SEMESTER 2 EC30910 ECONOMICS STATISTICS I Answer TWO questions Time allowed: ONE AND A HALF hours Mathematical and Statistical Tables are available Calculators may be used 1. (a) A company uses a sampling plan to accept or reject lots of components received from a supplier. The plan calls for taking a random sample of 6 items and rejecting the order if more than 1 defective is found in the 6. If a shipment of 24 items is received, a third of which are defective, what is the probability of acceptance? What assumption must be made if you were to use the binomial distribution? (b) The events A, B and C are such that A and B are independent and A and C are mutually exclusive. Given that P(A)=O.4, P(B)=O.2, P(C)=O.3 and P(B~C)=O.1, calculate P(AUB), P(C | B), P(B | AUC). Also calculate the probability that one and only one of the events B, C will occur. (c) Sebastian Coe is entered for both the 800 metres and 1500 metres races at the Commonwealth Games and the probability of his obtaining a medal in the 800 metres race is 0.4. If he does obtain a medal in the 800 metres he will run in the 1500 metres race but if he does not get a medal in the 800 metres, to protect his public image he will declare that he has a virus and withdraw from the 1500 metres race. Peter Elliot is entered in only the 1500 metres race and can be relied upon to run whatever happens. The probability that Elliot will win the gold medal if Coe also runs in the race is 0.8 whereas if Coe does not run the probability of Elliot winning is reduced to 0.7 (presumably the other athletes believe the media hype and concentrate on beating Coe if present rather than the real favour Elliot). Peter Elliot duly wins the 1500 metres gold medal; what is the probability that Sebastian Coe did not get a medal in the 800 metres race? (d) The demand for items from stock is often in the form of a Poisson distribution and for a particular wholesaler the mean demand for cases of a certain brand of soap is 8 per week. If the wholesaler replenishes the stock to a level of 10 at the beginning of each week, what is the probability that the demand in a particular week will exceed the supply? (e) State the axioms of probability and explain how probability can be described as a'measure'. 2. (a) The probability that an invoice contains a mistake is 0.1. In an audit a sample of 12 invoices is chosen from one department; what is the probability that fewer than two incorrect invoices are found? (b) An assembly line has 8 grinding machines each with an operator. At least 6 machines must be in operation in order to maintain adequate quantities of material for the assembly line to produce on schedule. The probability that a machine will be inoperative on any day because of mechanical failure is 0.15, and the probability that any operator will be absent is 0.10. Any operator can use any machine. What is the probability that the production line will not meet its schedule on any particular day? (c) A supermarket receives shipments of eggs once a week. Data show that sales are approximately normally distributed with a mean of 220 dozens per week and a standard deviation of 20 dozen. What is the probability that a manager who stocks 250 dozen eggs at the beginning of the week will sell out by the end of the week? How many would he have to stock in order to have a probability of only 0.02 of selling out? (d) In a very large population 40% of the people support a change in government policy on regional aid. If a sample of 1,000 people is chosen at random, what is the probability that this sample will contain fewer than 390 people who support a change in policy? (e) A company produces television tubes with a length of life that is normal with a standard deviation of 5 months. How large should the mean, U, be in order that 90% of the tubes last for the guarantee period of 18 months. 3. (a) A competitive firm has a total cost function : TC=50+2q where q is the output of its single product. The firm sells the product at a constant price, p, but sales (=output) is a random variable with mean 20 and variance 4. At what level of price can the firm expect to break even? What is the standard deviation of profit when this price is charged? (b) A discrete random variable X has the probability function x -2 -1 0 1 2 3 p(x) 0.10 0.15 0.20 0.30 0.20 0.05 Evaluate the mean and variance of X. Display the probability distribution of X2 (x squared) and find its mean and variance. (c) QUESTION UNAVAILABLE (d) The market research department of Tate and Lyle Sugar Company believe that the annual sugar consumption per household in Birmingham is normally distributed with a standard deviation of 3lbs; there is some doubt about the precise value of the mean consumption. A sample of 36 households is selected and the sugar usage is recorded for one year. What is the probability that the mean of the sample is within l lb of the population mean? What is the minimum size sample that must be selected for the probability to be 0.99 that the sample mean is within l lb of the population mean? (e) Experience shows that the sales of a certain firm are normally distributed, exceed 250 units 5% of the time and exceed 210 units 90% of the time. Find the expected value and the variance of the firm's sales. 4. (a) An experiment consists of rolling two fair six-sided dice and noting the number of spots showing on the uppermost faces. Define the random variable X as follows: X=l if the total number of spots is odd, X=2 if the total is even but the outcome is not a pair, and X=3 if the outcome is a pair. Derive the probability distribution of X. (b) In the experiment of part (a) above, Y is defined as 1 if the outcome of the throw of the first die is odd and 2 if it is even, while Z is defined as 1 if the second die shows 1, 2 or 3 spots and equals 2 if the second die shows 4, 5 or 6 spots. Determine the joint distribution of Y and Z. Are Y and Z independent? Are they uncorrelated? (c) QUESTION UNAVAILABLE (d) The weekly wages of employees in a particular industry are normally distributed with a mean of #172 and a standard deviation of #19. What is the probability that a random sample of 10 employees in the industry will have an average weekly wage of #180 or more? (e) A company selects a random sample of 100 individuals from a telephone directory for a survey concerning one of its products. If 20% of the people in the phone book use the product, what is the probability that 25 or more of the members of the sample will use the product? How large must the sample be for the probability to be 0.975 that less than 25% of the sample members use the product? 5. (a) Explain the inter-relationships between the Normal t and xsquared distributions; and discuss their implications when sampling from a Normal population. (b) Derive the 97% equal-tailed confidence interval for the mean u of a normal population with known variance o2, based on a random sample of size n with sample mean X. What effect will a lack of knowledge of o2 or the fact that the population is not normal have on your result? (c) A researcher wishes to estimate the mean weekly wage of the several thousand of workers employed in a firm; within plus or minus #10 and with a 99% degree of confidence. From past experience, the researcher knows that the weekly wages of these workers are normally distributed with a standard deviation of #20. What is the minimum sample size required? (d) A sample of 15 employees was randomly selected from a workforce in order to estimate the mean travel time to work. If the sample mean was 55 minutes and the sample standard deviation 13 minutes, find 90% and 98% confidence intervals for the population mean travelling time. (e) The standard deviation of the breaking strengths of a sample of 100 cables tested by a company was found to be 180 kg. Find a 90% equal-tailed confidence interval for the standard deviation of breaking strength of all such cables produced by the company, given that breaking strength has a normal distribution.