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class05-2-handouts

# class05-2-handouts - PSTAT 120B Probability Statistics...

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PSTAT 120B - Probability & Statistics Class # 05-2- Small-sample confidence intervals Jarad Niemi University of California, Santa Barbara 28 April 2010 Jarad Niemi (UCSB) Small-sample confidence intervals 28 April 2010 1 / 19

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Class overview Announcements Announcements Homework Homework 4 due Monday 3 May Do not turn homework into my mailbox - you will now get a zero Mid-term I Typo in answer key, correct answer for short answer 1b should be θ 2 3 n See me if you have this answer and had points marked off. Mid-term II Friday 14 May Friday Bring a calculator! Jarad Niemi (UCSB) Small-sample confidence intervals 28 April 2010 2 / 19
Class overview Goals Pivotal confidence interval approach Two-sided confidence interval One-sided confidence interval Finding the sample size Clinical trial example Small sample confidence intervals Difference in heights example Confidence intervals for variances Compare variances for the heights example Jarad Niemi (UCSB) Small-sample confidence intervals 28 April 2010 3 / 19

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Confidence interval approach Goal: For a parameter θ , find ˆ θ L and ˆ θ U such that P ( ˆ θ L θ ˆ θ U ) = 1 - α and call this a 100(1 - α )% confidence interval. Pivotal approach to finding confidence intervals Find α . Find a pivotal quantity Q = f ( X 1 , X 2 , . . . , X n , θ ). Find a and b such that P ( a Q b ) = 1 - α. Solve so that θ is left alone in the middle of the inequalities P ( ˆ θ L θ ˆ θ U ) = 1 - α Jarad Niemi (UCSB) Small-sample confidence intervals 28 April 2010 4 / 19
Confidence interval approach Exponential example (cont.) Suppose X 1 , . . . , X 10 iid Exp ( β ). Find a 90% two-sided CI for β . α = 1 - 0 . 90 = 0 . 10 Q = 2 β 10 i =1 X i χ 2 20 is a pivotal quantity for β P ( a Q b ) = 1 - α 0 10 20 30 40 50 0.00 0.02 0.04 0.06 0.08 Density for a χ 20 2 x f(x) α 2 α 2 1 - α So P (10 . 8508 Q 31 . 4101) = 1 - 0 . 10 = 0 . 90. Jarad Niemi (UCSB) Small-sample confidence intervals 28 April 2010 5 / 19

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Confidence interval approach Exponential example (cont.) Suppose X 1 , . . . , X 10 iid Exp ( β ). Find a 90% two-sided CI for β .
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