18. Two-sample problems for population means (σ unknown)

Transcription

1 8. Two-samle roblems for oulatio meas (σ ukow) The Practice of Statistics i the Life Scieces Third Editio 04 W.H. Freema ad Comay

2 Objectives (PSLS Chater 8) Comarig two meas (σ ukow) Two-samle situatios t-distributio for two ideedet samles Two-samle t test Two-samle t cofidece iterval Robustess

3 Two samles situatios We may eed to comare treatmets or coditios. It is imortat to determie if the samles are ideedet or ot. Ideedet samles: the idividuals i both samles are chose searately ( ideedetly )! Chater 8 Matched airs samles: the idividuals i both samles are related (for examle, the same subjects assessed twice, or sibligs)! Chater 7

4 t distributio for ideedet samles We have ideedet SRSs comig from oulatios with (µ,σ ) ad (µ,σ ) ukow. We use (,s ) ad (,s ) to estimate (µ,σ ) ad (µ,σ ) resectively. x x Both oulatios should be Normally distributed. I ractice, it is eough that both distributios have similar shaes ad that the samle data cotai o strog outliers.

5 The two-samle t statistic follows aroximately a t distributio with a stadard error SE (deomiator) reflectig variatio from both samles. The degrees of freedom of the t distributio are comuted as follows: t = ( x x ) ( µ µ ) s s + df = s + s + s s

6 Two-samle t test If you have ideedet radom samles ad wat to test H 0 : µ = µ <=> µ µ = 0 with either a oe-sided or a two-sided alterative hyothesis: Comute the t statistic ad aroriate df. t ( x x ) ( µ µ ) = s s + t = x x SE Obtai ad iterret the P-value (oe- or two-sided, deedig H a ).

7 Does smokig damage the lugs of childre exosed to aretal smokig? Forced Vital Caacity (FVC) is the volume (i milliliters) of air that a idividual ca exhale i 6 secods. FVC was obtaied for a samle of childre ot exosed to aretal smokig ad for a samle of childre exosed to aretal smokig. Paretal smokig Mea FVC s Yes No Is the mea FVC lower i the oulatio of childre exosed to aretal smokig? We test: H 0 : µ smoke = µ o <=> (µ smoke µ o ) = 0 H a : µ smoke < µ o <=> (µ smoke µ o ) < 0 (oe-sided)

8 Paretal smokig xbar s Yes No t = x s smoke smoke smoke x + o s o o = = df s s + ( ) = = s s (.9) + ( 7.6) I Table C, for df 40, we fid: t > 3.55 P < (oe-sided) Software gives P = , highly sigificat We reject H 0 Lug caacity is sigificatly imaired i childre exosed to aretal smokig, comared with childre ot exosed to aretal smokig.

9 Geckos are lizards with secialized toe ads that eable them to easily climb eve slick surfaces. Researchers wat to kow if male ad female geckos differ sigificatly i the size of their toe ads. I a radom samle of Tokay geckos, they fid that the mea toe ad area is 6.0 cm for the males ad 5.3 cm for the females. What is the aroriate ull hyothesis here? A.. H B.. H C.. H D.. H E.. H : : : : : x x µ µ µ male male male x x µ female female female differecem F differecem F = 0.7 = 0 = 0 = 0 = 0.7 Should the alterative hyothesis be oe-side or two-sided?

10 Two samle t-cofidece iterval Because we have ideedet samles we use the differece betwee ( x ) both samle averages to estimate (µ µ ). x C is the area betwee t* ad t* Fid t* i the lie of Table C for the comuted degrees of freedom The margi of error m is: C s s m = t * + = t * SE t* m m t*

11 How does esticide hel seedlig growth? Seeds are radomly assiged to be lated i ots with soil treated with esticide or i ots with utreated soil. Seedlig growth (i mm) is recorded after weeks. A 95% cofidece iterval for (µ - µ ) is: s ( x x) ± t * + Usig df = 30 from Table C, we get: s ( ) ± , or 9.96 ± 8.80 mm Table C

12 t-test: Two-Samle Assumig Uequal Variaces Excel for df = 38, m =.04* Treatmet grou Cotrol grou Mea Variace Observatios 3 Hyothesized Mea Differece - df 38 t Stat.3 P(T<=t) oe-tail 0.03 t Critical oe-tail.686 P(T<=t) two-tail 0.06 t Critical two-tail.04 t* We are 95% cofidet that usig esticide yields seeds that are. to 8.7 mm loger o average after weeks. Ideedet Samles Test SPSS ReadigScore Growth Equal variaces assumed Equal variaces ot assumed Levee's Test for Equality of Variaces F Sig. t df Sig. (-tailed) t-test for Equality of Meas Mea Differece 95% Cofidece Iterval of the Std. Error Differece Differece Lower Uer

13 Robustess The two-samle statistic is most robust whe both samle sizes are equal ad both samle distributios are similar. But eve whe we deviate from this, two-samle tests ted to remai quite robust. As a guidelie, a combied samle size ( + ) of 40 or more will allow you to work eve with the most skewed distributios.

14 Avoid the ooled two-samle t rocedures There are two versios of the two-samle t rocedures: oe assumig equal variace ( ooled ) ad oe ot assumig equal variace for the two oulatios. They have slightly differet formulas ad df. The ooled (equal variace) twosamle t test has degrees of freedom +. Two Normally distributed oulatios with uequal variaces However, the assumtio of equal variace is hard to check, ad thus the uequal variace test is safer.

15 REVIEW: t rocedures Oe-samle t rocedure Oe samle summarized by its mea x ad stadard deviatio s. Poulatio arameters µ ad σ ukow. Iferece about µ Matched airs t rocedure Two aired datasets (from a matched-airs desig). From the airwise differeces we comute ( x diff, sdiff ). Poulatio arameters µ diff ad σ diff ukow. Iferece about µ diff Two-samle t rocedure Two ideedet samles (urelated idividuals i the two samles). We summarize each samle searately with Poulatio arameters µ, σ, µ, σ ukow. ( x, s ; x, s ). Iferece about µ µ

16 Which tye of t iferece rocedure? A: oe samle, B: matched airs, C: two samles? Is blood ressure altered by use of a oral cotracetive? Comarig a samle of wome ot usig a OC with a samle of wome takig it. Does bread lose vitami with storage? Take a samle of bread loaves ad comare vitami cotet right after bakig ad agai after 3 days later. Average cholesterol level i geeral adult oulatio is 75 mg/dl. Take a samle of adults with high cholesterol arets. Is the mea cholesterol level higher i this oulatio? Does bread lose vitami with storage? Take a samle of bread loaves just baked ad a samle of bread loaves stored for 3 days ad comare vitami cotet.

17 Table C