Mathacle. PSet Stats, Concepts In Statistics Level Number Name: Date: Hypothesis Test We assume, calculate and conclude.

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1 Hypothesis Test We ssume, clculte d coclude. VIII. HYPOTHESIS TEST 8.. P-Vlue, Test Sttistic d Hypothesis Test [MATH] Terms for the mkig hypothesis test. Assume tht the uderlyig distributios re either Z-distributio or t-distributio. Test Sttistic The stdrdized differece of smple sttistic d hypotheticl popultio prmeter. It is usully the criticl z * or t *: Sttistic Prmetr Test Sttistic Vribility Test Sttistic is used i hypothesis test to determie the P- of the observed sttistic. P- The coditiol probbility of the observed sttistic occurs uder the ull hypothesis H :.) P Pr ( Z < Test Sttistic H ) or P Pr ( T Test Sttistic H, df ) < the probbility tht the s re less th the give Test Sttistic. b.) P Pr ( Z > Test Sttistic H ) or P Pr ( T Test Sttistic H, df ) >, the probbility tht the s re greter th the give Test Sttistic. 6

2 c.) P Pr ( Z > Test Sttistic H ) or P Pr ( T Test Sttistic H, df ) >, the probbility tht the s re greter th the Test Sttistic or less th the Test Sttistic, two-sided. Hypothesis Test Whe P, the evet of iterest is cosidered differet from the ull hypothesis, d hece the ull hypothesis is rejected, with level of sigificce α. If the decisio is wrog, error (Type I) is committed. So, α c be viewed s the probbility of committig Type I error. A sttisticl hypothesis is propositio bout popultio prmeter. The prmeters re usully the popultio me µ d some proportios p : Null Hypothesis: H : µ µ Altertive Hypothesis: H : µ µ, or µ > µ, or µ < µ Test Sttistic is the stdrdized differece of observed sttistic (smple me or smple proportio ˆp ) d hypotheticl popultio prmeter. It is usully the criticl z *: z* is the coditiol probbility of the observed sttistic occurs uder the ull P hypothesis H. The hypothesis test is used to decide if the ull hypothesis is true or flse: Reject ull hypothesis H if P CAN T reject ull hypothesis H if P > α 6

3 There re geerlly three cses to cosider whe it comes to clculte the p- for rdom vrible X. The bove wy to clculte the two-sided probbility is vlid oly for the symmetric distributios such s Z-distributio d t-distributio. For o-symmetric distributios such s χ distributio or F distributio test is usully used. [MATH] Procedures for Sttisticl Test: Step : Stte the pir of hypotheses Step : Idetify the test procedure d check the pproprite coditios Step 3: Clculte the test sttistic d P-Vlue. Step 4: Stte the coclusios i the cotet of study by usig the results of the test. Or CHAPS: C Coditios H Hypotheses A Alysis d Epltio P P- S Test Sttistic 6

4 8.. Oe - Smple Hypothesis Test Oe - Smple Z Test for Proportio Emple 8... Arthritis is piful, chroic iflmmtio of the joits. A eperimet o the side effects of pi reliever ibuprofe emied rthritis ptiets to fid the proportio of ptiets who suffer side effects whe usig ibuprofe to relieve the pi. I the eperimet, 44 rdomly selected subjects with chroic rthritis were give ibuprofe for pi relief; 3 subjects suffered from dverse side effects. If more th 3% of users suffer side effects, the Food d Drug Admiistrtio will put stroger wrig lbel o pckges of ibuprofe. Is there sufficiet evidece to coclude tht the proportio of ibuprofe users who suffer dverse side effects is greter th.3? Assume the sigifict level is t 5%. Use clcultor to verify your results. Use the followig chrt to solve the problem. Step Sttemet Questio.) Stte the pop proportio d smple size: p,.) Stte the sigifict level: α 3.) Null Hypothesis : H : p p 4.) Altertive Hypothesis:.). H : p < p : p > p : p p, two-sided Test Type: Coditios:.) Smple is rdom.) is less th % of pop 3.) Others 63

5 3 4 i, i Test Sttistic :.) H : p p : p p c.) : p p pˆ z* i i p pˆ p p ( p ), p ( p ) P Pr Z < z * <, ( ) >, P Pr ( Z > z *) H, P Pr ( Z z* ) > [Ti-84].). Stts ->TESTS-> 5: -PropZTest.) Iput p,,, select p, < p, > p 3.) Clculte or Drw Reject H, if P Solutio: Step Procedure Questio.) Stte the pop proportio d smple size: p, p.3, 44, α.5.) Stte the sigifict level: α 3.) Null Hypothesis : H : p >.3 4.) Altertive Hypothesis:.). H : p < p : p > p : p p, two-sided Test Type: Oe-proportio Z test Coditios:.) Smple is rdom.) is less th % of pop 3.) p >, q > Oe-proportio Z test From the questio, the smple is rdomly selected d ssumed to be less th % of the popultio. p 44(.3) >, ( p ) 44(.3) > 64

6 3 i, i pˆ p( p) p Test Sttistic : z*.) H : p p : p p c.) : p p i i, pˆ p p ( p ) P Pr Z < z * <, ( ) >, P Pr ( Z > z *) H, P Pr ( Z z* ) > i 3, p ( p ).3(.3) 44 p i 3 pˆ z pˆ p *.75 p p Pr( Z >.75) [Ti-84].). Stts ->TESTS-> 5: -PropZTest.) Iput p,,, select p, < p, > p 3.) Clculte or Drw Reject H, if P STAT > TESTS > 5 : propztest > { p :.3, : 3, : 44, prop : > p } > Clculte Sice the P- is less th the sigifict levelα, the ull hypothesis is rejected. Tht is, there is sufficiet evidece to coclude tht the proportio of ibuprofe users who suffer dverse side effects is greter th.3. Emple 8... Solve ech of the followig oe-smple proportio problems. Assume tht the smple is idepedet d the smple size is less th % of the popultio. The ull hypothesis H is p p. Cse # α p ) p z* pˆ p p ( p ) p > p p p p < p p p H P Decisio to reject H? 65

7 Solutio: Cse # α ˆp p z* pˆ p p ( p ) H P Decisio to reject H? p > p.8 Yes p p.54 No p < p. Yes p p.68 No Cse #: pˆ p.55.5 pˆ.55, z*., p 4(.5) >, 4 p ( p ).5(.5) 4 ( p ) 4(.5) P Pr Z > z * Pr Z >..8. >, ( ) ( ) Or use Ti-84: STAT > 5 : Pr opztest > { p :.5, :, 4, prop : > p } > Clculte H is rejected. Cse #: pˆ p pˆ.733, z 5 p ( p ).69(.3) 5 p 5(.69) >, ( p ) 5(.3) >, *.47, Or use Ti-84: ( ) ( ) P Pr Z > z * Pr Z >.47 (.56).54 > α, STAT > 5 : Pr opztest > { p :.69, :, 5, prop : p } > Clculte H is filed to be rejected. 66

8 Cse #3: 7 pˆ p.4.5 pˆ.4, z*.94, p ˆ 7(.4) >, 7 p ( p ).5(.5) 7 ( pˆ ) 7(.588) P Pr Z < z * Pr Z <.94., >, ( ) ( ) Or use Ti-84: STAT > 5 : propztest > { p :.5, : 7, 7, prop : < p } > Clculte H is rejected. Cse #4: Or use Ti-84: 4 pˆ p.5.3 pˆ.5, z 6 p ( p ).3(.7) 6 ) p ˆ 6(.5) >, ( p) 6(.75) >, *.38 ( ) ( ) P Pr Z < z * Pr Z <.38 (.837).68 > α, STAT > 5 : propztest > { p :.5, : 4, 6, prop : p } > Clculte, H is filed to be rejected. Emple [MC8] A eperimeter coducted two-tiled hypothesis test o set of dt d obtied p- of.44. If the eperimeter hd coducted oe-tiled test o the sme set of dt, which of the followig is true bout the possible p-(s) tht the eperimeter could hve obtied? (A) The oly possible p- is.. (B) The oly possible p- is.44. (C) The oly possible p- is.88. (D) The possible p-s re. d.78. (E) The possible p-s re. d.88. Solutio: The swer is D. The probbility o ech outside of the CI is.. So depedig o H testig greter or less th, the p could be. or

9 Oe-Smple Me Z Test Emple The popultio of ll verbl GRE scores is kow to hve stdrd devitio of 8.5. The uiversity Psychology deprtmet hopes to receive pplicts with verbl GRE scores over. This yer, the me of the verbl GRE scores for the 4 pplicts ws.79. Usig of α.5, is this ew me sigifictly greter th the desired me of? Step Sttemet Questio.) Stte the pop me, SD d smple size: µ,,.) Stte the sigifict level:α 3.) Null Hypothesis : 4.) Altertive Hypothesis:.). H : µ < µ : µ > µ : µ µ, two-sided 3 Test: Oe-Me Z Test Coditios:.) Smple is rdom.) is less th % of pop 3.) Smple size 3, kow pop i i, Test Sttistic : z*.) H : µ < µ, P Pr ( Z < z *) : µ > µ, P Pr ( Z > z *) P Pr Z > z*, ( ) 68

10 4 [Ti-84].) Stts ->TESTS-> : Z-Test.) Iput µ,,,, select µ, < µ, > µ 3.) Clculte or Drw Reject H, if P Solutio: Step Sttemet Questio.) Stte the pop me, SD d smple µ, 8.5, 4, size: µ,, α.5.) Stte the sigifict level:α 3.) Null Hypothesis : Null: : 4.) Altertive Hypothesis: lt : H : µ >.). H : µ < µ : µ > µ : µ µ, two-sided Test: Oe-Smple Me Z Test Coditios:.) Smple is rdom.) is less th % of pop 3.) Smple size 3, kow pop Oe-Smple Me Z Test. Smple is ssumed to be idepedet d rdom d less th % of popultio. 3, is kow. 3 i i, Test Sttistic : z*.) H : µ < µ, P Pr ( Z < z *) : µ > µ, P Pr ( Z > z *) P Pr Z > z*, ( ) [Ti-84].). Stts ->TESTS-> : Z-Test.) Iput µ,,,, select z* i i.79, Pr Z >.3 P.67 ( ) 69

11 4 µ, < µ, > µ 3.) Clculte or Drw Reject H, if P STAT > TESTS > : Z Test > { µ :, : 8.5, :.79, : 4, µ : > } > Clculte The ull hypothesis is rejected. Tht is, to clim tht the ew me is greter th the desired me of is sigifict. Oe-Smple Me t Test (Mtched Pirs t- Test) This test is ofte used i the mtched pir test for the smple size 5 < 3. Emple Evirometlists, govermet officils, d vehicle mufcturers re ll iterested i studyig the uto ehust emissios produced by motor vehicles. The mjor pollutts i uto ehust re hydrocrbos, mooide, d itroge oide (NOX). The followig tble gives the NOX levels i grms/mile for rdom smple of lightduty egies of the sme type. If the crs c oly pss ispectio if the NOX level is less th or equl to. grms per mile, is the dt sigifict to suggest tht this btch of egies eceeds the requiremet? Compre t both 5% d % level. 7

12 Step Sttemet Questio.) Stte the pop me, smple size d degrees of freedom: µ,, k.) Stte the sigifict level:α 3.) Null Hypothesis : H : µ µ 4.) Altertive Hypothesis:.). H : µ < µ : µ > µ : µ µ, two-sided Test: Oe-Me t Test Coditios:.) Smple is rdom..) is less th % of pop 3.) Smple size 5 3, ukow/kow pop 3 i i s i,, ( ) i Test Sttistic : t* s.) H P Pr T < t*, df <, ( ) >, P Pr ( T > t*, df ), P Pr ( T > t*, df ) s 4 [Ti-84].) Stts ->TESTS-> : T-Test.) Iput µ,, s,, select µ, < µ, > µ 3.) Clculte or Drw Reject H, if P 7

13 Solutio: Step Sttemet Questio 3.) Stte the pop me, smple size d degrees of freedom: µ,, k.) Stte the sigifict level:α 3.) Null Hypothesis : H : µ µ 4.) Altertive Hypothesis:.). H : µ < µ : µ > µ : µ µ, two-sided Test: Oe-Me t Test Coditios:.) Smple is rdom.) is less th % of pop 3.) Smple size 5 3, ukow/kow pop i i, s ( ) i, i s Test Sttistic : t s.) H <, P Pr ( T < t*, df ) >, P Pr ( T > t*, df ), P Pr ( T > t*, df ) µ., 46, k 45, α.5 or α.. ull: H : µ. lt: H : µ >. Pop stdrd devitio is ukow d is ssumed to be less th % of popultio. i i.39, s ( ) i.484 i s t.8.74 ( ) P Pr T >.8 df < <.5 P [Ti-84].). Stts ->TESTS-> : T-Test.) Iput µ,, s,, select 7

14 4 µ, < µ, > µ 3.) Clculte or Drw Reject H, if P STAT > TESTS > : T Test > { µ :.,.39, s.484, : 46 µ : > µ } > Clculte The ull is rejected t sigifict level α.5, d the result is isigifict t the level α.. Emple [MC75] Whe performig test of sigificce bout popultio me, t-distributio, isted of orml distributio, is ofte utilized. Which of the followig is the most pproimte epltio for this? (A) The smple size is ot lrge eough to ssume tht the popultio distributio is orml. (B) The smple does ot follow orml distributio (C) There is icrese i vribility of the test sttistic due to estimtio of the popultio stdrd devitio. (D) The smple stdrd devitio is ukow. (E) The popultio stdrd devitio is too lrge. Solutio: The swer is C. s is too lrge whe is smll. Choice A is icorrect becuse the distributio of popultio does ot eed to be ormlly distributed, the smple me will pproch to ormlly distributed whe the smple size is lrge. 73

15 Quick-Check 8... Oe-Smple Test QC 8... Accordig to the Ceter for Disese Cotrol (CDC), the percet of dults yers of ge d over i the Uited Sttes who re overweight is 69.%. Oe city s coucil wts to kow if the proportio of overweight citizes i their city is differet from this kow tiol proportio. They tke rdom smple of 5 dults yers of ge or older i their city d fid tht 98 re clssified s overweight. Is this city s proportio of overweight idividuls differet from.69? QC 8... Suppose you strt up compy tht hs developed drug tht is supposed to icrese IQ. You kow tht the stdrd devitio of IQ i the geerl popultio is 5. You test your drug o 36 ptiets d obti me IQ of.96. Usig lph of., is this IQ sigifictly differet from the popultio me of? QC (Mtched Pirs Test) The SGO test is used to mesure the studets uderstdig o specific topic. The pre-test is coducted t the begiig of the yer d post-test is coducted whe the studets fiish lerig the topic. The followig tble shows the scores of fiftee studets who took both tests lst yer. Pre-Test Post-Test Differece Is this differece of the results sigifict? Assume the sigifict level is.5. 74

16 Step Sttemet Questio.) Stte the pop proportio d smple size: p,.) Stte the sigifict level: α 3.) Null Hypothesis : H : p p 4.) Altertive Hypothesis:.). H : p < p : p > p : p p, two-sided 3 Test: Oe-Proportio Z Test Coditios:.) smple is rdom.) is less th % of pop 3.) p d ( p ) > > i, i Test Sttistic :.) H : p p : p p c.) : p p pˆ z* i i p pˆ p p ( p ), p ( p ) P Pr Z < z * <, ( ) >, P Pr ( Z > z *) H, P Pr ( Z z* ) > 4 [Ti-84].). Stts ->TESTS-> 5: -PropZTest.) Iput p,,, select p, < p, > p 3.) Clculte or Drw Reject H, if P 75

17 Step Sttemet Questio.) Stte the pop me, SD d smple size: µ,,.) Stte the sigifict level:α 3.) Null Hypothesis : H : µ µ 4.) Altertive Hypothesis:.). H : µ < µ : µ > µ : µ µ, two-sided 3 Test: Oe-Me Z Test Coditios:.) Smple is rdom.) is less th % of pop 3.) Smple size 3, kow pop i i, Test Sttistic : z*.) H : µ < µ, P Pr ( Z < z *) : µ > µ, P Pr ( Z > z *) P Pr Z > z*, ( ) 4 [Ti-84].). Stts ->TESTS-> : Z-Test.) Iput µ,,,, select µ, < µ, > µ 3.) Clculte or Drw Reject H, if P 76