Inference on One Population Mean Hypothesis Testing

Transcription

1 Iferece o Oe Popultio Me ypothesis Testig Scerio 1. Whe the popultio is orml, d the popultio vrice is kow i. i. d. Dt : X 1, X,, X ~ N(, ypothesis test, for istce: Exmple: : : : : : 5'7" (ull hypothesis: This is the origil belief : 5'7" (ltertive hypothesis : This is usully your hypothesis (i.e. wht you believe is true if you re coductig the test d i geerl, should be supported by your dt. The sttisticl hypothesis test is very similr to lw suit: 1

2 e.g The fmous O.J. Simpso tril : OJ is iocet ( iocet uless prove guilty : OJ is guilty ( supported by the dt: the evidece Jury s Decisio : OJ iocet The truth OJ guilty Right decisio Type II error Type I error Right decisio The sigificce level d three types of hypotheses. P(Type I error = α sigificce level of test (*Type I error rte 1. : : : :. : : : : 3. : : Now we derive the hypothesis test for the first pir of hypotheses. : Dt : : iid,, X N(, is kow d the give X 1, X ~, sigificce level α (sy,.5. Let s derive the test. (Tht is, derive the decisio rule

3 Two pproches (*equivlet to derive the tests: - Likelihood Rtio Test - Pivotl Qutity Method ***Now we will first demostrte the Pivotl Qutity Method. 1. We hve lredy derived the PQ whe we derived the C.I. for μ X ~ N(,1 is our P.Q.. The test sttistic is the PQ with the vlue of the prmeter of iterest uder the ull hypothesis ( iserted: X ~ (,1 is our test sttistic. N X Tht is, give : i true ~ N(,1 3. * Derive the decisio threshold for your test bsed o the Type I error rte the sigificce level For the pir of hypotheses: : versus : It is ituitive tht oe should reject the ull hypothesis, i support of the ltertive hypothesis, whe the smple me is lrger th. Equivletly, this mes whe the test sttistic is lrger th certi positive vlue c - the questio is wht is the exct vlue of c -- d tht c be determied bsed o the sigificce level α tht is, how much Type I error we would llow ourselves to commit. 3

4 Settig: P P(Type I error = P(reject = ( c : We will see immeditely tht below. c z from the pdf plot At the sigificce level α, we will reject i fvor of if 4

5 5 Other ypotheses : : (oe-sided test or oe-tiled test Test sttistic : ~ (,1 / X N c c P : ( : : (Two-sided or Two-tiled test Test sttistic : ~ (,1 / X N ( ( ( c P c P c P ( c P ( c P c Reject if

6 4. We hve just discussed the rejectio regio pproch for decisio mkig. There is other pproch for decisio mkig, it is pvlue pproch. *Defiitio: p-vlue it is the probbility tht we observe test sttistic vlue tht is s extreme, or more extreme, th the oe we observed, give tht the ull hypothesis is true. : : : : : : X Observed vlue of test sttistic ~ N(,1 p-vlue P( z (1 the re uder N(,1 pdf to the right of z p-vlue P( z ( the re uder N(,1 pdf to the left of z p-vlue P( z P( z (3 twice the re to the right of z 6

7 (1 : : ( : : 7

8 (3 : : The wy we mke coclusios is the sme for ll hypotheses: We reject i fvor of iff p-vlue < α The experimetl evidece gist the ull hypothesis tht Lucy did ot ct delibertely reches P-vlue of oe i te billio. Nevertheless, Chrlie Brow repets the experimet every yer. 8

9 Scerio. The lrge smple scerio: Ay popultio (*usully o-orml s the exct tests should be used if the popultio is orml, however, the smple size is lrge (this usully refers to: 3 Theorem. Let The Cetrl Limit Theorem X, X,, X be rdom smple from popultio 1 with me d vrice X * Whe is lrge eough ( 3, N(,1 X S ~ N(,1 (pproximtely by CLT d the Slutsky s Theorem Therefore the pivotl qutities (P.Q. s for this scerio: X X ~ N(,1 or ~ N(,1 S Use the first P.Q. if σ is kow, d the secod whe σ is ukow. 9

10 The derivtio of the hypothesis tests (rejectio regio d the p-vlue re lmost the sme s the derivtio of the exct -test discussed bove. : : : : : : X Test Sttistic ~ N(,1 S Rejectio regio : we reject i fvor of sigificce level if t the p-vlue P( z (1 the re uder N(,1 pdf to the right of z p-vlue P( z ( the re uder N(,1 pdf to the left of z p-vlue P( z P( z (3 twice the re to the right of z Scerio 3. Norml Popultio, but the popultio vrice is ukow 1 yers go people use -test This is OK for lrge ( 3 per the CLT (Scerio This is NOT ok if the smple size is smll. A Studet of Sttistics pe me of Willim Sely Gosset (Jue 13, 1876 October 16, 1937 The Studet s t-test X P.Q. T ~ t 1 S/ (Exct t-distributio with -1 degrees of freedom 1

11 A Studet of Sttistics pe me of Willim Sely Gosset (Jue 13, 1876 October 16,

12 Review: Theorem Smplig from the orml popultio i. i. d. Let X 1, X,, X ~ N(,, the 1 X ~ N(, W ( 1 S ~ 1 3 X d S (d thus W re idepedet. Thus we hve: T X S ~ t 1 Wrog Test for -sided ltertive hypothesis Reject if Right Test for -sided ltertive hypothesis Reject if, (Becuse t distributio hs hevier tils th orml distributio. Right Test * Test Sttistic : : T X ~ t 1 S * Reject regio : Reject t if the observed test sttistic vlue, 1

13 * p-vlue p-vlue = shded re * Further Review: 5. Defiitio : t-distributio T ~ W k ~ N(,1 t k W ~ k (chi-squre distributio with k degrees of freedom & W re idepedet. 6. Def 1 : chi-squre distributio : from the defiitio of the gmm distributio: gmm(α = k/, β = MGF: ( ( me & vrice: ( ( Def : chi-squre distributio : Let i,,, ~. i. d. (,1, 1 k N the W k i1 i ~ k 13

14 Exmple Jerry is plig to purchse sports good store. e clculted tht i order to cover bsic expeses, the verge dily sles must be t lest $55. Scerio A. e checked the dily sles of 36 rdomly selected busiess dys, d foud the verge dily sles to be $565 with stdrd devitio of $15. Scerio B. Now suppose he is oly llowed to smple 9 dys. Ad the 9 dys sles re $51, 537, 548, 59, 53, 49, 61, 499, 64. For A d B, plese determie whether Jerry c coclude the dily sles to be t lest $55 t the sigificce level of. 5. Wht is the p-vlue for ech scerio? Solutio Scerio A lrge smple (5 =36, x 565, s 15 : 55 versus : 55 *** First perform the Shpiro-Wilk test to check for ormlity. If orml, use the exct T-test. If ot orml, use the lrge smple - test. I the followig, we ssume the popultio is foud ot orml. x Test sttistic z 1. 6 s

15 At the sigificce level. 5, we will reject if z We c ot reject p-vlue p-vlue =.548 Altertively, if you c show the popultio is orml usig the Shpiro-Wilk test, it is better tht you perform the exct t-test. Solutio Scerio B smll smple Shpiro-Wilk test If the popultio is orml, t-test is suitble. (*If the popultio is ot orml, d the smple size is smll, we shll use the o-prmetric test such s Wilcoxo Siged Rk test. I the followig, we ssume the popultio is foud orml. x , s 53.9, 9 : 55 versus : 55 x Test sttistic t 1. s

16 At the sigificce level. 5, we will reject if t T8, We c ot reject p-vlue Wht s the p-vlue whe t 1.? Topics i ext lecture Power of the test Likelihood rtio test (for oe popultio me 16