4. Suppose that n light bulbs are burning simultaneously to determine the lengths of their lives....

 

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4. Suppose that n light bulbs are burning simultaneously to determine the lengths of their lives. We shall assume that they burn independently and that the lifetime of cach bulb has the exponential distribution with parameter B. Let X denote the lifetime (in thousand hours) of bulb i, for i = 1,..., n. The pdf, mean and variance of exponential distribution Exp(3) are -Bx 1 1 f(x)=e B, E(X) = Var(X) = B' 2 1/B (a) What does the central limit theorem say about the distribution of (x-1/3) when n is large? (b) Find a two-sided 95% confidence interval of the mean lifetime (1/3). Express your answer in terms of n, X and z (the z critical values). Interpret the confidence interval you obtain in context of the situation. (c) Now, suppose that we want to test whether or not the mean lifetime is 1000 hours, we would consider Ho=1 Ha : 1. : Derive the likelihood ratio statistic. (d) Denote the likelihood ratio statistic by A(X). Should the null hypothesis Ho be rejected when A(X) is large or small? (c) What is the large-sample distribution of -2 log A(X) when the null hypothesis Ho is true? (f) Explain what a Type I error would mean in context of this problem. 4. Suppose that n light bulbs are burning simultaneously to determine the lengths of their lives. We shall assume that they burn independently and that the lifetime of cach bulb has the exponential distribution with parameter B. Let X denote the lifetime (in thousand hours) of bulb i, for i = 1,..., n. The pdf, mean and variance of exponential distribution Exp(3) are -Bx 1 1 f(x)=e B, E(X) = Var(X) = B' 2 1/B (a) What does the central limit theorem say about the distribution of (x-1/3) when n is large? (b) Find a two-sided 95% confidence interval of the mean lifetime (1/3). Express your answer in terms of n, X and z (the z critical values). Interpret the confidence interval you obtain in context of the situation. (c) Now, suppose that we want to test whether or not the mean lifetime is 1000 hours, we would consider Ho=1 Ha : 1. : Derive the likelihood ratio statistic. (d) Denote the likelihood ratio statistic by A(X). Should the null hypothesis Ho be rejected when A(X) is large or small? (c) What is the large-sample distribution of -2 log A(X) when the null hypothesis Ho is true? (f) Explain what a Type I error would mean in context of this problem.