***SECTION 12.1*** Tests about a Population Mean

Transcription

1 ***SECTION 12.1*** Tests but Ppultin Men CHAPTER 12 ~ Significnce Tests in Prctice We begin by drpping the ssumptin tht we knw the ppultin stndrd devitin when testing clims but ppultin men. As with cnfidence intervls, this leds t the use f distributins. One-Smple t Sttistic nd the t Distributins Drw n SRS f size n frm ppultin tht hs the Nrml distributin with men nd stndrd devitin. The ne-smple t sttistic x t s n hs the t distributin with degrees f freedm. In Chpter 10 we used Tble C t find criticl vlues t cnstruct cnfidence intervl. Nw we will use Tble C t clculte P-vlues fr significnce test but. Exmple 1: Determining P-vlues Suppse yu crry ut significnce test f H : 5 versus : 5 0 H bsed n smple f size n = 20 nd btin t = Wht is the P-vlue? Exmple 2: Determining P-vlues Nw suppse yu crry ut significnce test f H : 5 versus : 5 0 H bsed n smple f size n = 37 nd btin t = Wht is the P-vlue? Ch. 12 ~ Pg. 1

2 The One-Smple t Test Drw n SRS f size n frm ppultin hving unknwn men. T test the hypthesis H : 0 bsed n n SRS f size n, cmpute the ne-smple t sttistic x t s n In terms f rndm vrible T hving the distributin, the P-vlue fr test f H 0 ginst H : is PT t H : is PT t H : is 2 PT t These P-vlues re if the ppultin distributin is nd crrect fr n in ther cses. Exmple 3: Sweet Cl Diet cls use rtificil sweeteners t vid sugr. These sweeteners grdully lse their sweetness ver time. Mnufcturers therefre test new cls fr lss f sweetness befre mrketing them. Trined tsters sip the cl lng with drinks f stndrd sweetness nd scre the cl n sweetness scle f 1 t 10. The cl is then stred fr mnth t high temperture t imitte the effect f fur mnths strge t rm temperture. Ech tster scres the cl gin fter strge. Our dt re the differences (scre befre strge minus scre fter strge) in the tsters scres. The bigger these differences, the bigger the lss f sweetness. Here re the sweetness lsses fr new cl, s mesured by 10 trined tsters: Mst re psitive. Tht is, mst tsters fund lss f sweetness. But the lsses re smll, nd tw tsters (the negtive scres) thught the cl gined sweetness. Are these dt gd evidence tht the cl lst sweetness in strge? STEP 1: Prmeter Identify the ppultin f interest nd the prmeter yu wnt t drw cnclusins but. Ch. 12 ~ Pg. 2

3 STEP 2: Hyptheses Stte the hyptheses nd significnce level STEP 3: Cnditins Chse the pprprite inference prcedure. Verify the cnditins fr using it. STEP 4: Clcultin Cmpute the test sttistic. STEP 5: P-vlue Drw the curve nd cmpute the P-vlue. STEP 6: Interprettin Interpret the prblem using stt tlk nd cntext. * NOTE: Check Pg. 749 in yur bk! This shws the vrius cmputer utputs tht might be given t yu n tests. Mke sure yu check it ut!! Yu wuld hte t wste time perfrming unnecessry clcultins. Ch. 12 ~ Pg. 3

4 Exmple 4: Diversify r be sued! An investr with stck prtfli with severl hundred thusnd dllrs sued his brker becuse lck f diversifictin in his prtfli led t pr perfrmnce (lw return). The tble belw gives the rtes f return fr the 39 mnths tht the ccunt ws mnged by the brker. An rbitrtin pnel cmpred these returns with the verge f the Stndrd & Pr s 500 stck index fr the sme perid. Cnsider the 39 mnthly returns s rndm smple frm the mnthly returns the brker wuld generte if he mnged the ccunt frever. Are these returns cmptible with ppultin men f 0.95%, the S&P 500 verge? STEP 1: Prmeter Identify the ppultin f interest nd the prmeter yu wnt t drw cnclusins but. STEP 2: Hyptheses Stte the hyptheses nd significnce level STEP 3: Cnditins Chse the pprprite inference prcedure. Verify the cnditins fr using it. Ch. 12 ~ Pg. 4

5 STEP 4: Clcultin Cmpute the test sttistic. STEP 5: P-vlue Drw the curve nd cmpute the P-vlue. STEP 6: Interprettin Interpret the prblem using stt tlk nd cntext. * NOTE: Check Pg. 752 in yur bk! This shws the vrius cmputer utputs tht might be given t yu n tests. Mke sure yu check it ut!! Yu wuld hte t wste time perfrming unnecessry clcultins. Pired t Tests In Exmple 3 the 10 ten tsters rted befre nd fter sweetness. Since the dt were pired by tster, we prefrmed ne-smple t test n the differences. Tht is, we used. Ch. 12 ~ Pg. 5

6 Exmple 5: Flrl Scents nd Lerning We her tht listening t Mzrt imprves students perfrmnce n tests. Perhps plesnt drs hve similr effect. T test this ide, 21 subjects wrked pper-nd-pencil mze while wering msk. The msk ws either unscented r crried flrl scent. The respnse vrible is their verge time n three trils. Ech subject wrked the mze with bth msks, in rndm rder. The rndmiztin is imprtnt becuse subjects tend t imprve their times s they wrk mze repetedly. The tble belw gives the subjects verge times with bth msks. Subject Unscented (secnds) Scented (secnds) Difference Unscented - Scented STEP 1: Prmeter Identify the ppultin f interest nd the prmeter yu wnt t drw cnclusins but. STEP 2: Hyptheses Stte the hyptheses nd significnce level Ch. 12 ~ Pg. 6

7 STEP 3: Cnditins Chse the pprprite inference prcedure. Verify the cnditins fr using it. STEP 4: Clcultin Cmpute the test sttistic. STEP 5: P-vlue Drw the curve nd cmpute the P-vlue. STEP 6: Interprettin Interpret the prblem using stt tlk nd cntext. * NOTE: Check Pg in yur bk! This shws the vrius cmputer utputs tht might be given t yu n tests. Mke sure yu check it ut!! Yu wuld hte t wste time perfrming unnecessry clcultins. Ch. 12 ~ Pg. 7

8 ***SECTION 12.2*** Tests but Ppultin Prprtin CHAPTER 12 ~ Significnce Tests in Prctice When three imprtnt cnditins re met,, nd the smpling distributin f ˆp is pprximtely Nrml with men nd stndrd devitin. Recll tht fr cnfidence intervls, we substituted ˆp fr p in the stndrd devitin t btin the stndrd errr. Hwever, fr significnce tests this is nt the cse. Remember tht the prbbility clcultins we d fr the inference prcess re bsed n the ssumptin tht is. The null hypthesis specifies vlue fr p, which we will cll p ( ). If we stndrdize ˆp by subtrcting its men p nd dividing by its stndrd devitin, we btin the fllwing z sttistic: estimte hypthesized prmeter vlue pˆ p z stndrd devitin f estimte p 1 p The One-Prprtin z Test Chse n SRS f size n frm lrge ppultin with unknwn prprtin p f successes. T test the hypthesis H : 0 p p, cmpute the z sttistic pˆ p z p 1 p In terms f rndm vrible Z hving the stndrd nrml distributin, the pprximte P-vlue fr test f H 0 ginst H : p p is PZ z n n H : p p is PZ z H : p p is 2 PZ z Nrmlity cnditin: Use this test when the expected number f successes filures n p 1 re bth. np nd Ch. 12 ~ Pg. 8

9 Exmple 6: Wrk Stress Accrding t the Ntinl Institute fr Occuptinl Sfety nd Helth, jb stress pses mjr thret t the helth f wrkers. A ntinl survey f resturnt emplyees fund tht 75% sid tht wrk stress hd negtive impct n their persnl lives. A rndm smple f 100 emplyees frm lrge resturnt chin finds tht 68 nswer Yes when sked, Des wrk stress hve negtive impct n yur persnl life? Is this gd resn t think tht the prprtin f ll emplyees in the chin wuld sy Yes differs frm the ntinl prprtin p 0.75? STEP 1: Prmeter Identify the ppultin f interest nd the prmeter yu wnt t drw cnclusins but. STEP 2: Hyptheses Stte the hyptheses nd significnce level STEP 3: Cnditins Chse the pprprite inference prcedure. Verify the cnditins fr using it. STEP 4: Clcultin Cmpute the test sttistic. STEP 5: P-vlue Drw the curve nd cmpute the P-vlue. Ch. 12 ~ Pg. 9

10 STEP 6: Interprettin Interpret the prblem using stt tlk nd cntext. * NOTE: Check Pg. 768 in yur bk! This shws the vrius cmputer utputs tht might be given t yu n tests. Mke sure yu check it ut!! Yu wuld hte t wste time perfrming unnecessry clcultins. In the previus exmple, we chse Yes nswer t the survey questin t be success nd N nswer s filure. If we hd reversed this nd tken N t be success we wuld hve btined the. The results f ur significnce test hve limited use in this exmple, s in mny cses f inference but single prmeter. Of curse, we d nt expect the experience f the resturnt wrkers t be exctly the sme s tht f the wrkers in the ntinl survey. If the smple f resturnt wrkers is sufficiently, we will hve sufficient t detect very difference. On the ther hnd, if ur smple size is very, we my be t detect differences tht culd be very. Fr these resns, we prefer t include s prt f ur nlysis. Cnfidence Intervls Prvide Additinl Infrmtin T see wht ther vlues f p re cmptible with the smple results, we will clculte cnfidence intervl. Ch. 12 ~ Pg. 10

11 Exmple 7: Estimting Wrk Stress The resturnt wrker survey in Exmple 6 fund tht 68 f rndm smple f 100 emplyees greed tht wrk stress hd negtive impct n their persnl lives. Find nd interpret 95% cnfidence intervl fr the true prprtin. STEP 1: Prmeter Identify the ppultin f interest nd the prmeter yu wnt t drw cnclusins but. STEP 2: Cnditins Chse the pprprite inference prcedure. Verify the cnditins fr using it. STEP 3: Clcultins If the cnditins re met, crry ut the inference prcedure. STEP 4: Interprettin Interpret yur results in the cntext f the prblem. The cnfidence intervl frm Exmple 7 is much mre infrmtive tht the significnce test f Exmple 6. We hve determined the tht re with the results. * Nte tht the stndrd errr used fr the cnfidence intervl is estimted frm the, wheres the denmintr fr the test sttistic z is bsed n the vlue in the. Ch. 12 ~ Pg. 11