10_09_09 - STAT 409 Fall 2009 Examples for Testing...

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STAT 409 Examples for 10/09/2009 Fall 2009 Testing Hypotheses about a Population Proportion p Null Alternative H 0 : p = p 0 vs. H 1 : p < p 0 Left – tailed. H 0 : p = p 0 vs. H 1 : p > p 0 Right – tailed. H 0 : p = p 0 vs. H 1 : p p 0 Two – tailed. 1. Let X have a Binomial distribution with the number of trials n = 8 and with probability of “success” p . We wish to test H 0 : p = 0.60 vs. H 1 : p < 0.60. Binomial distribution ( n = 8 ) CDF, P ( X x ): p n x 0.20 0.30 0.40 0.50 0.60 0.70 8 0 0.1678 0.0576 0.0168 0.0039 0.0007 0.0001 1 0.5033 0.2553 0.1064 0.0352 0.0085 0.0013 2 0.7969 0.5518 0.3154 0.1445 0.0498 0.0113 3 0.9437 0.8059 0.5941 0.3633 0.1737 0.0580 4 0.9896 0.9420 0.8263 0.6367 0.4059 0.1941 5 0.9988 0.9887 0.9502 0.8555 0.6846 0.4482 6 0.9999 0.9987 0.9915 0.9648 0.8936 0.7447 7 1.0000 0.9999 0.9993 0.9961 0.9832 0.9424 a) Suppose we observe X = 3. Find the p-value of this test. p-value = P ( value of X as extreme or more extreme than X = x observed | H 0 true ) = P ( X 3 | p = 0.60 ) = 0.1737 . b) Suppose we decided to use the rejection region “Reject H 0 if X 1.” Find the significance level α associated with this rejection region. significance level α = P ( Reject H 0 | H 0 is true ) = P ( X 1 | p = 0.60 ) = 0.0085 .

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c) Find the “best” rejection region with the significance level α closest to 0.05. Rejection Region for a Left – tailed test: Find a such that P ( X a ) α . Then the Rejection Region is “Reject H 0 if X a .” significance level α = P ( Reject H 0 | H 0 is true ) = P ( X a | p = 0.60 ). P
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10_09_09 - STAT 409 Fall 2009 Examples for Testing...

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