VIEWS: 15 PAGES: 6 CATEGORY: Government POSTED ON: 8/20/2009 Public Domain
B01.1305 Summer 2009 Homework 4 1. The rate of home sales at a small real estate agency is 1.3 per day. We’ll assume that a Poisson phenomenon can represent these home sales. (a) Find the probability that no homes will be sold on Monday. (b) Find the probability that one home will be sold on Monday. (c) Find the probability that two homes will be sold on Monday. The probabilities can be found with Minitab, but they are easy enough with just a good calculator. This “good calculator” will be needed to find e-1.3 0.272532. 2. At the Sweet Easter Company, internet orders for the very expensive Super Chocolate Bunny ($90 each) come in at a rate of 0.9 per day, and it is believed that this phenomenon can be described as a Poisson random variable with = 0.9. Find (a) the probability that there will be three or more orders on any day. (b) the probability that, in a five-day work week, there will be at least one day on which there are three or more orders. 3. The probability that a “cold call” will result in selling a home termite inspection service is 0.032. If 100 cold calls are made, find the probability that (a) no sales will be made (b) exactly one sale will be made (c) exactly two sales will be made (d) exactly three sales will be made This problem is much easier with Minitab than with a calculator. 4. The number of customer complaints, per month, at a large brokerage firm follows a Poisson distribution with parameter = 3.2. Find the probability that, over the next month, (a) no customer complaints will be made (b) exactly one customer complaint will be made (c) exactly two customer complaints will be made (d) exactly three customer complaints will be made These can be done with a calculator, but Minitab will be easier. (e) How would you compare the numbers here to the corresponding numbers from the previous problem? 1 gs2009 B01.1305 Summer 2009 Homework 4 5. A bowl contains 18 bags of potato chips. Of these, 12 bags are standard potato chips, and six bags are onion-flavored chips. If Lucy grabs four of these bags without looking, what is the probability that she will get two bags of the onion-flavored chips? NOTE: “get two” means exactly two. 6. For each of the following situations, indicate whether the model should be binomial or Poisson or hypergeometric. (a) The number of major forest fires to strike Colorado in calendar year 2006. (b) The number of trading days in the month of October that the stock of General Electric will go up in value. (c) The number of plays at craps, out of 50 attempted, that are winners. (d) The number of prize coupons, out of 800 inserted into cereal boxes, that are returned to collect the prizes. (e) The number of visitors to your web site on 25 FEB 2007. (f) The number of dead squirrels found on one mile of highway 93, on May 15. (Such schemes are actually used to estimate animal populations.) (g) The number of expense account claims with inadequate documentation, in a sample of 10 selected from a master file of 280. (h) The number of mattresses, out of 140 sold during the month of May, returned by the customers. (i) The number of diamonds in a single hand at hearts. (In the game of hearts, a single hand consists of 13 cards dealt from the deck of 52.) (j) The number of customers, out of 418 who made a purchase at Windham Supermarket, who purchased milk. 7. The multivariate hypergeometric applies in a situation in which the set of N items has more than two types. Suppose that there are K types, and that N1 is the number of type 1, N2 is the number of type 2, … , and NK is the number of type K. The accounting equation N = N1 + N2 + … + NK assures that we’ve counted everything correctly. Suppose that you take a sample of size n (without replacement, as usual) and that X1 is the random number of these of type 1, X2 is the random number of these of type 2, …, XK is the random number of these of type K. The accounting equation here is X1 + X2 + … + XK = n. It can be shown without difficulty (but with a lot of notation) that N1 N 2 N 3 N K x x x ... x P[ X1 = x1 , X2 = x2 , …, XK = xK ] = 1 2 3 K N n As an example, suppose that you take a deck of 52 cards and select 13 cards at random. The probability that you get exactly 5 spades, 4 hearts, 1 diamond, and 3 clubs is 2 gs2009 B01.1305 Summer 2009 Homework 4 13 13 13 13 5 4 1 3 = 1,287 715 13 286 0.0053878 52 635,013,559,600 13 NOTE: Some of these numbers will be useful below. This applies the multivariate hypergeometric with N = 52, N1 = N2 = N3 = N4 = 13. Minitab is not set up to do multivariate hypergeometric problems, so this required a calculator. In the game of bridge, each player receives 13 of the 52 cards. (a) Find the probability that a bridge player will receive exactly four spades, three hearts, three diamonds, and three clubs. (b) Argue that this is also the probability of receiving three spades, four hearts, three diamonds, and three clubs. (c) Find the probability of a “flat hand.” A “flat hand” has 4-3-3-3 suit distribution, meaning there are four cards of one suit and three cards of each of the other suits. 8. The ethnic makeup of American juries is a contentious issue. Suppose that a jury pool of 117 has 62 women and 55 men. Suppose that the jury of 12 ends up as 10 men and 2 women. Is this evidence of some form of discrimination? NOTE: Juries are not selected randomly. Still, we make statistical calculations on the assumption of randomness. Here that assumption would be that the selection was random with respect to gender. (a) Let X be the random number of women selected for the jury. Find P[X = 2], assuming that selection is random with regard to gender. This calculation is easy with Minitab. (b) The value of P[X = 2] by itself is not indication of prejudicial selection. The correct comparison is P[X 2]. Find this value. Do you consider this jury unfairly selected with regard to gender? 3 gs2009 B01.1305 Summer 2009 Homework 4 9. Suppose that a jury pool (meaning people available to serve on a jury) is categorized as White 48 Black 32 Hispanic 26 Asian 14 TOTAL 120 It is claimed by the prosecution that the 12 jury members will be selected without regard to ethnicity. Suppose that the jury consists of 6 whites, 2 blacks, 4 Hispanics, and no Asians. Find the probability that this happened by chance alone. As a computational 48 120 hint, = 12,271,512 and = 10,542,859,559,688,820. 6 12 10. How would you count the homeless people in New York? Here is an approximate version of the methodology. The city picks a target date, say April 10. Then the city then recruits two types of people, counters and poseurs. The counters go through a training session to get them ready for their job. The job consists of the following steps. C1. On April 10, approach people who seem to be homeless (in subway stations, on street corners, food kitchens, and other spots). C2. Identify yourself as a city survey worker. C3. Ask the person whether or not he or she is homeless. C4. Ask if some previous city survey worker has already interviewed them. C5. Report back to the city on April 11 with information in the form “I interviewed 45 people, of whom 17 identified themselves as homeless and 28 identified themselves as non-homeless.” This could represent the information in line 1 of the table below. The “17” and the “28” in this statement are both first encounters. This counter perhaps questioned 29 non-homeless people, of whom one had been questioned before. This counter perhaps also questioned 21 homeless, but 4 of them had previously been questioned before. The poseurs will also go through a training session. The poseur job has the following steps. P1. Dress in shabby clothes on April 10. P2. Hang out in places frequented by the homeless (in subway stations, on street corners, food kitchens, and other spots). 4 gs2009 B01.1305 Summer 2009 Homework 4 P3. If questioned by a counter, the poseur is required to keep his or her status a secret. (After the first meeting with a counter, the poseur becomes irrelevant and can go home.) P4. Report back to the city on April 11, with a simple yes-or-no result. Was the poseur approached by one of the counters? Suppose that there are H homeless people in the city. Of course H is unknown, and our desire is to estimate H. On April 10, we will know P, the number of poseurs, and C, the number of counters. (The value of C is irrelevant. This exercise could, in theory, be done with one very busy counter.) On April 11, we will know (from the poseurs) X, the number of poseurs found by the counters. On April 11, we will also know n, the total number of homeless people found by the counters. This value of n will include both poseurs and genuine homeless Counter Claimed Claimed Non-homeless TOTAL number Homeless 1 17 28 45 2 11 26 37 C 20 55 75 TOTAL n Only the “Claimed Homeless” column of this table will be used. The P poseurs will report to us information that we express as X = number of poseurs approached by counters. This will enable us to assemble this table: Genuine Poseurs TOTAL Homeless Interviewed Yes X n by Counter? No TOTAL P H N=P+H ˆ Let’s suppose that we want to estimate N. Call the estimate N . Since we know P, the ˆ ˆ ˆ number of poseurs, the estimate for H (call it H ) can be found as H = N - P. All work will be done after we know the value of n, the number of “claimed homeless” people interviewed by the counters. (a) What is the distribution of the random variable X? (b) What is the expected value of X? 5 gs2009 B01.1305 Summer 2009 Homework 4 (c) Suppose that you set up P = 200 poseurs. Now suppose that the counters interview 1,000 people, of whom 80 claim to be homeless. Then suppose that we learn from the poseurs that 25 were interviewed. This leads to Genuine Poseurs TOTAL Homeless Interviewed Yes X = 25 n = 80 by Counter? No TOTAL P = 200 H N = 200 + H Use any reasonable argument to estimate N, and then give the corresponding estimate of H. (This is for a city smaller than New York.) 11. Suppose that Z represents a standard normal random variable. Don’t forget that “standard” here says = 0 and = 1. (a) Find the probability P[ Z > 1.42 ]. (b) Find the probability P[ -0.22 < Z < -0.13 ]. (c) Find the probability P[ | Z | 0.90 ]. (d) Find the value h for which P[ | Z | > h ] = 0.08. 12. Suppose that X is a normal random variable with mean 4,500 and with standard deviation 1,000. Find the probability (a) P[ X < 5,000 ] (b) P[ X > 3,500 ] (c) P[ 4,000 X 5,000 ] (d) P[ | X – 4,000 | > 800 ] 13. It is maintained that, in a quiet equity market with no news, the daily number of shares trades of EquiNimbus Corporation will be approximately normally distributed with mean 280,000 and with standard deviation 32,000. Find the probability that the number of shares traded tomorrow will be at most 325,000. 6 gs2009