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ams572_notes_5

# ams572_notes_5 - AMS 572 Lecture Notes#5 Quiz 1 Let be a...

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AMS 572 Lecture Notes #5 September 24, 2010 Quiz 1. Let be a random sample from a normal population with mean , variance . Please (1) Derive the distribution of (2) Derive the 100(1- α )% confidence interval for μ Solution: (1) First we derive the mgf of Now we derive the mgf of 1

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m.g.f. of Thus we have shown that (2) Based on the PDF of the pivotal quantity for μ We have: Thus 2
the 100(1- )% C.I. for is α μ <Today’s Topic> Power Calculation (Inference on one population mean) Truth Decision Type II error Type I error = P(Type II error) = P(Fail to reject |) Power Calculation (Inference on one population mean) 3

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1. 1 st scenario, Normal population, is known. Test statistic : At the significance level , we reject if Power of the test = P(reject |) = 1- Power = 1- = P(reject |) = P( = , If , = = , 4
2 nd scenario, Normal population, is un known. Test statistic : At the significance level , we reject if Power of the test = P(reject |) = 1- Power = 1- = P(reject |) = P( = = = , Here 5

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Recall that the Shapiro-Wilk test : can be used to determine whether the population is normal.
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