sample1 - 1 ·· X n be a random sample from the...

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Sample Problems Statstics 517 1. The length of life of brand X lightbulbs in hours is assumed to have a mean μ 1 and a variance σ 2 1 = 400. The length of life of brand Y lightbulbs in hours is assumed to have amean μ 2 and a variance σ 2 2 = 480. To compare μ 1 and μ 2 , we take a random sample of size n = 50 from brand X lightbulbs and a random sample of size m = 60 from brand Y lightbulbs. Two samples are independent. The sample statistics are ¯ x = 900 hours, and ¯ y = 910 hours. (a) We do not know the distributions of X and Y . To construct a confidence interval for μ 1 - μ 2 , what distribution will you use? By which theorem is this legitimate? (b) Find a 95% confidence interval for μ 1 - μ 2 . (c) Suppose we want to decide the sample size for the future study based on this sample. Assume that we only can take 50 observations from brand X , but we can take as many obervations as we want from brand Y in the future study. If we want the length of 95% confidence interval for μ 1 - μ 2 to be at most 14 hours, how large should be the size of the sample from brand Y ? 2. Let X
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Unformatted text preview: 1 , ··· , X n be a random sample from the distribution with the following probability density function. f ( x ; θ ) = 3 x 2 θ 3 , ≤ x ≤ θ, θ > (a) Show that the method of moments estimator of θ is ˜ θ = 4 3 ¯ X . (b) Show that ˜ θ is an unbiased estimator of θ . (c) Show that the maximum likelihood estimator of θ is ˆ θ = max 1 ≤ i ≤ n X i . 1 (d) ˆ θ is a biased estimator. What is the bias? (Hint: Remember the distribution of order statistics. Use the pdf of Y n = max 1 ≤ i ≤ n X i .) (e) Show that 3 n +1 3 n ˆ θ is an unbiased estimator of θ . We now have two unbiased estimators, ˜ θ and 3 n +1 3 n ˆ θ , found in (a) and (e). Since both of them are unbiased, one with a smaller variance is more efficient than the other with a larger variance. We will compare their variances. (f) Calculate var( ˜ θ ) and var( ˆ θ ). (g) Suppose n > 1. Which one is more efficient, ˜ θ or 3 n +1 3 n ˆ θ ? 2...
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This note was uploaded on 09/10/2009 for the course STATS 517 taught by Professor Song during the Fall '07 term at Purdue.

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sample1 - 1 ·· X n be a random sample from the...

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