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Homework9_solution - σ =0.35(ppm A sample of ten fish was...

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Homework 9 Due Date: 11/29/2010, but you are welcome to submit it next Wednesday. Problem #1: Suppose that the current measured in a thin copper wire follows a uniform distribution with a range of [ ] θ , 0 and the probability density function ( 29 = elsewhere x x f 0 0 / 1 θ θ A sample of five measurements is taken and the data are shown below (in milliamperes): 11.5, 3.2, 7.3, 19.2, 16.7 Show that the maximum likelihood estimate of θ is 2 . 19 max = = x θ . The likelihood function is ( 29 ( 29 = elsewhere x x L 0 0 ,..., 0 / 1 5 1 5 θ θ θ θ Since L( θ ) is a decreasing function of θ , the lower value θ is of, the larger the likelihood function. We should set θ as low as possible, on the other hand, we have to make sure that θ is larger than all the observed values. Therefore we have 2 . 19 max = = x θ . Problem #2: Assume that the mercury contamination in a largemouth bass follow a normal distribution with unknown mean
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Unformatted text preview: σ =0.35 (ppm). A sample of ten fish was selected from Florida lakes and mercury concentration in the muscle tissue was measured (ppm). 1.230 0.490 1.080 0.590 0.280 1.200 0.710 0.190 0.730 0.340 (a) Find the maximum likelihood estimate of the unknown population mean µ . The log likelihood function is ( 29 ( 29 ( 29 ∑ = ----= 10 1 2 2 35 . 2 1 35 . ln 2 10 2 ln 2 10 i i x L μ π Taking of the first derivative of ( 29 L versus µ and solve for µ. 684 . = = ∧ x (b) Construct a 95% confidence interval for µ . Ans: (0.467, 0.901) (c) How large a sample is required if we want to be 95% confident that our estimate of µ is off by less than 0.05 ppm. Ans: 189 Problem #3: 9.12 Problem #4: 9.25 Practice problem: Not required for anyone....
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