- August 28, 2020

BAX-400 Homework 2 Mehul Rangwala August 2020 Total Points: 100 – Due Date: August 30, 2020 11:59 PM Instructions 1. There are 13 questions. Some have multiple parts. 2. Please complete each question for full or partial credit. 3. Please don’t leave any questions unattempted. Zero credit will be awarded for unattempted questions. 4. Some questions might require a lot more thought than others. 5. Submit either: (a) an RMD file showing your R code and the results. OR (b) a PDF file containing your outputs, results, and interpretations AND your file containing the R code. 6. Email me if you have any questions. [email protected] 1 Question 1 (3 points) An oil firm plans to drill 20 wells, each having probability 0.2 of striking oil. Each well costs $20,000 to drill; a well which strikes oil will bring in $750,000 in revenue. Find the expected gain from the 20 wells. Question 2 (2 points) Toll booths on the New York State Thruway are often congested because of the large number of cars waiting to pay. A consultant working for the state concluded that if service times are measured from the time a car stops in line until it leaves, service times are exponentially distributed with a mean of 2.7 minutes. What proportion of cars can get through the toll booth in less than 3 minutes? Question 3 (3 points) Leslie loves to swim and compete in races. The time it takes Leslie to swim 100 yards in a race follows a normal distribution with mean of 62 seconds and standard deviation of 2 seconds. In her next five races, what is the probability that she will swim under a minute exactly twice? Question 4 (10 points) Sampling is a very common practice in quality control. Sampling plans are cre- ated to control the quality of manufactured items that are ready to be shipped. To illustrate the use of a sampling plan, suppose that a chip manufacturing company produces and ships electronic computer chips in lots, each lot consist- ing of 1000 chips. This company’s sampling plan specifies that quality control personnel should randomly sample 50 chips from each lot and accept the lot for shipping if the number of defective chips is four or fewer. The lot will be rejected if the number of defective chips is five or more. Find the probability of accepting a lot as a function of the actual fraction of defective chips. In particular, let the actual fraction of defective chips in a given lot equal any of 0.02, 0.04, 0.06, 0.08, 0.10, 0.12, 0.14, 0.16, 0.18. Then compute the lot acceptance probability for each of these lot defective fractions. Create a bar plot showing how the acceptance probability varies with the frac- tion defective. Comment on what you observe. A revised sampling plan call for accepting a given lot if the number of defective chips found in the random sample of 50 chips is five or fewer.Repeat the above exercise and summarize any notable differences between the two bar plots. 2 Question 5 (15 points) Many transportation vehicles such as airplanes are built with redundant sys- tems for safety. Many components have backup systems so that if one or more components fail, backup components take over and this assures the safe, unin- terrupted operation of the component and the vehicle as a whole. For example, consider one main component of an international trans-Atlantic airplane that has n duplicated systems (i.e., one original system and n− 1 backup systems). Each of these systems functions independently of the others, with probability 0.98. This component functions successfully if at least one of the n systems functions properly. 1. 5 points Find the probability that this airplane component functions successfully if n = 2. 2. 5 points Find the probability that this airplane component functions successfully if n = 4. 3. 5 points What is the minimum number n of duplicated systems that must be incorporated into this airplane component to ensure at least a 0.9999 probability of successful operation? Question 6 (4 points) The Internal Revenue Service is studying the category of charitable contribu- tions. A sample of 25 returns is selected from young couples between the ages of 20 and 35 who had an adjusted gross income of more than $100,000. Of these 25 returns, five had charitable contributions of more than $1,000. Four of these returns are selected for a comprehensive audit. 1. 2 points What is the probability exactly one of the four audited had a charitable deduction of more than $1,000? 2. 2 points What is the probability at least one of the audited returns had a charitable contribution of more than $1,000? 3 Question 7 (4 points) The sales of Mercedes automobiles in the Sacramento area follow a Poisson distribution with a mean of three per day. 1. 2 points What is the probability that no Mercedes is sold on a particular day? 2. 2 points What is the probability that for five consecutive days at least one Mercedes is sold? Question 8 (6 points) The shoplifting sensor at a certain Fry’s Electronics store exit gives an alarm 0.5 times a minute. 1. 2 points Find the median waiting time until the next alarm. 2. 2 points Find the first quartile of waiting time before the next alarm. 3. 2 points Find the 30th percentile of waiting time until the next alarm. Question 9 (6 points) A fire-detection device uses three temperature-sensitive cells acting indepen- dently of one another so that any one or more can activate the alarm. Each cell has a probability of p = 0.8 of activating the alarm when the temperature reaches 135oF or higher. Let x equal the number of cells activating the alarm when the temperature reaches 135oF. 1. 2 points Find the probability distribution of x. 2. 2 points Find the probability that the alarm will function when the temperature reaches 135oF. 3. 2 points Find the expected value and the variance for the random variable x. 4 Question 10 (10 points) Suppose after graduating from MSBA, you work for a survey research company. In a typical survey, you mail questionnaires to 150 companies. Some of these companies might decide not to respond. Assume that the nonresponse rate is 45%; that is, each company’s probability of not responding, independently of the others, is 0.45. Suppose your company does this survey in two “waves.” It mails the 150 questionnaires and waits a certain period for the responses. Assume that the nonresponse rate for this first wave is 45%. However, after this initial period, your company follows up (by telephone, say) on the nonrespondents, asking them to please respond. Suppose that the nonresponse rate on this second wave is 70%; that is, each original nonrespondent now responds with probability 0.3, independently of the others. Your company now wants to find the probability of obtaining at least 110 responses total. What is the probability (fraction of successes) of getting this required number of returns from both waves? Question 11 (8 points) TWA Flight 800 crashed in 1996. What caused this crash? Hailey and Helfand, two Physics professors of Columbia University believe there is a reasonable pos- sibility that a meteor hit Flight 800. They reason as follows. On a given day, 3000 meteors of a size large enough to destroy an airplane hit the earth’s at- mosphere. Approximately 50,000 flights per day, averaging two hours in length, have been flown from 1950 to 1996. This means that at any given point in time, planes in flight cover approximately two-billionths of the world’s atmosphere. Determine the probability that at least one plane in the last 47 years has been downed by a meteor. Use the Poisson approximation to the binomial distri- bution. This approximation says that if n is large and p is small, a binomial distribution with parameters n and p is approximately Poisson distributed with µ = np). Question 12 (2 points) The demand for a daily newspaper at a newsstand at a busy intersection is known to be normally distributed with a mean of 150 and a standard deviation of 25. How many newspapers should the newsstand operator order to ensure that he runs short on no more than 20% of days? 5 Question 13 (27 points) A leading pizza vendor has a contract to supply pizza at all home baseball games in Sacramento. Before each game begins, a constant challenge is to determine how many pizzas to make available at the games. Ken Binlard, a business an- alyst, has determined that his fixed cost of providing pizzas is $1,000. Ken believes that this cost should be equally allocated between two types of pizzas. Ken will supply only two types of pizzas: plain cheese and veggie-and-cheese combo. It costs Ken $4.50 to produce a plain cheese pizza and $5.00 to pro- duce a veggie-and-cheese pizza. The selling price for both pizzas at the game is $9.00. Left over pizzas will have no value and will be donated to the homeless. From experience, Ken has arrived at the following demand distributions for the two types of pizza at home games: 1. 12 points For both plain cheese and veggie-and-cheese combo, determine the profit (or loss) associated with producing at different possible demand levels. For instance, determine the profit if 200 plain cheese pizzas are produced and 200 are demanded. What is the profit if 200 plain cheese pizzas are produced but 300 were demanded, and so on? Summarize your results in a two-way data table using R. A two-way data table is one in which rows correspond to one variable (say, demand) and columns correspond to another variable (say, production). The body of the table contains the data. You will create two such tables – one for plain cheese and another for veggie-and-cheese pizza. 2. 10 points Compute the expected profit associated with each possible production level (assuming Ken will only produce at one of the possible demand levels) for each type of pizza. Hint: This would be a vector of expected values. You will need two such vectors of expected values– one for plain cheese and another for veggie-and-cheese pizza 6 3. 5 points If Ken wants to maximize the expected profit from pizza sales at the game, then how many of each type of pizza should he produce? Hint: The answer to this question is based on the results of part 2 above. 7