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2S1: Assignment 4 27th February 2009 Due date: Tuesday 31st March 1. A random variable X has a Poisson distribution with unknown parameter λ. Given a random sample x1 , . . . , xn , show that the maximum likelihood estimate for λ is the sample mean x̄. 2. Let X be a continuous random variable with an exponential distribution. The probability density function for X is λ e−λx if x > 0 f (x) = 0 otherwise where λ is an unknown parameter. Given a random sample x1 , . . . , xn , show that the method of moments estimate for λ is x̄1 . 3. A population follows a normal distribution with unknown variance σ 2 . If a random sample of size 20 has sample variance s2 = 1.1, find a 95% confidence interval for the variance of the total population. 4. A scientist samples a chemical and finds that in 6 observations the levels of phosporous are 3, 5, 7, 4, 5, 6 A second scientist takes a sample of 8 observations and obtains the results 8, 7, 9, 11, 7, 8, 10, 12 Carry out a test with a significance level of 5% to determine if there is a significant difference between the population means. (The variance of both populations are assumed to be the same). 5. Use a χ2 goodness of fit test to determine if the following data follows a binomial distribution with p = 0.5: x frequency 0 5 1 6 2 9 1 3 7 4 2