(a).40 (b).05 (c).20 (d).10 (e) none of these

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1 STT 421 Some Example Multiple Choice Questions October 24, 2016 Correction to Answer for Prob 50 November 1, Here are the ages for children in a population of 5 children: {2, 2, 4, 5, 7}. An SRS of size n = 2 is to be chosen. What is the probability that the sample average is 6, that is, calculate P(X = 6). (a).40 (b).05 (c).20 (d).10 (e) none of these 2. The average of a large number of repeated measurements using a biased methodology will produced a trustworthy estimate of the parameter. (a) True (b) False 3. Generally speaking, an unbiased estimate is more trustworthy the larger the sample size. (a) True (b) False 4-6. Here is a Venn diagram of a sample space with nine equally likely outcomes. Two events A, B are shown. A B A 4. P ( A B) (a) 1/9 (b) 6/9 (c) 7/9 (d) 3/9 (e) none of these 5. P(A c B c ) = (a) 4/9 (b) 3/9 (c) 4/5 (d) 2/9 (e) none of these 6. P(A B c ) = (a) 5/7 (b) 1/3 (c) 4/9 (d) 4/6 (e) none of these 7. A Department has 52 employees: 7 Females and 45 Males. Three persons are selected at random from the Department, one after the other, without replacement. What is the probability of selecting at least one Male? (a) (b) (c) (d) (e) none of these 8. A basketball player has an 80% chance of making the first free-throw he shoots. If he makes the first free-throw shot, then he has a 90% chance of making the second free-throw he shoots. If he misses the first free-throw shot, then he only has a 70% chance of making the second free-throw he shoots. Suppose this player has been awarded two free-throw shots. Find the probability that he makes at least one of the two shots. (a).88 (b).90 (c).76 (d).94 (e) none of these 9. Suppose there is a 40% chance that a risky stock investment will end up in a total loss of your investment. Because the rewards are so high, you decide to invest in three independent risky stocks. What is the probability that all three stocks end up in total losses? (a).133 (b).188 (c).064 (d).124 (e) none of these From a group of 10 armadillos, 6 Female and 4 Male, Noah selects a SRS of size 2 to go on the ark. 10. What is the probability that both are female? (a).33 (b).36 (c).76 (d).30 (e) What is the probability that both genders are represented in the sample? (a).88 (b).50 (c).76 (d).94 (e).53

2 A large population of armadillos is 60% Female and 40% Male. Noah selects a SRS of size 2 to go on the ark. 12. What is the probability that both are female? (a).33 (b).36 (c).76 (d).30 (e) What is the probability that both genders are represented in the sample? (a).88 (b).50 (c).76 (d).94 (e) Five True-False questions are given to a student. Suppose that the student selects his answers by 5 independent tosses of a fair coin with say, H = True, T = False. Let X denote the number of correct answers. Under this model, X is distributed B(5,.50). 14. What is the probability that the student gets all five questions correct, i.e., what is P(X = 5)? (a).50 (b).25 (c).125 (d).0625 (e) What is the probability that the student gets exactly 3 questions correct, i.e., what is P(X = 3)? (a).3125 (b).25 (c).6000 (d).0625 (e) What is the probability that the student gets at least one correct answer, i.e., what is P(X 1)? (a).50 (b).888 (c).9687 (d).5625 (e) Here is the probability distribution for a discrete random variable X. x p(x) What is the expected value of X? (a) 4.0 (b) 3.5 (c) 3.9 (d) 3.7 (e) none of these 18. P(1 < X 5) = (a).20 (b).70 (c).50 (d).40 (e) none of these In a given region, the probability that a person is exposed to an advertisement of a paper company is A random sample of ten persons is chosen from the population in the region. Let X denote the count of number of persons in the sample who were exposed to the ad. The sample statistic X has the binomial distribution B(10,.20). 19. What is the probability that the sample has at least 2 persons who were exposed to the advertisement, i.e., what is P(X 2)? (a).9999 (b).6242 (c).7316 (d).3020 (e) none of these 20. What is the probability that the sample has exactly 2 persons who were exposed to the advertisement, i.e., what is P(X = 2)? (a).2000 (b).8794 (c).0887 (d).3020 (e) none of these 21. What is the expected value of the number exposed to the advertisement, i.e., what is µ X? (a) 1.5 (b) 2.0 (c) 2.5 (d) 3.0 (e) none of these 22. What is the standard deviation of the number exposed to the advertisement, i.e., what is µ X? (a) 2.76 (b) 3.0 (c) 1. 6 (d).84 (e) none of these A large population of parts is 10% defective and 90% nondefective. Four parts are drawn at random one after. Let D denote defective part and let N denote nondefective part and, for example, let NNND denote the sample outcome nondefective followed nondefective followed by nondefective followed by defective. Let X denote the number of defective parts selected by the sampling process. 23. P(NNND) = (a).0402 (b).0729 (c).2916 (d).5990 (e) none of these 24. X has a normal distribution. (a) True (b) False 25. P(X = 1) = (a).0402 (b).0729 (c).2916 (d).5990 (e) none of these

3 26. Let X have the triangular probability density function given below. Find the probability that X will be between.1 and.5. f(x) = 2x 0 for 0 x 1 otherwise (a).10 (b).24 (c).50 (d).75 (e) none of these Let X have the continuous uniform [0, 9] distribution. 27. What is the mean (expected value) of X? (a) 6.0 (b) 6.5 (c) 3 (d) 4.5 (e) none of these 28. P(2 < X < 4) = (a).33 (b) 4/9 (c) 2/9 (d).11 (e) none of these A random variable X has the distribution tabled below. For this random variable, = SD(X) = 2.1. x P(x) = E(X) = (a) 3.0 (b) 3.7 (c) 3.4 (d) 4.0 (e) none of these 30. What is the value of - 2? (a) 0 (b) 2.5 (c) 1.2 (d) 0.5 (e) none of these 31. What is the value of + 2? (a) 4.8 (b) 6.7 (c) 7.2 (d) 9.4 (e) none of these 32. Chebyshev states that the probability P( - 2 < X < + 2) (a).75 (b).89 (c).94 (d).96 (e) none of these 33. The exact value P( - 2 < X < + 2) = (a).65 (b).75 (c).85 (d).95 (e) none of these Following are the personal incomes in thousands of dollars for an SRS from a population of industrial workers: 43, 44, 46, 42. Here n = 4, the sample average is x 44. 0, and the sample standard deviation is s = What is the point estimate for the average (mean) personal income of the population? (a) 5.0 (b) 44.0 (c) 1.58 (d) 46.0 (e) none of these 35. What is the point estimate for the standard deviation of the population? (a) 5.0 (b) 44.0 (c) 1.58 (d) 46.0 (e) none of these 36. A market research worker interviewed a random sample of 18 people about their use of a certain product. The results, in terms of Y or N (for Yes = a user of the product, or No = not a user of the product), are as follows: Y N N Y Y Y N Y N Y Y Y N Y N Y Y N. Estimate the population proportion of users of the product. (a).48 (b).53 (c) 1/2 (d).84 (e) Consider a large population with the following probability distribution for a characteristic x. An SRS of size n = 2 is to be drawn. x p(x) What is the expected value of the sample average X? (a) 2.5 (b) 2.0 (c) 3.0 (d) 2.7 (e) none of these 38. P( X 4.5) = (a).01 (b).02 (c).03 (d).04 (e).05

4 You interview for positions at four companies Company A, Company B, Company C and Company D. Let A, B, C and D also denote the events you get offers from the Companies A, B, C, D, respectively. You model your prospects as P(A) =.5, P(B) =.5, P(C) =.5, P(D) =.5 with A, B, C, D independent events. 39. What is the probability of getting exactly three offers? (a).133 (b).188 (c).064 (d).124 (e) What is the probability of getting at least one offer? (a).9375 (b).1025 (c).0625 (d).1875 (e) We model the outcome X for the roll of a fair die as P(X = k) = p(k) = 1/6, k = 1, 2, 3, 4, 5, 6. One can show that μ X = 3.5 and σ X 2 = Consider the random variable Y = 2X 1 + 4X where X 1 and X 2 are the outcomes from two independent rolls of the die. 41. Calculate μ Y. (a) 21 (b) 24 (c) 13 (d) 27 (e) none of these 42. Calculate σ Y 2. (a) (b) (c) (d) (e) none of these Suppose a random sample of size n = 90 is to be selected from a large dichotomous population with population proportion p = Let Pˆ be the corresponding sample proportion. 43. What is the expected value of the sample proportion Pˆ? (a).20 (b).16 (c).40 (d).042 (e) none of these 44. What is the standard deviation of the sample proportion Pˆ? (a).20 (b).16 (c).031 (d).042 (e) none of these 45. What is the approximate probability that the sample proportion will be at least 0.15? (Use normal approximation.) (a).48 (b).95 (c).88 (d).80 (e) Suppose a random sample of size n = 400 is selected from a population with mean µ= 53 and standard deviation σ = What is the expected value of the sample average X? (a) 53 (b) 10 (c) 400 (d) 0.50 (e) none of these 47. What is the standard deviation of the sample average? (a) 0.45 (b) 10 (c) 400 (d) 0.50 (e) none of these 48. What is the approximate probability that the sample average will be between 52 and 54? (Use normal approximation.) (a).48 (b).90 (c).95 (d).80 (e) Classify the time X until failure for a light bulb as a discrete or continuous random variable. (a) discrete (b) continuous A chemical plant has an emergency alarm system for a given batch reaction. Based on investigations and records, an emergency situation with the batch reaction is a rare event; it has probability When an emergency situation exists, the alarm sounds with probability When an emergency situation does not exist, the alarm system sounds with probability (Here sensitivity of the alarm system is.90 and the specificity of the alarm system is.99.) 50. When the batch reaction is run, what is the probability that the alarm sounds? (a).0040 (b) (c).0018 (d).3556 (e) Given that the alarm has sounded, what is the probability that an emergency situation exists? Here you are computing the positive predictive value PPV. (a).0674 (b).5000 (c).1601 (d).112 (e) Given that the alarm has not sounded, what is the probability that an emergency situation does not exist? Here you are computing the negative predictive value NPV. (a).9326 (b).9998 (c).9601 (d).9112 (e) none of these

5 An electronic device is in position at the exit of a store. The device senses whether a customer departing the store has shoplifted items in his/her possession. The device captures visual and behavioral data on a subject and sounds an alarm under certain readings. Let S denote the presence of a shoplifted item and let A denote the sounding of the alarm. The system is not perfect. In fact testing shows that for a customer leaving the store P(A S) = 0.90 (sensitivity) and P(A c S c ) =.95 (specificity). Assume that the shoplifting rate is 1% for the following calculations, that is, P(S) = Calculate the probability that the alarm sounds when a customer leaves the store, P(A) = (a).0585 (b).0090 (c).9910 (d).0408 (e) none of these 54. Calculate the conditional probability of the customer carrying shoplifted items given that the alarm sounds P(S A) = (a).009 (b).154 (c).050 (d).336 (e) none of these 55. Calculate the conditional probability of the customer carrying shoplifted items given that the alarm does not sound P(S A c ) = (a) (b) (c) (d) (e) none of these 56. If a large population has mean 1,247, variance 10,000, and the sample size is 400, what is the approximate probability that X will be less than 1,240? (Use the normal distribution for X.) (a).02 (b).59 (c).32 (d).76 (e) An economist wishes to estimate the average family income in a certain large population. The population standard deviation is known to be $4,500, and the economist uses a random sample of size n = 225. What is the probability that the sample mean will fall within $600 of the population mean? (Use the normal distribution for X and for your calculations specify any value for the population mean µ.) (a).76 (b).80 (c).95 (d).84 (e) Shimano mountain bikes are displayed in chic clothing boutiques in Milan, Italy, and the average price for the bike in the city is $700. Suppose that the standard deviation of bike prices is = $100. If a random sample of 60 boutiques is selected, what is the probability that the average price for a Shimano mountain bike in this sample will be between $680 and $720? (Use the normal distribution for X.) (a).48 (b).52 (c).94 (d).88 (e) Suppose you are sampling from a large population with mean = 1065 and standard deviation = 500. The sample size is n = 100. What are the expected value and the standard deviation of the sample mean X? (a) 1065 and 100 (b) 500 and 100 (c) 1065 and 50 (d) 1065 and 500 (e) none of these Suppose it is known that 40% of the people who inquire about investment opportunities at a brokerage house end up purchasing stock, and 50% end up purchasing bonds. It is also known that 28% of the inquirers end up getting a portfolio with both stocks and bonds. The population proportions being given are p(s) =.40, p(b) =.50 and P(S B) =.28. A person is selected at random from the population of those who make inquiries. 60. What is the probability that she or he will get stocks or bonds (i.e., open any portfolio)? Calculate P(S B). (a).20 (b).90 (c).62 (d).80 (e) none of these 61. Are the events "purchase stock" S and "purchase bonds" B independent events? (a) Yes (b) No 62. What is the probability that the selected person will get bonds given that he/she gets stocks? Calculate P(B S). (a).20 (b).90 (c).68 (d).70 (e) none of these 63. The manager of a large restaurant knows from experience that 80% of the people who make reservations actually show up for dinner. The manager decides one evening to overbook and accept 100 reservations when only 85 seats are available. What is the probability that more than 85 of the 100 persons actually show up for dinner? (Use the Binomial model. If you have a calculator that has the Binomial distribution on it, you may use that function to calculate the probability; otherwise use the Normal distribution with mean µ = np = 100(.80) = 80 and standard deviation σ = np(1 p) = 100(. 80)(.20) = 4 to answer the question P(X > 85) =? Pick the closest. (a).3 (b).2 (c).001 (d).1 (e) A computer system contains 50 identical microchips. The probability that any microchip is in working order at any given time is.70. A certain operation requires that at least 30 chips be in working order. What is the probability that the operation will be carried out successfully? (Use the Binomial model B(50,.70). If you have a calculator that has the Binomial distribution on it, you may use that function to calculate the probability otherwise use the Normal distribution with mean µ = np = 50(.70) = 35 and standard deviation σ = np(1 p) = 50(. 70)(.30) = 3.24 to answer the question P(X 30) =?.) Pick the closest. (a).6 (b).5 (c).9 (d).8 (e).7

6 Here is the Venn diagram of a sample space of 23 equally likely outcomes. Events A, B and C are depicted. A A B C 65. P(B C) = (a) 13/23 (b) 10/23 (c) 12/23 (d) 0 (e) none of these 66. P(A c B) = (a) 1/23 (b) 1/7 (c) 0 (d) 7/23 (e) none of these 67. P(A B) = (a) 1/7 (b) 8/7 (c) 0 (d) 2/7 (e) none of these 68. P(A B C) = (a) 20/23 (b) 18/23 (c) 0 (d) 12/23 (e) none of these 69. P(A c B c C c ) = (a) 3/23 (b) 21/23 (c) 1 (d) 5/23 (e) none of these 70. P(B C) = (a) 1/6 (b) 1/7 (c) 0 (d) 1/23 (e) none of these Answers (Please report any errors you find to me. Thanks, DG) 1d,2b,3a,4c,5d,6d,7d,8d,9c,10a,11e,12b,13e,14e,15a,16c,17d, 18d,19b,20d,21b,22c,23b,24b,25c,26b,27d,28c,29b,30d,31e(7.9), 32a,33d,34b,35c,36e,37d,38e,39e,40a,41d,42a,43a,44d,45c,46a, 47d,48c,49b,50b,51e,52b,53a,54b,55d,56e,57c,58d,59c,60c,61b, 62d,63d,64c,65c,66d,67c,68c,69a,70a