Business Mathematics ISample Final ExaminationThis test is only presented as a sample of the types of questions that might appear on an examination. The instructor of each section will set his or her own examinations, which may differ from the sample in both length and content.Show all of the steps and all of the work that you use to solve each problem. A list of formulas that you may find useful is provided on the last page of this test.1. (12 points) An experiment consists of picking, at random, one slip from the following set of 10 slips.Use S = {1, 2, 3} as a sample space. (i) Assign realistic probabilities to each outcome in S.(ii) Let E be the event that the number on the selected slip is odd. Use your work in part i to compute P(E).2. (12 points) Let S be the event that the S & P 500 goes up on December 18, 2005 and let D be the event that the Dow Jones Industrials go up on that day. Assume that P(S) = 0.5 and P(D) = 0.6. The probabilityof either S or D (possibly both) occurring is 0.7.(i) What is the probability that both the S & P 500 and the Dow go up on December 18, 2005?(ii) In real world terms, what does it mean to state that the probability of the Dow going up on the given day is 0.6?3. (12 points) Let X be the pay-off of a lottery ticket, which assumes only the values of $0, $2, $5, or $1,000; with the following probabilities.x $0 $2 $5 $1,000P(X = x) 0.90 0.05 0.04 0.01(i) What price would you pay for a lottery ticket, if you wanted to break even over the purchase of many tickets? Explain your answer.(ii) If tickets are $12, what is the probability that a given ticket pays back less than its purchase price of $12?1122222333© 2001 by The Arizona Board of Regentsfor The University of Arizona.All rights reserved.- Business Mathematics I, Sample Final Examination: page 2 -4. (12 points) Delta and American Airlines both have flights from Tucson to Chicago that depart at approximately the same time. Let D be the event that the Delta flight is full and A be the event that the American flight is full. Suppose that D and A are independent events, and that P(D) = 0.3 and P(A) = 0.2.(i) What is the probability that both flights are full (that is, D and A both happen)?(ii) In the real world are D and A likely to be independent? Explain your answer.5. (12 points) Coca-Cola is considering the production of a new soda called SLOSH. Let T be the eventthat SLOSH sells well in Tucson and D be the event that it sells well in Denver. We assume that P(T) = 0.7, P(D) = 0.8, and P(T D) = 0.6.(i) Compute P(T|D).(ii) Based on their probabilities, are T and D independent? Why or why not?6. (14 points) Consider the soda SLOSH, from Problem 5. Let L be the event that a person, picked at random form the entire population, likes SLOSH, D be the event that he or she is an adult, and Y be the event that he or she is a young person. Surveys show that P(L|D) = 0.4, P(L|Y) = 0.6, P(D) = 0.8, P(Y) = 0.2. Note that D and Y partition a sample space. Use this information to compute P(L). 7. (14 points) The Bigbill & Dolittle School of High Finance knows that 75% of the students in Accounting 101, pass the course, and that the other 25% fail the course. Records show that 80% of those who pass the course passed a placement test, but only 30% of those who failed the course passed a placement test. Let PT, FT, PC, and FC be the events that a student passed the placement test, failed the placement test, passed the course, and failed the course, respectively. Thus, P(PC) = 0.75, P(FC) = 0.25, P(PT|PC) = 0.80, and P(PT|FC) = 0.30. Use this information and Bayes’ Theorem to compute P(PC|PT).8. (12 points) A small sample of bank records, similar to those used in Project 1, is shown below. (i) What function would you use to have Excel count the number of loans in the entire set of records that had a successful workout in normal times?ResultCustomer NumberFormer BankYears In BusinessEducation LevelState Of EconomyLoan Paid Back?1 BR 14 no2 Cajun High School no3 DuPont Normal no4 DuPont Boom no5 DuPont Boom yes6 DuPont Recession noBank InformationBorrower- Business Mathematics I, Sample Final Examination: page 3 -(ii) Where would you find this function in Excel?9. (10 points) Interest is compounded daily (365 times per year) at an annual rate of 5%. How much money (present value) would you need to deposit now in order to have $10,000 after 4 years?10. (10 points) If interest is compounded continuously at 7%, find the future value of $12,000 after 10 years.11. (10 points) A random variable X can assume only the following values.x 0 1 2 3 4P(X = x) 0.1 0.2 0.3 0.3 0.1(i) Let fX be the p.m.f. for X. fX(2) = ___ .(ii) Let FX be the c.d.f. for X. FX(2) = ___ .12. (10 points) The graph of the probability density function for a continuous random variable, X, with auniform distribution on the interval from 0 to 2 is shown below.(i) P(X 1) = ?(ii) Let FX be the c.d.f. for X. FX(0.5) = ___ .13. (10 points) X is an exponential random variable whose p.d.f. is given by fX(x) = 4e4x, for x 0. (i) Find a formula for FX(x) and us this to compute P(X 0.2), rounded to 3 decimal places.(ii) X = ?14. (10 points) Let A be the random variable that gives the percentage of its revenue that a small business spends on advertising. A random sample of 5 businesses found the following values for A.3%, 2%, 3%, 5%, 4%(i) Use the sample data to estimate E(A).1 0.5 0 0.5 1 1.5 2 2.5 30.50.51- Business Mathematics I, Sample Final Examination: page 4 -(ii) Use the sample data to estimate P(A 3.5%)15. (10 points) 100 observations of a continuous random variable, X, produced the following values.Range Frequency0 x <1 201 x <2 302 x <3 403 x 4 10Use this sample data to plot a histogram that approximates the probability density function for X.16. (10 points) Explain how you would use the RAND and IF functions to have Excel simulate a single toss of a fair coin.17. (10 points) A small portion of stock price data, similar to that used in Project 2, is shown below.Stock HistoryDate Closing Price Normalized RatioFriday, July 13 $178.07 0.9880Friday, July 6 $179.86 1.0034Friday, June 29 $178.90 0.9950Friday, June 22 $179.44 1.0448Friday, June 15 $171.41 0.9831Friday, June 8 $174.01(i) Starting with $178.07, use this data to simulate one possible closing price after 2 weeks, on July 27.. Explain what you are doing and show all of your work.(ii) Suppose
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