Introduction to mathematical Statistics

Transcription

1 More ttistics tutoril t wwwdumblittledoctorcom Itroductio to mthemticl ttistics Chpter 7 ypothesis Testig Exmple (ull hypothesis) : the verge height is 5 8 or more ( : μ 5'8" ) (ltertive hypothesis) : the verge height is less th 5 8 ( : μ p 5'8" ) Exmple (lw suit) OJ impso cse : OJ is iocet (origil belief) : OJ is guilty Decisio Correct Decisio Type I error Truth Type II error Correct Decisio P(Type I error) = P(Reject ) = This probbility is clled the sigificce level of the test cerio : Iferece o oe popultio me μ (orml popultio, is kow) There re two methods Pivotl Qutity Method (We will ler this method first) Likelihood Rtio Test (We will ler this method lter)

2 More ttistics tutoril t wwwdumblittledoctorcom Now we itroduce the geerl procedures i the Pivotl Qutity method Before derivig the test, you must kow your Pivotl Qutity for the give problem: Z = ~ N(,) Now you c strt your hypothesis test First write dow your hypotheses There re 3 cses (or choices) You eed to choose the oe tht is most suitble for your give problem or e : μ μ (oe-sided test) quivletly, : μ f μ : μ f μ : μ μ b (oe-sided test) or equivletly, : μ p μ : μ p μ : μ = μ c (two-sided test) : μ μ I the followig we itroduce the remider of the test derivtio procedures for ech pir of hypotheses Cse A uppose tht we re testig : μ f μ 3 Now you write dow your Test ttistic (which is your pivotl qutity with the vlue of the prmeter uder plugged i ere we plug i μ = μ ): Test sttistic: : μ= μ Z = ~ N(,) 4 Now, we derive the decisio rule bsed o ( is the sigificce level of the test) = P(Reject ) = P( Z c )

3 More ttistics tutoril t wwwdumblittledoctorcom Thus we will reject i fvor of if Z Z For exmple, = 5 Reject if Z Z5 = 645 Cse B, ow suppose we re testig, : μ p μ Test ttistic : : μ= μ Z = ~ N(,) = P(Reject ) = P( Z c ) 3

4 More ttistics tutoril t wwwdumblittledoctorcom We will reject i fvor of if Z Z Cse C, ow suppose we re testig, : μ μ Test ttistic : : μ= μ Z = ~ N(,) = P(Reject ) = P( Z c ) = PZ ( c ) + PZ ( c = PZ ( c ) ) = PZ ( c ) or c= Z At the sigificce level, we reject i fvor of if (For exmple, = 5, Z 96 ) Z Z 4

5 More ttistics tutoril t wwwdumblittledoctorcom cerio : Iferece o oe popultio me μ (lrge smple, y popultio typiclly ot orml, is kow or ukow) mple is usully clled lrge smple if 3 Pivotl Qutity : X μ X μ Z = ~ N(,) or Z = ~ N(,) s The rest is the sme s cerio cerio 3 : Iferece o oe popultio me μ (orml popultio, is ukow) iid uppose we hve smple X, X, L, X N( μ, ), is ukow First we review how to derive the pivotl qutity T = ~ t Poit Estimtor for μ : ~ X ~ N( μ, ) X is NOT pivotl qutity sice Z = ~ N(,) is ukow This is lso NOT pivotl qutity sice is ukow 3 Theorem mple from orml popultio Z ~ N(,) ( ) W = χ ~ Z Defiitio T = ~ t T = ~ t W ( ) ( Z d W re idepedet) 5

6 More ttistics tutoril t wwwdumblittledoctorcom T is pivotl qutity for μ 4 Now we re redy to derive the test for ech pir of hypotheses A : μ f μ Test ttistic : T = ~ t = (Reject ) = ( ) P P T c ( is usully 5) c = t, At the sigificce level, we reject i fvor of if T t, B : μ p μ Test ttistic : T = ~ t = (Reject ) = ( ) P P T c Reject i fvor of if T t, 6

7 More ttistics tutoril t wwwdumblittledoctorcom c = t, C : μ μ Test ttistic : T = ~ t = P(Reject ) = PT ( c ) = PT ( c ) = PT ( c ) At the sigificce level, we reject i fvor of if T t, 7

8 More ttistics tutoril t wwwdumblittledoctorcom c = t, Exmple Prospective slespeople for ecyclopedi compy re ow offered sles triig progrm Previous dt idicte tht the verge umber of sles per moth for those who do ot prticipte is 33 To determie whether the triig progrm is effective, rdom smple of 35 ew employees is give the sles triig d the set Oe moth lter, the me d stdrd devitio for the umber of ecyclopedi sold re 35 d 84, respectively Do the dt preset sufficiet evidece to idicte tht the triig progrm ehces sles? Use = 5 olutio This is cerio : lrge smple ( 3 ) Pivotl Qutity: Z = ~ & N(,) : 3 μ = 3 : μ f 33 Test ttistic: Z = = = At = 5, we would reject i fvor of if Z Z5 = 645 We do ot reject 8

9 More ttistics tutoril t wwwdumblittledoctorcom If you re told the popultio is orml, the you should use the t-test i cerio 3 becuse the t-test is exct while the z-test is pproximte i this situtio The exct test is lwys preferred over the pproximte test Exmple uppose tht the popultio is orml i the bove exmple olutio : 3 μ = 3 : μ f 33 Test ttistic : T = = = At = 5, we would reject i fvor of if T t, = t34,5 = 69 exct test Becuse T = 4 is ot greter th 69, we cot reject *** Plese red the etire Chpter 7 *** 9