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Unformatted text preview: THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT 0302 Business Statistics Assignment 2 (Do all. Hand in solutions to the FOUR starred questions on or before 27.2.09.) *1. (a) Circular metal discs are manufactured in a factory in large quantities. The target diameter of the discs is 16.00 cm. It is known that this dimension follows a normal distribution with a standard deviation of 0.03 cm. An item is randomly drawn. Determine a 90% prediction interval for its diameter. (b) A large batch of electrical equipment has been produced. The quality in question is the resistance, which is normally distributed with a mean of 175 ohms and a standard deviation of 0.7 ohms. Any piece whose resistance lies between 170 and 180 ohms is considered to satisfy customer’s speciﬁcations. Determine the sixsigma process limits. How well does the producer’s quality satisfy the customer’s speciﬁcations? c) A lift has the following speciﬁcations “Capacity: 20 persons, Maximum load: 1250 kg”. It is known that the weights (including their personal belongings) of the passengers taking the lift have a normal distribution with a mean of 62.5 kg and a standard deviation of 8.5 kg. What is the probability that the lift will be overloaded when it is ﬁlled to capacity? (d) Two assembly parts, A and B , have N(8.45, 0.032 ) and N(8.60, 0.042 ) distributions (unit of measurement: cm), respectively. Assembly is random. It is speciﬁed that the clearance should lie between 0.08 and 0.20. Find the proportion of rejects due to too-tight or too-loose ﬁttings. 2. (a) The length of a certain kind of item is distributed normally with mean 8.5 cm and standard deviation 0.02 cm. Any item whose length exceeds 8.565 cm must be scrapped. Find the proportion of scraps in a large batch of such items. (b) The lengths of the sardines received by a cannery have a normal distribution with mean 11.52 cm and standard deviation 0.5 cm. What percentage of all these sardines are: (i) shorter than 10.00 cm, (ii) from 10.50 to 12.00 cm long? (c) On a certain Bank of China 24-hour banking machine which keeps only $100 bank notes, the demand for cash in any ordinary weekend has a normal distribution with mean $135,000 and standard deviation $15,000. If the bank wants to be at least 85% sure that enough $100 bank notes are available to cope with clients’ demands, how much money must it keep in the machine just before 1:00 p.m. on Saturday? (d) Let x ∼ N (µ, σ 2 ). Find k such that P(µ − kσ < x < µ + 2kσ ) = 0.8. S&AS: STAT 0302 Business Statistics 2 *3. (a) The probability that parents with a certain type of blue-brown eyes will have a child with blue eyes is 3/7. In a such family, there are four children. (i) Calculate the probability that at least three of them have blue eyes. (ii) Find the expected value and standard deviation of the number of children who have blue eyes. (b) The day after placing a very large shipment of canned spinach on the shelves a grocer receives notice from the packer that 20% of the cans are spoiled. The grocer has sold 14 cans. (i) What is the probability that more than two but fewer than ﬁve cans were spoiled? (ii) Find the mean and standard deviation of the number of spoiled cans. (c) A plays B in the ﬁnal of a snooker tournament. Whoever wins four games ﬁrst gets the gold medal. Both perform independently from game to game. In each game, A’s chance of winning is 2/3. Find the probability that A gets the gold medal. (d) A station has two mini-buses, A and B , awaiting, each already having 12 seats occupied. Passengers come singly, selecting A with probability 0.45, and selecting B with probability 0.55. Find the probability that A is ﬁlled up before B and moves out ﬁrst. (A mini-bus has 16 seats.) 4. (a) A dating service ﬁnds that 22.5% of the couples that it matches eventually get married. In the next 125 matches that the service makes, ﬁnd the probability that (i) at least 20 couples get married; (ii) at most 37 couples get married. (b) In a very large batch of marbles, 7.5% are cracked. A sample of 60 marbles is observed. What is the probability that not more than four cracked marbles will be included in the sample? (Use a suitable approximation method.) (c) If r ∼ Bin(12, 0.35), what is the most probable value of r that will occur? What is this probability? (d) If r ∼ Bin(n, p), and it is known that the expected value and variance of r are 35 and 10.5 respectively, ﬁnd n and p. (e) If
15 2k = 15 3+k , ﬁnd k . 5. A sum of money, $10,000, is deposited in a bank, which oﬀers a nominal interest rate of 11% p.a. If computed on continuous growth basis, how much will the sum become at the end of the 6th year? S&AS: STAT 0302 Business Statistics 3 6. The sizes of two large batches, A and B , of an electrical component produced on two diﬀerent machines are in the ratio of 2 : 1. The resistances of the items in each batch have a normal distribution with mean 300 ohms and standard deviation 25 ohms for A, and mean 330 ohms and standard deviation 20 ohms for B . (a) The two batches are combined and mixed. Find the resulting mean, µ, and standard deviation, σ . (b) Any item whose resistance is more than 2σ (ohms) above or below µ is regarded as defective. Find the proportion of defective items in the combined batch. 7. (a) The number of persons going to watch the FB Cup (ﬁnal) football match this Saturday afternoon can be described by a normal distribution, whose mean is 15,000 and whose standard deviation is 1,800. (i) The sponsoring association will run a deﬁcit if the number of watchers is less than 15,450. What is the probability that this deﬁcit situation will arise? (ii) If the number of watchers exceeds 15,450, then on every person in excess of this number, the sponsoring association makes a proﬁt of $60. What is the probability that it will make a proﬁt of at least $90,000? (b) A die is loaded so that, in each roll, the probability of getting an odd number is 0.36. What is the probability of getting an even number between 15 and 22 times inclusive, out of 27 rolls of the die? (c) A circular box consists of a lid and a base. The diameter (in mm) of the lid has a N(35.52, 0.042 )-distribution. The diameter (also in mm) of the base has a N(35.42, 0.032 )-distribution. Lids and bases are randomly assembled to form boxes. The clearance should be at least 0.03 mm and at most 0.15 mm. Find the proportion of acceptable boxes. *8. (a) In a grocery shop it is found that on the average 8.0 boxes of a certain cake mix are sold per week, and that the probability of the number of boxes sold can be closely described by a Poisson distribution. Stock is re-ordered each week. To what number of boxes must the stock be topped up in order that there should be a less than 5% chance of running out of stock? (b) A large batch of items is 5.75% defective. A sample of 115 items is chosen. What is the probability of ﬁnding not more than 5 defective items? (Use Poisson approximation to binomial.) (c) What is the probability of getting at least 30 replies to questionnaires mailed to 100 persons, when the probability is 0.35 that any one of them will reply? S&AS: STAT 0302 Business Statistics 4 9. (a) If 73% of all persons ﬂying across the Atlantic Ocean experience the symptoms of jet lag for at least 24 hours, what is the probability that among 80 persons ﬂying across the Atlantic Ocean, at most 50 will experience the symptoms of jet lag? (b) What is the probability of getting more than 5 replies to questionnaires mailed to 92 persons, when the probability is only 0.065 that any one of them will reply? (c) Two contestants, A and B , are independently given the same multiple choice quiz which has 18 questions, each with four alternative answers of which only one is correct. A knows 14 of the correct answers and B knows 16. Both of them answer the remaining questions by sheer guesswork. Find the probability that A beats B ? (d) It is given that r ∼ Bin(80, 0.35). Find µ and σ 2 for r. 10. Five friends contributed equally to play the game of mark-six. They ﬁrst selected eight numbers, 1, 2, 3, 4, 5, 6, 7, and 8. They then bought one ticket of each combination of six numbers taken from these eight. Upon the lottery draw, the seven winning numbers were as follows: Regular Numbers: 2, 3, 4, 5, 48, and 49; Extra Number: 1 What was each person’s net win, after deducting his/her stake? 11. On the average ﬁve copies per issue of an unpopular weekly magazine are sold in a bookstore. The probability distribution of sales can be closely described by a Poisson distribution. The magazine costs $15.5 per copy to stock and is sold at $19.0. Unsold copies are not returnable to the publisher and hence result in a loss of $15.5 each. What is the most proﬁtable number of copies per issue to be stocked by the bookstore? What is the expected proﬁt for this order level? *12. A salesman regularly sells umbrellas or hot-dogs on Saturday afternoons at football games. He makes decision on Thursday on one of the following three possible actions: A = sell only umbrellas; B = sell some umbrellas and some hot-dogs; C = sell only hot-dogs. The various proﬁts (in dollars) that he can expect for the three kinds of weather, sunny, cloudy, and rainy, are as follows: Weather Sunny Cloudy Rainy A -300 500 2500 B 1200 750 1000 C 2000 600 -750 This Thursday he estimates the probability distribution of the weather for Saturday to be 0.40, 0.25 and 0.35 for sunny, cloudy and rainy condition, respectively. What is his optimum decision to make? What is the maximum expected proﬁt? ...
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- Spring '09