# Lecture22.pdf - STAT 311 LECTURE 22 Effect size power and...

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STAT 311: LECTURE 22Effect size, power and sample size calculations
HousekeepingNext week is week 9Last week of new materialFocus is on statistical inference for measures of associationLast labFocus is on inference for linear regressionAnd…Memorial Day weekend coming upAll week 9 homework/lab due dates shifted to the following Tuesday (wk 10)2Stat 311 Spring 2018 : Prof Morris
This weekMonday (UH Ch 12)Hypothesis tests: overviewType 1 and Type 2 errorsWednesday (UH Ch 13)Hypothesis tests: proportionsHypothesis tests: meansFriday Comparing HTs & CIsCalculating the power of a testEffect sizes for comparing means3Stat 311 Spring 2018 : Prof Morris
Review: the logic of testingWe use a sample to estimate a parameter, A mean, proportion, or differences in theseWe test H0: 𝜃 = 𝜃0Against one of three possible Ha: 𝜃 > 𝜃0𝜃 < 𝜃0𝜃 ≠ 𝜃0Then, based on the p-value of 𝜃in the H0distributionWe use the zscore transformation of 𝜃for our test statisticDecide: reject H0 or do not reject H04Stat 311 Spring 2018 : Prof Morris
Setting up the test5Calculate the test statistic: 𝜃−𝜃0?0(𝜃)Find the p-value of the test statistic?. ?????(𝜃) = 𝑃(𝜃 | 𝐻0, 𝐻𝑎, ?)= 𝑃? ?? 𝑇>𝜃−𝜃0?0(𝜃)????? ??𝑖? ????<𝜃−𝜃0?0(𝜃)????? ??𝑖? ????>𝜃−𝜃0?0(𝜃)??? ??𝑖? ????Stat 311 Spring 2018 : Prof Morris * for a two-tail test multiply the p-value by 2Just a z-score in the standardized H0distnTable lookup, or software
Which tail to use?6Simple: It just depends on HaH0HaTestH0:𝜃 = 𝜃0Ha: 𝜃 > 𝜃0Upper tail testHa: 𝜃 < 𝜃0Lower tail testHa:𝜃 ≠ 𝜃0Two tail testStat 311 Spring 2018 : Prof Morris a%a%?2%?2%
Test 𝐻0??. 𝐻𝑎7Reject 𝐻0if: ?. ?????𝜃< ααis the “significance level” of the testαis the type 1 error rate of this testFor example: an upper tail z-testIf: 𝜃−𝜃0?0(𝜃)> ?𝑐?𝑖?Then: ?. ?????𝜃< αso reject 𝐻0Stat 311 Spring 2018 : Prof Morris a%(1-a)%?𝑐?𝑖?Reject H0Note: the p-value test isALWAYS LESS THAN
Same test: alternative statementsStat 311 Spring 2018 : Prof Morris 8Using the value of 𝜃: 𝜃−𝜃0?0(𝜃)> ?𝑐?𝑖?in the standardized 𝐻0distribution𝜃 > 𝜃𝑐?𝑖?in the 𝐻0distribution (original measurement scale)Using the the p-value: ?. ?????𝜃−𝜃0?0(𝜃)< αin the standardized 𝐻0distribution?. ?????𝜃< αin the 𝐻0distribution (original measurement scale)Or < for a lower tail test
0Stat 311 Spring 2018 : Prof Morris 9𝑐?𝑖?
Reject H0if ? > ?𝑐?𝑖?a%1-a%Example: Solving for ?𝑐?𝑖??𝑐?𝑖?= ?0+ ?0.05 ∗ ?0(?)= 0.5 + 1.65 ∗0.5(1−0.5)100= 0.58H0: p = 0.5Ha: p > 0.5a: 5%n=10010Stat 311 Spring 2018 : Prof Morris ?𝑐?𝑖?= 0.58So if ? > 0.58we can reject H0Upper tail test for single proportion
Type 2 error and power for specific alternative hypothesesPower of a test11Stat 311 Spring 2018 : Prof Morris
Hypotheses, and errorsWhen H0is true:ais the probability of a false positivePopulationcondition:Test Result:Fail to reject H0Reject H0H0is trueP(correct decision) = (1-a)True negativeP(Type 1 error) = aFalse positiveH0is falseP(Type 2 error) = bFalse negativeP(correct) = (1-b)True positive12Stat 311 Spring 2018 : Prof Morris