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MSc Regulation and Competition Quantitative Techniques 1 Exam questions for January 2004 Time allowed: two hours Attempt ALL questions. The points for each answer are shown in square brackets [like this] 1. What are the main sources of data errors? What can be done to reduce errors in data you did not collect yourself? [10] 2. B: For each of the following statements about the probabilities of outcomes A and (a) Say whether they are true, false, or uncertain (b) If uncertain, spell out conditions under which they are true (i) (ii) (iii) (iv) (v) P(A or B) = P(A) + P(B) P(AB) = P(A) + P(B) P(AB)=P(A).P(B) P(AB) = P(A).P(B) P(A | B)= P (AB)/P(B) [3] [3] [3] [3] [4] 3. Suppose x is a continuous random variable with the probability density function (pdf): f(x)= x for 0x1 2 - 2x for 1 x 2 0 elsewhere NB This question does not work: see answers file! a) Draw a graph of this function [4] b) Explain how you know this is a valid pdf. [4] c) Comment on the relative position of the mean, median and mode. [4] d) Calculate the probability that 0.5 x 1.5 [4] 1 4. You are organising a concert and believe that attendance will depend on the weather. You believe the following possibilities are appropriate: Weather Terrible weather Mediocre weather Good weather Probability 0.2 Attendance 500 0.6 0.2 1200 2000 a) What is the expected attendance? [4] b) Suppose each ticket costs £5 and the fixed costs are £2,000. What are the expected profits? [4] c) Graph the probability distribution for profits [4] d) What is the most you could pay for the fixed costs and still have an 80% chance of making a profit on the event? (to nearest £) [4] 5. Suppose that heights in a population are normally distributed with a mean of 78 inches and a standard deviation of 5 inches. a) What is the probability that an individual selected at random will have a height between 68.2 and 79.8 inches? [6] b) Construct a 95% confidence interval for the average height in a random sample of four individuals. [6] 6. Suppose we wanted to conduct a survey. It is desired that we produce an interval estimate of the population mean that is within 5 from the true population mean with 99% confidence. Based on a historical planning value of 15 for the population standard deviation, how big should the sample be? [6] 7. Explain in simple terms the differences (and similarities if any) between the following approaches to estimation; a) method of moments [4] b) maximum likelihood [4] c) least squares [4] 8. Under what conditions will the ordinary least squares estimator be a) unbiased b) efficient? [4] [4] c) What does it mean to say an estimate is consistent? [4] 2