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Unformatted text preview: 03/2010 THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1302 PROBABILITY AND STATISTICS II Assignment 4 Section A Hypotheses & likelihood ratios In this section, you are required to (i) specify the null and alternative hypotheses in terms of unknown parameters, and (ii) calculate the likelihood ratio for the hypotheses for the following problems. A1. A property developer suggests to build a private residential estate by the lake if no more than 21,000 fish live in the lake. This project is expected to be highly profitable. However, an environmentalist objects strongly to the project as the living environment of the fish will be seriously damaged. Field investigators took 1,000 fish from the lake, marked them, and returned them. Over the next month, they caught 2,000 fish and found that 100 of them were marked. Give your answers (a) if you were the property developer; (b) if you were the environmentalist. A2. A group of 108 patients suffering from a certain illness is divided into two subgroups of the same size; the first subgroup receives ordinary treatment, the remainder receives a new treatment. Suppose the probabilities of substantial, mild and no improvements for the two treatments are as follows: Ordinary treatment New treatment Substantially improved p s q s Mildly improved p m q m Not improved p n = 1- p s- p m q n = 1- q s- q m . The following results were observed: 1 Ordinary treatment New treatment Substantially improved 12 18 Mildly improved 6 12 Not improved 36 24 Total 54 54 A doctor has to decide whether to adopt the new treatment. A3. A coin has a probability p to turn up a head when tossed. It was tossed 100 times and 60 heads turned up. You want to determine if the coin is fair. A4. Assume that the interarrival times of customers to a service centre are i.i.d. exponential random variables with rates θ and ψ for the periods 4:00-4:30pm and 7:00-7:30pm respectively. During the half hour from 4:00pm to 4:30pm, customers arrive at a service centre at times 4:08pm, 4:12pm, 4:15pm, 4:23pm and 4:29pm. During the half hour from 7:00pm to 7:30pm, customers arrive at the same centre at times 7:02pm, 7:06pm, 7:10pm, 7:13pm, 7:16pm, 7:17pm, 7:19pm, 7:22pm, 7:24pm, 7:28pm and 7:29pm. The manager wants to find out if customers arrive more frequently during the latter period in general. A5. Assume the number of bus breakdowns on a single day is a Poisson ( λ ) random variable. A bus company experienced three bus breakdowns on a single day, and six on the next day. The company claimed that the probability that there is no bus breakdown on a single day is bigger than 0 . 75. A6. Assume the height of a student is normally distributed, with mean μ 1 , variance σ 2 1 for boys, and mean μ 2 , variance σ 2 2 for girls. The heights of 9 randomly selected students were measured to be 1.72, 1.76, 1.80, 1.65 m for males, and 1.56, 1.48, 1.62, 1.55, 1.58 m for females. You are interested in knowing whether male students are taller in general.interested in knowing whether male students are taller in general....
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