A random sample of size n is to be used to test thenull hypothesis that the parameter θ of an exponentialpopulation equals θ0 against the alternative that it doesnot equal θ0.(a) Find an expression for the likelihood ratio statistic.(b) Use the result of part (a) to show that the criticalregion of the likelihood ratio test can be written as x · e−x/θ0 F K
A random sample of size n is to be used to test the
null hypothesis that the parameter θ of an exponential
population equals θ0 against the alternative that it does
not equal θ0.
(a) Find an expression for the likelihood ratio statistic.
(b) Use the result of part (a) to show that the critical
region of the likelihood ratio test can be written as
x · e−x/θ0 F K
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Can you use these results and Theorem 12.2 (see image) to test the null
hypothesis that ? = 150 thousand miles against the alternative hypothesis ? < 150 thousand miles at the 0.01 level of significance for the following: the times to failure of a power train component in a certain car make and model are (in thousands of miles) 225, 459, 54, 180, 585, 558, 36, 342, 270, 378, 108, 396, 18, 252,135, 18, 162, 297, 171, and 27. Consider these data to be a random sample from an exponential (?) population.
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