class10-1-handouts

class10-1-handouts - PSTAT 120B Probability Statistics...

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PSTAT 120B - Probability & Statistics Class # 10-1- Hypothesis testing review Jarad Niemi University of California, Santa Barbara 2 June 2010 Jarad Niemi (UCSB) Hypothesis testing review 2 June 2010 1 / 20
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Class overview Announcements Announcements Homework Homework 7 due now Final exam Monday, 7 June @ 8am-11am in HFH 1104 (here) Allowed/required items: one (front and back) 8 1 2 × 11 cheat sheet calculator #2 pencil Jarad Niemi (UCSB) Hypothesis testing review 2 June 2010 2 / 20
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Class overview Goals P-values Review of hypothesis testing Jarad Niemi (UCSB) Hypothesis testing review 2 June 2010 3 / 20
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P-values Definition If W is a test statistic, the p-value is the smallest level of significance α for which the observed data indicate that the null hypothesis should be rejected. Definition The p-value is the probability of obtaining a test statistic W at least as extreme as the one observed assuming the null hypothesis is true. Jarad Niemi (UCSB) Hypothesis testing review 2 June 2010 4 / 20
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Hypothesis testing decisions Any kind of decision you make under uncertainty can have 4 possible outcomes. H 0 True False Test result Reject H 0 Type I error True positive Do not reject H 0 True negative Type II error α = P(type I error) β = P(type II error) 1 - β = power of the test Jarad Niemi (UCSB) Hypothesis testing review 2 June 2010 5 / 20
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Large sample tests Large-sample α -level hypothesis tests H 0 : θ = θ 0 H a : θ > θ 0 (upper-tail alternative) θ < θ 0 (lower-tail alternative) θ 6 = θ 0 (two-tailed alternative) Test statistic : z = ˆ θ - θ 0 σ ˆ θ Rejection region : z > z α (upper-tail RR) z < - z α (lower-tail RR) | z | > z α/ 2 (two-tailed RR) p-values : P ( Z > z ) (upper-tail) P ( Z < z ) (lower-tail) 2 P ( Z > | z | ) (two-tailed) Jarad Niemi (UCSB) Hypothesis testing review 2 June 2010 6 / 20
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Sample size Typically, the type I error probability α is given, and the type II error probability β can be chosen by selecting the appropriate sample size n . To obtain a specific β find the smallest integer n , such that n ( z α + z β ) 2 σ 2 ( θ 0 - θ a ) 2 Jarad Niemi (UCSB) Hypothesis testing review 2 June 2010 7 / 20
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Sample size Relationship to confidence intervals Type Large-sample CIs Large-sample hypothesis testing Pivotal Interval Rejection region Upper-tail ˆ θ - θ σ ˆ θ < z α ˆ θ - θ 0 σ ˆ θ > z α Lower-tail ˆ θ - θ σ ˆ θ > - z α ˆ θ - θ 0 σ ˆ θ < - z α Two-tail ˆ θ - θ σ ˆ θ < z α/ 2 ˆ θ - θ 0 σ ˆ θ >
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