1Lecture20PairedttestComparingtheMeansofTwoNormalDistributionsReviewSupposethatX1,…,Xnformarandomsamplefromanormaldistributionforwhichthemeanmisunknown,butthevarianceisknown.Considertestingthehypotheses:

ororLet.Youcansimilarlyderivethe criticalregionsas:ReviewSupposethatX1,…,Xnformarandomsamplefromanormaldistributionforwhichthemeanmisunknown,butthevarianceisunknown.Considertestingthehypotheses:

ororLet.Youcansimilarlyderivethe criticalregionsas:Review:p-值：衡量拒絕原假設(shè)的強(qiáng)度通過計(jì)算一個(gè)概率，即p-值，來判斷我們觀測(cè)到的樣本在H0成立的條件下，是否是偏向H1的。

如果p-值很小，則認(rèn)為我們的樣本不太可能在原假設(shè)的情況下產(chǎn)生的，應(yīng)該拒絕原假設(shè)。

教學(xué)改革以前，MBA教學(xué)評(píng)估的長(zhǎng)期平均水平為4.2。當(dāng)前收集的教學(xué)評(píng)估涉及118門課程，其中有52門MBA課程，它們教學(xué)評(píng)估的平均水平為4.34，標(biāo)準(zhǔn)差為0.54。問題：MBA教學(xué)質(zhì)量比以往提高了嗎？H0：MBA教學(xué)平均分等于4.2。Ha：MBA教學(xué)平均分大于4.2（MBA教學(xué)質(zhì)量比以往提高了）。求p-值。Example用p-值得好處p-值是根據(jù)我們觀察到的數(shù)據(jù)，我們將拒絕H0的最小水平。如果p-值比顯著性水平小，我們拒絕原假設(shè)。統(tǒng)計(jì)軟件的標(biāo)準(zhǔn)輸出。不用預(yù)先指定顯著性水平。研究者A采用顯著性水平0.01，而研究者B采用顯著性水平0.05。Matched-sampledesign:e.g.Wewanttocompare2productionmethods.Before/after: e.g.Inaproductionexperiment,eachworkerfirstusesoneproductionmethodandthenusesproductionmethod2. Variationbetweenworkersiseliminatedasasourceofsamplingerror.Paired: e.g.Inaproductionexperiment,eachalloyisdividedinto2parts.Productionmethod1isusedononepart,andproductionmethod2isusedonanotherpart.

Variationbetweenalloysiseliminatedasasourceofsamplingerror.ThePairedtTestConsiderAssuming

then

Assumeyouworkinthefinancedepartment.Isthenewfinancialpackagefaster(0.05level)?Youcollectthefollowingdataentrytimes:PairedtTest:Example

User

CurrentLeader(1)

NewSoftware(2)

Difference

DiC.B. 9.98Seconds 9.88Seconds .10T.F. 9.88 9.86 .02M.H. 9.84 9.75 .09 R.K. 9.99 9.80 .19M.O. 9.94 9.87 .07D.S. 9.84 9.84 .00S.S. 9.86 9.87 -.01C.T. 10.12 9.86 .26K.T. 9.90 9.83 .07S.Z. 9.91 9.86 .05Isthenewfinancialpackagefaster(0.05level)?Alevela0

=.05ttestrejectH0ifTheteststatisticiscalculatedasReject

a0=.051.8331Decision:RejectH0Conclusion:

Thenewsoftwarepackageisfaster.ComparingtheMeansof

TwoNormalDistributionsSupposethatX1,…,Xmformarandomsampleofmobservationsfromanormaldistributionforwhichboththemeanm1andthevarianceareunknown,supposealsothatY1,…,Ynformarandomsampleofnobservationsfromanormaldistributionforwhichboththemeanm2andthevarianceareunknown.Supposethefollowinghypothesesaretobetestedataspecifiedlevelofsignificancea0

:DefineTheteststatisticwewilluseisWewillconsiderthetestthatrejectsH0whenWhenm1=m2,wehaveBecauseareindependent,ZandW

areindependent.Sowhenm1=m2,thestatistic

willhaveatm+n-2distribution.Notethat SothesizeofthetestisGivenasignificancelevela0,wecanchooseaconstantcsuchthat

Arandomsampleof8specimensoforeiscollectedfromacertainlocationinacoppermine,andthattheamountofcopperineachspecimenismeasuresingrams.Wedenotethese8amountsbyX1,…,X8.Supposethatitisfoundthatand .Anotherrandomsampleof10specimensororeiscollectedfromanotherpartofthemine,andtheamountsofcopperinthesespecimensaredenotedbyY1,…,Y10.Supposethatitisfoundthatand

ExampleLetm1denotethemeanamountofcopperinalltheoreatthefirstlocationinthemine,andletm2denotethemeanamountofcopperinalltheoreatthesecondlocation.WeshallcarryoutanttestofthehypothesesExample Solution:Assumethatalltheobservationshaveanormaldistributionandthatthevarianceisthesameatbothlocationsinthemine.Then Itcanbefoundthatthep-valueis Sothenullhypothesiswillberejectedforanyspecifiedlevelofsignificancea0>0.00167.TheOtherOne-SidedHypotheses

andTwo-SidedHypothesesIfwewanttotestthehypotheses Thelevela0

ttestrejectsH0whenIfwewanttotestthehypotheses Thelevela0

ttestrejectsH0when案例：老字號(hào)

從20世紀(jì)80年代中期以后開始，至今沒有停止，老字號(hào)在西方先進(jìn)經(jīng)營(yíng)理念和多種業(yè)態(tài)的沖擊下，相繼崩潰。有資料顯示，國(guó)內(nèi)貿(mào)易部曾于1990年認(rèn)定的1600多家中華老字號(hào)企業(yè)，現(xiàn)在勉強(qiáng)維持的占70%；長(zhǎng)期虧損、面臨倒閉或已經(jīng)倒閉的占20%；生產(chǎn)經(jīng)營(yíng)達(dá)到一定規(guī)模、效益良好的僅有10%。很多老字號(hào)企業(yè)由于沒有及時(shí)調(diào)整內(nèi)部機(jī)制、進(jìn)而逐步喪失了競(jìng)爭(zhēng)優(yōu)勢(shì)，市場(chǎng)份額日趨縮減，導(dǎo)致生存困難。

記得北京城的那首民謠：頭戴馬聚源，腳踩內(nèi)聯(lián)升，身穿瑞蚨祥，腰纏四大恒。到如今，這已成為歷史寫照。老字號(hào)的消失，實(shí)際上是一種歷史文化的消失，它就像文物古跡一樣需要后人的精心關(guān)心和愛護(hù).揚(yáng)州的謝馥春鴨蛋香粉

謝馥春背景創(chuàng)建于1830，已有近200年歷史是中國(guó)第一家化妝品企業(yè)1915年與“茅臺(tái)酒”同獲巴拿馬國(guó)際博覽會(huì)大獎(jiǎng)

如何最大限度地發(fā)揮懷舊營(yíng)銷的溢出？廣告文本:A廣告文本:B廣告文本:C廣告文本:D非常不可能比較不可能一般比較可能非常可能您購(gòu)買的可能性12345抽樣結(jié)果TreatmentGroupSampleSizeSampleMeanSampleSDControl903.200.90A913.881.19B943.221.00C923.271.18D954.210.97討論各個(gè)廣告的側(cè)重點(diǎn)是什么?哪個(gè)最好?你確信嗎?總體和樣本?你的決策？Controlvs.D.樣本容量:樣本均值:樣本均值差異:實(shí)際工作中常常需要在兩個(gè)對(duì)立的決策中選擇一個(gè)！?。臉颖局形覀冇^察到的樣本均值差異:對(duì)于總體，我們需要決策：樣本均值差異是什么原因產(chǎn)生的？Bonus

Problem：Bamberger’s百貨公司Bamberger‘s百貨公司成立于1947年，是一家為社區(qū)提供大眾性商品的零售商店，長(zhǎng)期以來，Bamberger’s的口碑一直很好。它的顧客主要是中高收入的工薪家庭。Bamberger's的顧客服務(wù)有著極好的聲譽(yù)，針對(duì)每一位顧客的需要