STAT 2655 Assignment 2 - Due in class October 21, 2015 1) A city commissioner claims that 80% of the people living in the city favour contracting out garbage collection to a private company. To test the commissioner's claim, 25 city residents are randomly selected, yielding 22 who prefer contracting to a private company. (a) If the commissioner's claim is correct, what is the probability that the sample would contain at least 22 who prefer contracting to a private company? (b) If the commissioner's claim is correct, what is the probability that exactly 22 would prefer contracting to a private company? (c) Based on observing 22 in a sample of size 25 who prefer contracting to a private company, what do you conclude about the commissioner's claim that 80% of city residents prefer privatization of garbage collection? 2) Of a population of consumers, 60% are reputed to prefer a particular brand, A, of toothpaste. If a group of randomly selected consumers is interviewed, what is the probability that exactly ve people have to be interviewed to encounter the rst consumer who prefers brand A? 3) Suppose that Y is a binomial random variable based on n trails with success probability p and consider Z = n Y . (a) Argue that for z = 0, 1, . . . , n P (Z = z) = P (n Y = z) = P (Y = n z). (b) Use the result from part (a) to show that P (Z = z) = n pnz q z = nz n z nz q p . z (c) The result in part (b) implies that Z is a binomial r.v. based on n trails and success probability p = q = 1 p. Explain why this result is \"obvious\". 4) If Y is a geometric r.v. as dened in lecture, let Z = Y 1. If Y is interpreted as the trail on which the rst success occurs, then Z can be interpreted as the number of failures before the rst success. If Z = Y 1, P (Z = y) = P (Z = y + 1) for y = 0, 1, 2, . . . . Show that P (Z = y) = q y p, y = 0, 1, 2, . . . and derive the mean and variance of Z. 5) Consider the negative binomial distribution as dened in lecture. (a) Show that if y r + 1, P (Y =y) P (Y =y1) = y1 yr q. This establishes a recursive relation- ship between successive probabilities since P (Y = y) = P (Y = y 1) y1 yr q. STAT 2655 (b) Show that if y > P (Y =y) P (Y =y1) = y1 yr q > 1 if y < r1 . 1q Similarly, P (Y =y) P (Y =y1) = y1 yr q<1 r1 . 1q (c) apply the result in part (b) for case r =7, p to determine values of y which (y =y) > P (Y = y 1). 6) If Y is a binomial r.v. based on n trails and probability of success p, show that P (Y > 1|Y 1) = 1 (1 p)n np(1 p)n1 . 1 (1 p)n 7) One concern of a gambler is that she will go broke before achieving her rst win. Suppose she plays a game in which the probability of winning is 0.1 (which is unknown to her). It costs her $10 to play and she receives $80 for a win. If she starts playing with $30, what is the probability that she wins exactly once before she losses her initial capital? 8) You are trying to recruit 5 individuals that suer from a particular disease for a study you wish to conduct. Suppose Health Canada has a list of all individuals who suer from this ailment. You know from past studies that only 10% of individuals contacted will be agreeable to take part in studies of this type. In order to get your sample you randomly start calling people on the list and asking them if they would be willing to be part of your study. (a) What is the probability that it will take more than 50 calls to recruit enough people? (b) What is the minimum number of calls required such that the probability of successfully recruiting the individuals is greater than 0.9? (c) Suppose each call will take 5 minutes of your time. How much time would you expect to have to spend to recruit the 5 people? (d) How reliable is (c)? Justify your