TESTING MEANS. we want to test. but we need to know if 2 1 = 2 2 if it is, we use the methods described last time pooled estimate of variance

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1 Introdction to Statistics in Psychology PSY Profess Greg Francis Lectre 6 Hypothesis testing f two sample case Planning a replication stdy TESTING MENS we want to test H : µ µ H a : µ µ 6 bt we need to know if if it is, we se the methods described last time pooled estimate of ariance s (n )s +(n )s n + n ts + n n and se the t-distribtion REVISED TESTING MENS when 6 we mst make two changes. di erent estimate of standard err of the di erence. adjstment of degrees of freedom still se the t distribtion B s + s n n ts X + s X 3 DEGREES OF FREEDOM when 6 we calclate df as: s /n + s /n! (s /n ) /(n ) + (s /n ) /(n s s X + s X /(n ) s X /(n ) looks (and is) messy jst a matter of plgging in nmbers careflly still se the t-test as befe! We call it Welch s test EXMPLE researcherwantstoknowifsingle married parents are me satisfied with their stats. She randomly samples 6 single and 6 married parents. Each parent rates her/his marital stats satisfaction, with higher sces indicating greater satisfaction. The researcher wants to know if there is a di erence between the poplation means of single erss married parents. data smmary Variable n X s s X Grop Grop HYPOTHESES H : µ µ indicating there is no di erence in satisfaction between the two grops H a : µ µ 6 indicating there is a di erence in satisfaction between the two grops not an dered hypothesis becase we do not know who might be me satisfied leel of significance is set at

2 WORRY BOUT HOMOGENEITY We do not know the tre ales of and,btwenoticethatn <n and that s >s. This makes s wry that maybe Type I err rate will be o (and maybe too big), so we se Welch s t-test The online calclat in the textbook ses Welch s test nless n n TEST STTISTI pooled standard err estimate B B (.6) 6 s + s n n (.46) 6 TEST STTISTI the fmla f the test statistic is Test statistic Statistic Parameter Standard Err t (X X ) (µ µ ) ( ) () t TEST STTISTI adjsted degrees of s X + s s X /(n ) s X /(n ) (.77) +(.36) ((.77) ) /(6 ) + ((.36) ) /(6 ) p VLUE From the t-distribtion calclat, we find (f a two-tailed test with ) that p.8455 >.5 we cannot reject H indicates that the means across poplations do not di er there is no eidence that satisfaction with marital stats di ers f married erss single parents the probability that the obsered ( me extreme) di erence in means wold occr by chance if µ µ isgreaterthan.5 ONLINE LULTOR s always, it is best to se a compter. We can enter the smmary statistics.

3 ONLINE LULTOR Yo need to nderstand how to pll ot the infmation yo want ONFIDENE INTERVL Basic fmla f all confidence interals: I statistic±(critical ale)(standard err) f a di erence of sample means I (X X ) ± t c We already hae most of the terms (we get t c from the Inerse t-distribtion calclat, so I 95 ( )±(.9879)(.75683) I 95 (.4635,.5455) POWER Power is treated mch the same as f the one-sample case We jst hae to keep track of whether we are sing the standard t-test Welch s test The on-line calclat of textbook does this f yo atomatically Power is ery imptant when designing an experiment REPLITION INTERESTING STUDY INTERESTING STUDY n imptant characteristic of science is replication Show that the same methods and measres prodce the same reslts Hard sciences are ery good at this (e.g., physics, chemistry) Sciences that depend on statistics face challenges We always face a risk of making a Type I a Type II err Ths, sccessfl replication is not expected een f real e ects Yo can mitigate these problems by designing good replication stdies that se the same methods, bt hae high power onsider a stdy on how nonconfmity an indce higher stats in certain enironments Participants were 5 shop assistants wking in downtown Milan, Italy botiqes (rmani, Brberry, hristian Di, La Perla, Les opains, and Valentino) Two grops of 6 each read a ignette: Imagine that a woman is entering a lxry botiqe in downtown Milan dring smmer. She looks approximately 35 years old. Nonconfming condition (Grop ): She is wearing plastic flip-flops and she has a Swatch on her wrist. onfming condition (Grop ): She is wearing sandals with heels and she has a Rolex on her wrist. Rate the stats of the woman on a scale of 7 (bigger means higher stats) The reslts are: Nonconfming X 4.8 onfming X 4. t(5). p

4 REPLITION Yo want to repeat the stdy, bt it is not easy to get shop assistants from high end stes (yo might hae to go to hicago f y sbjects) The online power calclat reqires yo to enter estimates of: H : µ µ H : µ a µ a X X.6, f the standard deiations, we se some algebra. We know that f the repted t-test: X X. t s X X.6 s X X so s.6 X X..857 We can assme the standard t-test was sed, so.ths s X s t + s t X n n s(.7735) so s which we can se f both and 9 REPLITION Oftentimes researchers jst se the same sample size as apreiosstdy.fterall,thatstdywked,soit mst be an appropriate sample size, right? no. If we se n n 6,theon-linepowercalclat gies power.5397 this shold make sense becase the p.4 in the iginal stdy is jst below the.5 criterion if we take a di erent random sample, we will get a di erent p ale, almost half the time it will be bigger than REPLITION Sppose yo want 8% power The calclat tells yo that yo need n n 48 participants. Nearly twice as big as the iginal stdy! if yo do the replication crectly, yo typically rn a better stdy than the iginal That is common in science, where new experiments are better than old experiments EFFET SIZE The power calclat comptes a term called.thisisanestimateofd between the nll and specific alternatie distribtions. Bigger ales of mean it is easier to notice a di erence. It can be compted from the means and standard deiation estimates that yo proide to the power calclat. n estimate, d, canalsobecompted from the t ale and sample sizes d t t + (.) n n Often called ohen s d t EFFET SIZE Yo might wry that the e ect size of the iginal stdy is an oerestimate fter all, if the researchers had not fond a significant di erence, they might not hae pblished their paper (pblication bias) conseratieapproachistodiidethe estimated e ect size half, and do the power calclation from that new e ect size. Ths, we can directly enter: d The power calclat now tells s that to hae 8% power, we need n n 87sbjects This cold be a ery di clt experiment to rn ONLUSIONS Welch s test Power Replication 3 4

5 NEXT TIME hypothesis testing f dependent samples sampling distribtion standard err Jst relax. 5