: : 8.2. Test About a Population Mean. STT 351 Hypotheses Testing Case I: A Normal Population with Known. - null hypothesis states 0

Transcription

1 8.2. Test About Popultio Me. Cse I: A Norml Popultio with Kow. H - ull hypothesis sttes. X1, X 2,..., X - rdom smple of size from the orml popultio. The the smple me X N, / X X Whe H is true. X Descriptio of Test Procedure. Null hypothesis: H : x Test sttistic vlue: z / Altertive Hypothesis H H H : : :. Rejectio Regio for level Test z z z (upper-tiled test) (lower-tiled test) either z z/2 or z z/2 z (two tiled test) 1

2 Sequece of steps recommeded whe testig hypotheses bout prmeter. 1. Idetify the prmeter of iterest d describe it i the cotext of the problem situtio. 2. Determie the ull vlue d stte the ull hypothesis. 3. Stte the pproprite ltertive hypothesis. 4. Give the formul for the computed vlue of the test sttistic (substitutig the ull vlue d the kow vlues of y other prmeters, but ot those of y smple bsed qutities). 5. Stte the rejectio regio for the selected sigificce level. 6. Compute y ecessry smple qutities, substitute ito the formul for the test sttistic vlue, d compute tht vlue. 7. Decide whether H should be rejected, d stte this coclusio i the problem cotext. The formultio of hypotheses (Steps 2 d 3) should be doe before exmiig the dt. Exmple (Exmple 8.6 pp textbook.) A mufcturer of sprikler systems used for fire protectio i office buildigs clims tht the true verge system-ctivtio temperture is 13. A smple of 9 systems, whe tested, yields smple verge ctivtio temperture of F. If the distributio of ctivtio times is orml with stdrd devitio 1.5 F, does the dt cotrdict the mufcturer s clim t sigificce level.1? Solutio. 1. Prmeter of iterest: verge ctivtio temperture. 2. Null hypothesis: H : 13 (ull vlue 13) 3. Altertive hypothesis: H : 13 ( deprture from the climed vlue i either directio is of cocer). 4. Test sttistic vlue: x x 13 z / Rejectio regio: The form of H implies use of two-tiled test with rejectio regio either z z.5 or z z.5. From Sectio Appedix Tble A.3, z , so we reject H if either z 2.58or z

3 6. Substitutig 9 d x 131.8, z / 9.5 Tht is, the observed smple me is bit more th 2 stdrd devitios bove wht would hve bee expected were H true. 7. The computed vlue z 2.16 does ot fll i the rejectio regio so H cot be rejected t sigificce level.1. The dt does ot give strog support to the clim tht the true verge differs from the desig vlue of Cse II: Lrge-Smple Tests Whe the smple size is lrge, the z tests for cse I re esily modified to yield vlid test procedures without requirig either orml popultio distributio or kow. x x Isted of usig test sttistic z, hs to be used z. / s/ (Popultio stdrd devitio is replced by smple stdrd devitio s ). Exmple Is me hum body temperture relly 98.6 degrees, or it is lower? Smple dt: 11, x degrees, s.73 degrees. Solutio. H : 98.6 degrees H : 98.6 degrees A test sigifict level.5 rejects H whe z ( lower tiled test) Accordig to the smple dt, z Sice H hs to be rejected (t the sigificce level 5%). 3

4 8.3. Tests Cocerig Popultio Proportio. Test procedure described for the popultio me c be pplied to the testig hypotheses bout the popultio proportio just by usig correspodig formul for test sttistic Z. Me Proportio Z X / Z p pˆ p 1 p Exmple Reserch questio: Is the proportio of bbies bor mle differet from.5? Smple dt: 2, 96 were mle. Solutio. H :.5 p H : p.5 Test Sttistic: Pˆ p Z ; pˆ.48; z.566 p1 p With sigificce level.5 the rejectio regio is z or z Coclusio: Sice z (ot i the rejectio regio), we do ot hve sufficiet evidece to stte tht i the popultio the proportio of bbies bor mle is differet from.5. 4

5 8.4. p-vlue Exmple Retur to the bbies exmple from the previous prt. We were determied the vlue of Z sttistic for the prticulr smple s.48.5 z Now let use z-tble to determie the probbility tht correspods to this z vlue. Accordig to the z-tble p This is the probbility tht Z.566 or Z.566.Becuse this is two-tiled test we must tke ito ccout both the left d right tils. So, i this cse the probbility to be i the rejected re is Forml writig: p , so H do ot rejected MINITAB outputs (slightly differet) Decide Betwee the Null d Altertive Hypotheses. From the begiig, we determied two tiled rejectig regio hvig the re of.5. Accordig to our smple fidig, it should be much wider. Coclusio is the sme s with other method of solutio: There is o sufficiet evideces to stte tht i the popultio the proportio of bbies bor mle is differet from.5. 5

6 This secod pproch of decisio mkig whe testig hypotheses (for exmple bout the popultio me or proportio) is clled p -vlue pproch. A p-vlue c be thought of s the followig coditiol probbility: Defiitio P rejectig H H is true The p-vlue is the probbility, clculted ssumig tht the H is true, of obtiig vlue of test sttistic t lest s cotrdictory to H, s the vlue clculted from the vilble smple Decisio Rule Bsed o p-vlue. Select sigificce level (the desired type I error). Reject Hif p-vlue Do ot reject H if p-vlue Exmple Suppose tht we wt to test the hypothesis with sigificce level of.5 tht the climte hs chged sice idustriliztio. Suppose tht the me temperture throughout history is 5 degrees. Durig the lst 4 yers, the me temperture hs bee 51 degrees d suppose the popultio stdrd devitio is 2 degrees. Wht c we coclude? H : 5; H : We compute the z score: z / 4 From z tble p.9992 the re of til Are of two tils Coclusio: We c coclude tht there hs bee chge i temperture. 6

7 The Rules for Computig p-vlues for z Tests. 1 z for upper-tiled z test p-vlue z for upper-tiled z test 2 1 z for two-tiled z test Grphicl Illustrtio of p-vlues for z Test. 7

8 Exmple The dimeter of spidle i smll motor is supposed to be 5 mm. If the spidle is either too smll or too lrge, the motor will ot perform properly. The mufcturer mesures the dimeter i smple of 6 motors to determie whether the me dimeter hs moved wy from the trget. They foud the me dimeter of the smple to be mm. From pst studies, it is kow tht the popultio stdrd devitio is.2 mm. At 5%, do you hve eough evidece to coclude tht the me dimeter is ot 5mm? At the 1% level? Solutio. H : 5; H : 5 Coditios: 3, is kow. Test sttistics d p-vlue: z / 6 p-vlue: 2[ Coclusio. At the 5% sigificce level, we do ot hve eough evidece to reject the ull hypothesis ( p-vlue ). p p 3 At the 1% sigificce level, ( p-vlue.1), thus we hve eough evidece to reject the ull hypothesis. C you test the clim usig pproprite CI? Sice we hve two-sided test ( i the ltertive hypothesis), we c costruct two-sided CI % CI: x z / z , 5.16, climed vlue 5 is IN CI we cot reject.2.2 9% CI: x z / z H , , climed vlue 5 is NOT i CI we c reject H. 8

9 Grphicl Illustrtio of p-vlues for t-test The p-vlue is the smllest sigificce level t which the ull hypothesis c be rejected. p-vlue is ltertively referred to s the observed sigificce level (OSL) for the dt. Note tht smll p-vlues will result i rejectio of H d lrge p-vlues will result i filig to reject H. 9

10 Exmple A idustril compy clims tht the me ph level of the wter i erby river is 6.8. You rdomly select 19 wter smples d mesure the ph level of ech. The smple me d stdrd devitio re 6.7 d.24 respectively. Is there eough evidece to reject the compy s clim t.5? Assume tht the popultio is ormlly distributed. Is your coclusio the sme t.1level? Solutio. H : 6.8; H : 6.8 Coditios: 19, smll, x6.7, s.24. The popultio is ormlly distributed. Test sttistics d p-vlue. x t t s/.24 / 19 p-vlue.86 t Coclusio. At the 5% sigificce level, we do ot hve eough evidece to reject the ull hypothesis ( p-vlue.5). At the 1% sigificce level, ( p-vlue.1), thus we hve eough evidece to reject the ull hypothesis. Costructio of correspodig CI. Sice we hve two-sided test ( i the ltertive hypothesis), we c costruct two-sided CI. s % CI: x t /2, t.25, , , climed vlue 6.8 is IN CI we cot reject s % CI: x t /2, t.5, , , climed vlue 6.8 is NOT i CI we c reject H. H. 1

11 Choosig Approprite Iferece Procedure: Cofidece Itervl or Hypothesis Test? Use cofidece itervl to estimte ukow popultio prmeter. Use testig hypothesis to swer mybe yes or mybe o questio whe some guess bout vlue of ukow popultio prmeter is vilble. I my cses whe testig hypothesis is doe, it is good ide to costruct cofidece itervl s well to estimte the vlue of ukow popultio prmeter. 11