Chapter 7 - Hypothesis Tests Applied to Means

Transcription

1 Chapter 7 - Hypothei Tet Applied to Mea 7.1 Ditributio of 100 radom umber: mea(dv) = 4.46 t. dev(dv) =.687 var(dv) = Doe the Cetral Limit Theorem work? The mea ad tadard deviatio of the ample are 4.46 ad.69. The mea ad tadard deviatio are very cloe to the other parameter of the populatio from which the ample wa draw (4.5 ad.7, repectively.) The mea of the ditributio of mea i 4.45, which i cloe to the populatio mea, ad the tadard deviatio i 1.0. Populatio Parameter Predictio from Cetral Limit Theorem Empirical Samplig ditributio µ = 4.5 = 4.5 = 4.45 σ = 7. σ 7. = 1.44 = = = The mea of the amplig ditributio i approximately correct compared to that predicted by the Cetral Limit theorem. The variace of the amplig ditributio i almot exactly what we would have predicted. 7.5 The tadard error would have bee maller, becaue it would be etimated by itead of I ued a two-tailed tet i the lat problem, but a oe-tailed tet could be jutified o the groud that we had o iteret i howig that thee tudet thought that they were below average, but oly i howig that they thought that they were

2 above average. 7.9 While the group that wa ear the bottom certaily had le room to uderetimate their performace tha to overetimate it, the fact that they overetimated by o much i igificat. (If they were i the bottom quartile the bet that they could have cored wa at the 5 th percetile, yet their mea etimate wa at the 68 th percetile.) 7.11 Everitt data o weight gai: The Mea gai = 3.01, tadard deviatio = t =.. With 8 df the critical value =.048, o we will reject the ull hypothei ad coclude that the girl gaied at better tha chace level Std. Dev = 7.31 Mea = N = Weight Gai (i poud) 7.13 a. Performace whe ot readig paage µ µ t = = = = = b. Thi doe ot mea that the SAT i ot a valid meaure, but it doe how that people who do well at gueig at awer alo do well o the SAT. Thi i ot very urpriig Cofidece limit o µ for Exercie 7.14:

3 CI.95 = ± t = 4.39 ±.03 = 4.39 ± = µ 5.73 A iterval formed a thi oe wa ha a probability of.95 of ecompaig the mea of the populatio. Sice thi iterval iclude the hypotheized populatio mea of 3.87, it i coitet with the reult i Exercie We ued a paired-t tet i Exercie 7.16 becaue the data were paired i the ee of comig from the ame ubject. Some ubject howed geerally more betaedorphi at both time tha other, ad we wated to elimiate thi ubject-toubject variability that had othig to do with tre. I fact, there i t much of a relatiohip betwee the two meaure, but we ca t fairly igore it after the fact Paired t tet o marital atifactio: 1 D D t = = = D 1 D = = = We caot reject the ull hypothei that male ad female are equally atified. A paired-t i appropriate becaue it would ot eem reaoable to aume that the exual atifactio of a hubad i idepedet of that of hi wife. 7.1 Correlatio betwee hubad ad wive: cov Y r = = = = = Y ( 1.357)( 1.167) The correlatio betwee the core of hubad ad wive wa.334, which i igificat, ad which cofirm the aumptio that the core would be related. 7.3 The importat quetio i what would the amplig ditributio of the mea (or differece betwee mea) look like, ad with 91 pair of core that amplig ditributio would be ubtatially cotiuou with a ormal ditributio of mea.

4 7.5 Sulliva ad Bybee tudy = 5.03 = 1.01 = 135 it it it = 4.61 = 1.13 = 130 ctrl it it t it ctrl = = it ctrl it ctrl = = = = p( t > ab(.545)) =.011 The quality of life wa igificatly better for the itervetio group. 7.7 Paired t-tet o before ad after itervetio quality of life = 4.47 = 5.03 = 1.30 = 135 before after diff D t = = = = diff p <.000 Cofidece limit o weight gai i Cogitive Behavior Therapy group: CI.95 = D ± t.05(8) D ( )( ) = 3.0 ± = 3.0 ± µ 5.80 The probability i.95 that thi procedure ha reulted i limit that bracket the mea weight gai i the populatio. 7.9 Katz et al (1990) tudy a. Null hypothei there i ot a igificat differece i tet core betwee thoe who have read the paage ad thoe who have ot. b. Alterative hypothei there i a igificat differece betwee the two coditio.

5 c. t = where = ( 1) ( 1) 1 ( ) ( ) = = = t = = = = t = 8.89 o 43 df if we pool the variace. Thi differece i igificat. (d) We ca coclude that tudet do better o thi tet if they read the paage o which they are goig to awer quetio t = where = ( 1) ( 1) 1 ( ) ( ) = = = t = = = = = A t o two idepedet group = o 53 df, which i ot igificat. Cogitive behavior therapy did ot lead to igificatly greater weight gai tha the Cotrol coditio. (Variace were homogeeou.) 7.33 If thoe mea had actually come from idepedet ample, we could ot remove differece due to couple, ad the reultig t would have bee omewhat maller The differece betwee the two awer i ot greater tha it i becaue the correlatio betwee hubad ad wive wa actually quite low.

6 7.37 a. I would aume that the experimetal hypothei i the hypothei that mother of chizophreic childre provide TAT decriptio that how le poitive paret-child relatiohip. b. Normal Mea = 3.55 = = 0 Schizophreic Mea =.10 = = 0 t = = = = = [t.05 (38)=.0] Reject the ull hypothei Thi t i igificat o 38 df, ad I would coclude that the mea umber of picture portrayig poitive paret-child relatiohip i lower i the chizophreic group tha i the ormal group There i o way to tell caue ad effect relatiohip i Exercie It could be that people who experiece poor paret-child iteractio are at rik for chizophreia. But it could alo be that chizophreic childre dirupt the family ad poor relatiohip come a a reult % cofidece limit CI.05 = ( 1 ) ± t.05 ( µ µ ) = ( ) ± (.131) = ±

7 7.43 Repeatig Exercie 7.4 with time a the depedet variable: t = t = = = = The variace are very differet, but eve if we did ot adjut the degree of freedom, we would till fail to reject the ull hypothei If you take the abolute differece betwee the obervatio ad their group mea ad ru a t tet comparig the two group o the abolute differece, you obtai t = Squarig thi you have F = 0.391, which make it clear that Levee tet i SPSS i operatig o the abolute differece. (The t for quared differece would equal 0.13, which would give a F of ) 7.47 Differece betwee male ad female o axiety ad depreio: (We caot aume homogeeity of regreio here.) Equal variace ot aumed Idepedet Sample Tet DEPRESST ANT t-tet for Equality of Mea 95% Cofidece Iterval of the Mea Std. Error Differece t df Sig. (-tailed) Differece Differece Lower Upper Effect ize for data i Exercie 7.5 After Before 3.0 d = = = Before I choe to ue the tadard deviatio of the before therapy core becaue it provide a reaoable bae agait which to tadardize the mea differece. The cofidece iterval o the differece, which i aother way to examie the ize of a effect,

8 were give i the awer to Exercie a. The cale of meauremet i importat becaue if we recaled the categorie a 1,, 4, ad 6, for example, we would have quite differet awer. b. The firt exercie ak if there i a relatiohip betwee the atifactio of hubad ad wive. The ecod imply ak if male (hubad) are more atified, o average, tha female (wive). c. You could adapt the uggetio made i the text about combiig the t o idepedet group ad the t o matched group. d. I m really ot very comfortable with the t tet becaue I am ot pleaed with the cale of meauremet. A alterative would be a raked tet, but the umber of tie i huge, ad that probably worrie me eve more.