviation of 3 degrees F. Based on these records, determine: (a) the probability of a daily temperature between 79 degrees F and 85 degrees F (b) the probability that the daily temperature exceeds 90 degrees F (c) the probability that the daily temperature is below 76 degrees F ANSWER: (a) P(7990) = 0.00379 (c) P(X<76) = 0.02275 2.158 Using the table for finding the areas under normal curves, find the area under a normal curve with a mean of 200 and a standard deviation of 10 between the values of: (a) 200 to 205 (b) 195 to 205 (c) 200 to 215 (d) 195 to 215 (e) 186.5 to 217 ANSWER: (a) 0.19146 (b) 0.38292 (c) 0.43319 (d) 0.62465 (e) 0.86692 2.159 ABC Manufacturing has 6 machines that perform a particular task. Breakdowns occur frequently for this machine. Past records indicate that the number of breakdowns that occur each day is described by the following probability distribution: Number of Breakdowns Probability 0 0.4 1 0.3 2 0.2 3 0.1 More than 3 0.0 (a) What is the expected number of breakdowns in any given day? (b) What is the variance for this distribution? (c) What is the probability that there will be at least 2 breakdowns in a day? ANSWER: (a) expected value = 1.0 (b) variance = 1.0 (c) P(2 or more) = 0.2 + 0.1 = 0.3 2.160 Arrivals in a university advising office during the week of registration are known to follow a Poisson distribution with an average of 4 people arriving each hour. (a) What is the probability that exactly 4 people will arrive in the next hour? (b) What is the probability that exactly 5 people will arrive in the next hour? ANSWER: (a) P(X=4) = 0.1952 (b) P(X=5) = 0. 1563 2.161 The time required to complete a project is known to be normally distributed with a mean of 46 weeks and a standard deviation of 4 weeks. (a) What is the probability that the project is finished in 40 weeks or less? (b) What is the probability that the project is finished in 52 weeks or less? (c) There is an 80 percent chance that the project will be finished in less than how many weeks? ANSWER: (a) 0.06881 (b) 0.93319 (c) 46 + 0.84(4) = 49.36 2.163 An urn contains 7 blue and 3 yellow chips. If the drawing of two (or 3 in part a) chips in succession is done with replacement, determine the probability of: (a) drawing three yellow chips (b) drawing a blue chip on the first draw and a yellow chip on the second draw (c) drawing a blue chip on the second draw given that a yellow chip was drawn on the first draw (d) drawing a yellow chip on the second draw given that a blue chip was drawn on the first draw (e) drawing a yellow chip on the second draw given that a yellow chip was drawn on the first draw ANSWER: (a) 0.027 (b) 0.210 (c) 0.700 (d) 0.300 (e) 0.300 2.167 A new television program was viewed by 200 people (120 females and 80 males). Of the females, 60 liked the program and 60 did not. Of the males, 60 of the 80 liked the program. (a) What is the probability that a randomly selected individual disliked the program? (b) If a male in this group is selected, what is the probability that he disliked the program? (c) What is the probability that a randomly selected individual is a female and disliked the program? ANSWER: (a) 80/200 = 0.40 (b) 20/80 = 0.25 (c) 60/200 = 0.30 Chapter 3 Practice Questions 3.97 A concessionaire for the local ballpark has developed a table of conditional values for the various alternatives (stocking decision) and states of nature (size of crowd). STATES OF NATURE (size of crowd) Alternatives Large Average Small Large Inventory $22,000 $12,000 $2,000 Average Inventory $15,000 $12,000 $6,000 Small Inventory $ 9,000 $ 6,000 $5,000 If the probabilities associated with the states of nature are 0.30 for a large crowd, 0.50 for an average crowd, and 0.20 for a small crowd, determine: (a) the alternative that provides the greatest expected monetary value (EMV) (b) the expected value of perfect information (EVPI) ANSWERS: (a) maximum EMV = $12,200 (b) EVPI = 13800  12200 = 1,600 3.98 A concessionaire for the local ballpark has developed a table of conditional values for the various alternatives (stocking decision) and states of nature (size of crowd). States of Nature (size of crowd) Alternatives Large Average Small Large Inventory $22,000 $12,000 $2,000 Average Inventory $15,000 $12,000 $6,000 Small Inventory $ 9,000 $ 6,000 $5,000 If the probabilities associated with the states of nature are 0.30 for a large crowd, 0.50 for an average crowd, and 0.20 for a small crowd, determine: (a) the opportunity loss table (b) minimum expected opportunity loss (EOL) ANSWERS: (a) Opportunity Loss Table States of Nature Alternatives Large Average Small Large 0 0 8,000 Average 7,000 0 0 Small 13,000 6,000 1,000 (b) minimum EOL = $1,600 3.99 Given the following conditional value table, determine the appropriate decision under uncertainty using: (a) maximax (b) maximin (c) equally likely (d) minimax States of Nature Alternatives Very Favorable Market Average Market Unfavorable Market Large Plant $275,000 $100,000 $150,000 Small Plant $200,000 60,000 $ 10,000 Overtime $100,000 $ 40,000 $ 1,000 Do Nothing 0 0 0 ANSWERS: (a) Large Plant (b) Do Nothing (c) Small Plant (d) Small Plant 3.101 The ABC Co. is considering a new consumer product. They have no idea whether or not the XYZ Co. will come out with a competitive product. If ABC adds an assembly line for the product and XYZ does not follow with a competitive product, their expected profit is $40,000; if they add an assembly line and XYZ does follow, they still expect $10,000 profit. If ABC adds a new plant addition and XYZ does not produce a competitive product, they expect a profit of $600,000; if XYZ does compete for this market, ABC expects a loss of $100,000. Calculate Hurwicz’s criterion of realism using ’s of 0.7, 0.3, and 0.1. ANSWERS: Criterion of Realism Decision  = 0.7  = 0.3  = 0.1 add assembly line $31,000 $19,000 $13,000 plant addition $390,000 $110,000 $30,000 do nothing $0 $0 $0 3.104 The following payoff table provides profits based on various possible stocking decisions and various demand situations. States of Nature Demand Alternatives Low Medium High Stock 12 800 800 800 Stock 13 700 900 900 Stock 14 600 800 1000 Based on current information, it is believed that the probabilities of the three demand states are each 1/3. If you wished to minimize the expected opportunity loss, what decision should be made and what would the minimum expected opportunity loss be? ANSWER: Opportunity loss table: States of Nature Demand Alternatives Low Medium High Stock 12 0 100 200 Stock 13 100 0 100 Stock 14 200 100 0 EOL (12) = 100 EOL (13) = 66.7 ====> Therefore, stock 13. EOL (14) = 100 3.105 The following payoff table provides profits based on various possible decision alternatives and various levels of demand. States of Nature Demand Alternatives Low Medium High Alternative 1 80 120 140 Alternative 2 90 90 90 Alternative 3 50 70 150 The probability of a low demand is 0.4, while the probability of a medium and high demand is each 0.3. (a) What decision would an optimist make? (b) What decision would a pessimist make? (c) What is the highest possible expected monetary value? (d) Calculate the expected value of perfect information for this situation. ANSWER: (a) Alternative 3 (b) Alternative 2 (c) maximum EMV = 110 (d) EVPI = 117  110 = 7 3.106 Norman L. Flowers holds the exclusive university contract for donut sales. The demand (based on historical records) appears to follow the following distribution: Daily Demand (Dozens) Probability 4 0.15 5 0.25 6 0.30 7 0.25 8 0.05 The cost of producing these is $1.20 per dozen while the selling price is $4.20 per dozen. Based on a marginal analysis of this situation, how many donuts should Norman produce each day? ANSWER: P > 1.20/4.20 = 0.286. Therefore, Norman should produce seven dozen. 3.107 Orders for clothing from a particular manufacturer for this year’s Christmas shopping season must be placed in February. The cost per unit for a particular dress is $20 while the anticipated selling price is $50. Anything not sold during the season can be sold for $15 to a discount store. Demand is projected to be either 50, 60, or 70 units. There is a 40 percent chance that demand will be 50 units, a 50 percent chance that demand will be 60 units, and a 10 percent chance that demand will be 70 units. If the company decides to use the EMV criterion, how many units should be ordered in February? ANSWER: Payoff table: States of Nature Demand Alternatives 50 60 70 EMV Order 50 $1,500 $1,500 $1,500 $1,500 Order 60 $1,450 $1,800 $1,800 $1,660 Order 70 $1,400 $1,750 $2,100 $1,645 0.4 0.5 0.1 Therefore, order 60. Chapter 4 Practice Questions 4.94 Bakery Products is considering the introduction of a new line of products. In order to produce the new line, the bakery is considering either a major or minor renovation of the current plant. The following conditional values table has been developed by the bakery. Alternatives Favorable Market $ Unfavorable Market $ Major Renovation 100,000 90,000 Minor Renovation 40,000 20,000 Do Nothing 1,000 0 Under the assumption that the probability of a favorable market is equal to the probability of an unfavorable market, determine: (Use a Decision Tree to do this problem...see page 111 in your text for an example) (a) the EMV of a major renovation. (b) the EMV of a minor renovation. (c) the EMV of the do nothing option. (d) the best alternative using EMV. ANSWER: (a) 5000 (b) 10000 (c) 500 (d) choose the minor renovation alternative