Mathacle. Hypothesis Tests We assume, calculate and conclude.

Transcription

1 Level ---- Number --- Hypothesis Tests We ssume, clculte d coclude. V. HYPOTHESIS TESTS A sttisticl hypothesis is rgumet or ssumptio bout popultio prmeter, d the rgumet my or my ot be vlid. Hypothesis tests refer to procedure bsed o the give propositios to vlidte sttisticl hypotheses. Exmple 5... Assume tht you use bised rdom umber geertor to geerte digits of,,3,, 9. The results is formulted s histogrm give below. The grph c be viewed s pproximtio of true distributio of the umber geertor. The grph shows tht mjority of the digits re, 3, 4, 5, 6, d it is very rre for the geertor to get, 7, 8, d 9. Use hypothesis test to prove if this is true.

2 Level ---- Number --- Solutio: Let the uiversl set U {,,3,4,5,6,7,8,9}. A prtitio of U is H {,3,4,5,6} d H {,7,8,9}. Tht is, U H H d H H {}. Whe digit x is geerted, x H or x H but ot both. The probbility Px is clled p-vlue. Bsed o the distributio is give, whe digit x is geerted, the hypotheses c be mde s Null Hypothesis: obtiig x H is ot commo Altertive Hypothesis: obtiig x H is ot commo Whe sigifict level is set, the hypothesis test proceeds s follows: If P x H be rejected. Tht is, obtiig should ot be rejected. Whe.5, the p-vlue is less th the sigifict level, the ull hypothesis will H x H is very rre. Otherwise, the ull hypothesis 76 6 H H is very rre., P x.95% rejected. So, obtiig {,7,8,9}, so the ull hypothesis is 76 6 Whe., P x H.95%, d the ull hypothesis should ot be rejected. Tht is, there is ot eough evidece to sy obtiig H {,7,8,9} is more rre th obtiig H {,3,4,5,6}. For geerl prctice, ormlized sttistic NSttistic : NSttistic Sttistic Prmeter Vribilit y is used to trsform the kow or to pproximte the ukow distributio to stdrd distributio such s the Z-distributio or the t-distributio. Sttistic is the smple proportio or smple me; Prmeter is the popultio proportio or me; d Vribilit y is the stdrd devitio/error. NSttistic is expressed s either z or t.

3 Level ---- Number --- There re geerlly three cses to cosider whe it comes to clculte the p-vlue for rdom vrible X :.) X NSttistic P, the probbility tht the vlues re less th the give NSttistic, oe-sided. b.) X NSttistic P, the probbility tht the vlues re greter th the give NSttistic, oe-sided. c.) X NSttistic P, the probbility tht the vlues re greter th hlf of the NSttistic or less th hlf of NSttistic, two-sided. This 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 distributio or F distributio, oe-sided test is usully used. Exmple 5... Let x s Sttistic, Prmeter, T Vribilit y, z NSttistic d pvlue p-vlue. Give the sigifict level. 5, clculte p vlue, d compre p vlue d i ech cse. Note tht T is the fil clculted vritio for the orml trsformtio i ech cse, so there is o eed to fctor i the smple size i some cses..) x, 9,, Norml Distributio, P( Z z) s T p vlue b.) x 6, 9,, Norml Distributio, P( Z z) s T p vlue 3

4 Level ---- Number --- c.) x 75, 9,, Norml Distributio, P( Z z ) s T p vlue d.) x, 9,, 5, t-distributio, P( T z) s T p vlue e.) x 6, 9,, 5, t-distributio, P( T z) s T p vlue f.) x 75, 9,, 5, t-distributio, P( Z z ) s T p vlue Solutio: x 9.) s z, p vlue P( Z z) ( Z ) T x 6 9 b.) s z 3, p vlue P( Z z) ( Z 3). 34. T x 75 9 c.) s z. 5, T p vlue P( Z z ) ( Z.5) ( Z.5) (.668). 336 x 9 d.) s z, df 5 4, T p vlue x 6 9 e.) s z 3, df 5 4, T p vlue P( Z z) 4( Z ). 674 P( Z z) 4( Z 3). 997 x 75 9 f.) s z. 5, df 5 4 p vlue T P( Z z ) 4( Z.5) 4( Z.5) (.7337)

5 Level ---- Number Oe - Proportio Z Test Testig Procedures Step :.) Stte the popultio proportio d smple size: p, b.) Stte the sigifict level: c.) Check the coditios: is less th % of popultio, p, ( p ). xi i d.) Clculte the smple proportio: pˆ Step : Stte the ull hypothesis: H : p p Step 3: Stte the possible ltertives:.) H : p p, oe-sided ltertive b.) H : p p, oe-sided ltertive c.) H : p p, two-sided ltertive Step 4: p ( p ), p Test Sttistic: Step 5: z pˆ p p ( p ) P-vlues d Decisios:.) H : p p, pˆ p P vlue ( Z z) Z p ( p ), reject H, if P vlue 5

6 Level ---- Number --- b.) H : p p, c.) H : p p, pˆ p P vlue ( Z z) Z p p ( ) pˆ p P vlue ( Z z ) Z p ( p ), reject H, if, reject H, if P vlue P vlue Exmple 5... Arthritis is piful, chroic iflmmtio of the joits. A experimet o the side effects of pi reliever ibuprofe exmied rthritis ptiets to fid the proportio of ptiets who suffer side effects whe usig ibuprofe to relieve the pi. I the experimet, 44 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%. Solutio: Step :.) the popultio proportio d smple size: p.3, 44 b.) the sigifict level:.5 c.) the coditios: p 44(.3), ( p ) 44(.3) d) the smple proportio: p( p).3(.3) xi 3, p.8 44 Step Step 3: the ull hypothesis: H : p.3 the ltertive: H : p.3 x i 3 pˆ

7 Level ---- Number --- Step 4: pˆ p.53.3 Test Sttistic: z.75.8 Step 5: P-vlues d Decisios p p ( Z z) ( Z.75).3 vlue Sice the P-vlue is less th the sigifict level, 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. Exmple 5.. 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? Solutio: Step :.) the popultio proportio d smple size: p.69, 5 b.) the sigifict level:.5 c.) the coditios: p 5(.69), ( p) 5(.69) d) the smple proportio: p( p).69(.69) 98, p x i 98 pˆ

8 Level ---- Number --- Step Step 3: the ull hypothesis: H : p.69 the ltertive: H : p.69 Step 4: pˆ p Test Sttistic: z Step 5: P-vlues d Decisios p p ( Z z ) ( Z.984) ( Z.984).364 vlue The ull hypothesis is filed to be rejected. There is ot sufficiet evidece to stte tht the proportio of citizes of this city who re overweight is differet from the tiol verge of Oe-Smple Me Z - Test Testig Procedures Step :.) Stte the popultio me, stdrd devitio d smple size:,, b.) Stte the sigifict level: c.) Check the coditios for testig: smple size 3 d is less th % of popultio xi i d.) Clculte the smple me d stdrd devitio: x Step : Stte the ull hypothesis H :, x 8

9 Level ---- Number --- Step 3: Stte the possible ltertives:.) H :, oe-sided ltertive. b.) H :, oe-sided ltertive. c.) H :, two-sided ltertive. Step 4: x Test Sttistic z x Step 5: P-vlues d Decisios x.) H :, P vlue ( Z z) Z x b.) H :, P vlue ( Z z) Z x c.) H :, P vlue ( Z z ) Z, reject H, if, reject H, if P. vlue, reject H, if P. vlue P. vlue Exmple 5.. The popultio of ll verbl GRE scores re 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 verbl GRE scores for the 4 pplicts ws.79. Usig vlue of α =.5 is this ew me sigifictly greter th the desired me of? Solutio: Step :.) the popultio me, stdrd devitio d smple size:, 8.5, 4 9

10 Level ---- Number --- b.) the sigifict level:.5 c.) the coditios for testig: 3 d is less th % of popultio d.) the smple me d stdrd devitio: Step - Step 3: x i xi.79, the ull hypothesis: H : oe-sided ltertive : H : Step 4: x.79 Test Sttistic z.3.3 Step 5: P-vlues d Decisios x P vlue ( Z z) Z x.3 4 The ull hypothesis is rejected. Tht is, the ew me sigifictly greter th the desired me of. Exmple 5.. 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 vlue of., is this IQ sigifictly differet from the popultio me of? Solutio: Step :.) the popultio me, stdrd devitio d smple size:, 5, 36 b.) the sigifict level:. c.) the coditios for testig: 3 d is less th % of popultio d.) the smple me d stdrd devitio:

11 Level ---- Number --- Step - Step 3: x i xi.96, 5 x.5 36 the ull hypothesis: H : oe-sided ltertive : H : Step 4: x.96 Test Sttistic z.8 x.5 Step 5: P-vlues d Decisios P vlue ( Z z ) Z.8.38 The ull hypothesis c ot be rejected. Tht is, The IQ of drug ptiets (M =.96) is ot sigifictly differet from Oe-Smple Me t - Test Whe smple size 5 3, the me c be cosidered s X ~ T( k), where T(k) is the t-distributio with pdf (x) d degrees of freedom k. Testig Procedures Step : k.) Stte the popultio me, smple size d degrees of freedom:,, k b.) Stte the sigifict level. c.) Check the coditios for testig: smple size 5 3, or stdrd devitio is ukow, d is less th % of popultio. d.) Clculte the smple me d stdrd devitio: x i x i s xi x, x, i s

12 Level ---- Number --- Step : Stte the ull hypothesis H : Step 3: Stte the possible ltertives:.) H :, oe-sided ltertive. b.) H :, oe-sided ltertive. c.) H :, two-sided ltertive. Step 4: x Test Sttistic t x Step 5: P-vlues d Decisios x.) H :, P vlue k ( T t) k T s x b.) H :, P vlue k ( T t) k T s x c.) H :, P vlue k ( T t ) k T s, reject H, if, reject H, if P. vlue P. vlue, reject H, if P. vlue Exmple 5.3. Evirometlists, govermet officils, d vehicle mufcturers re ll iterested i studyig the uto exhust emissios produced by motor vehicles. The mjor pollutts i uto exhust re hydrocrbos, mooxide, d itroge oxide (NOX). The followig tble gives the NOX levels i grms/mile for smple of light-duty egies of the sme type.

13 Level ---- Number --- 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 exceeds the requiremet? Compre t both 5% d % level. Solutio:.) the popultio me, smple size d degrees of freedom:., 46, k 45 b.) the sigifict level.5 or.. c.) Check the coditios for testig: stdrd devitio is ukow d is less th % of popultio. d.) Clculte the smple me d stdrd devitio: x i Step Step 3: xi.39 the ull hypothesis: H :. oe-sided ltertive: H :. s s.484 xi x.484, x.74 46, i Step 4: x.39. Test Sttistic t.8 x.74 Step 5: P-vlues d Decisios: P vlue k ( T t) 45 T.8.385,. P vlue.5. The ull is rejected t sigifict level.5, d the result is ot sigifict t the level.. 3

14 Level ---- Number Two - Proportio Z Test A more commo situtio is to compre two popultio proportios. Let us ssume.) There re idepedet Beroulli rdom vribles X, X,, X tht tke either oe or zero, with success probbility of p for ech tril, d there re idepedet Beroulli rdom vribles Y, Y,, Y tht tke either oe or zero, with success probbility of p for ech tril.....). The ubised smple proportio estimtor ˆ X X X X P hs the me ˆ E[ P] pd vrice ˆ p ( p ) Vr[ P ], d ubised smple proportio estimtor... ˆ Y Y Y Y P hs the me E[ P ˆ ] pd vrice ˆ p( p) Vr[ P ]. 3.) The popultio is t lest times s lrge s the smple size d -- the % rule. 4.) p, ( p) d p, ( p). Some texts use 5 isted of. This could be flwed sice p d p could follow two differet distributios. The differece of the two proportios p p is ubised estimtor, d the vrice is E[ Pˆ Pˆ ] E[ Pˆ] E[ Pˆ] p p Vr[ Pˆ Pˆ ] Vr[ Pˆ] Vr[ Pˆ ] p ( p ) p( p) Whe the smple size is lrge, the smplig distributio is 4

15 Level ---- Number --- p ˆ pˆ ~ N p p, p( p) p ( p ) The stdrd error is estimted by ˆ ˆ pˆ ( pˆ ) pˆ ( pˆ ) SE[ P P]. Ad the cofidece itervl is pˆ pˆ z SE[ Pˆ Pˆ ] 5.) Whe the two proportios hve the sme distributios, d hece proportios re the X Y sme p p, the pooled proportio Pˆpooled is better pproximtio th ˆp or ˆp. The stdrd error is i this cse ˆ ˆ SE ˆ ˆ pooled [ P P] ppooled ( ppooled ). Testig Procedures Step :.) Stte the smple sizes d proportios:,, p, p b.) Stte the sigifict level: c.) Check the coditios:, re less th % of popultio d p, ( p) d p, ( p). d.) Clculte the smple proportios whe the uderlyig prmeters re the sme: pˆ i x i, pˆ i y i, pˆ pooled x i i i y i, SE ˆ ( ˆ pooled ppooled ppooled ). Step : Stte the ull hypothesis: H p p p. Step 3: Stte the possible ltertives: : 5

16 Level ---- Number ---.) p p p, oe-sided ltertive. H : H : p p p H : p p p b.), oe-sided ltertive. c.), two-sided ltertive. Step 4: Test Sttistic: Step 5: z pˆ pooled pˆ pˆ ( pˆ pooled ). P-vlues d Decisios.) H : p p p, P vlue ( Z z), reject H, if b.) H : p p p, P vlue ( Z z), reject H, if P. vlue P. vlue c.) H : p p p, ( Z z ) Z z P vlue, reject H, if P. vlue Exmple 5.4. It is believed tht sweeteer clled xylitol helps prevet er ifectios. I rdomized experimet, 65 childre took plcebo d 68 of them got er ifectios. Aother smple of 59 childre took xylitol d 46 of them got er ifectios. We believe tht the proportio of er ifectios i the plcebo group will be greter th the xylitol group. Test this hypothesis t.5. Solutio:.). the smple sizes d proportios: 65, 59, p, p (ot give) b.) Stte the sigifict level:.5 c.) Check the coditios:, re less th % of popultio. Ad p 5, ( p) 5d p 5, ( p) 5 (See estimte of p pooled below) d.) the smple proportios whe the uderlyig prmeters re the sme: 6

17 Level ---- Number --- x 68 i i pˆ.4 65 x, y y 46 i i pˆ i i i i pˆ pooled , Step Step 3: SE ˆ ( ˆ pooled ppooled ppooled ).35(.35) the ull hypothesis: H p p p : the possible ltertives: p p p Step 4: H : pˆ pˆ.4.89 Test Sttistic: z.3.53 pˆ ( ˆ pooled ppooled ) Step 5: P-vlues d Decisios P vlue ( Z z) ( Z.3). The ull is rejected. Tht is, tht the proportio of er ifectios i the plcebo group is greter th the xylitol group. 7

18 Level ---- Number Two-Smple Me Z Test Let us ssume.) Idepedet rdom vribles X, X,, X, with me of d vrice for ech tril, d idepedet rdom vribles Y,, Y, Y, with me of d vrice for ech tril..) The ubised smple me estimtor X X X... X hs the me E[ X ] d vricevr[ X ]. The ubised smple me estimtor Y Y... Y Y hs the me E[ Y] d vricevr[ Y]. 3.) Whe smple size 3 d 3, d X Y X or ~ N(,) ~ N(, ) Y or ~ N(,) ~ N(, ) 4.) The differece of the two mes X Y mes: is ubised estimtor the differece of the E[ X Y ] E[ X ] E[ Y ] d the vrice of the differece X Y is 8

19 Level ---- Number --- Vr[ X Y ] Vr[ X ] Vr[ Y ] Whe the smple size is lrge, the distributio for X Y is X Y ~ N, The stdrd error is by SE[ X Y ]. Testig Procedures Step :.) Stte the mes, stdrd devitios d smple sizes:,,,,,. b.) Stte the sigifict level. c.) Check the coditios for testig: 3 d 3 d, re less th % of the popultio. b.) Clculte the smple mes d stdrd error for X Y : x i x i, y i y i, SE[ X Y ]. Step : Stte the ull hypothesis H Step 3: Stte the possible ltertives: :.) H, oe-sided ltertive. : 9

20 Level ---- Number --- b.) H, oe-sided ltertive. : : c.) H, two-sided ltertive. Step 4: Test Sttistic Step 5: z x y P-vlues d Decisios.) H, : P. vlue b.) H, : P. vlue c.) H, : P. vlue P vlue ( Z z) Z P vlue ( Z z) Z x y x y P vlue ( Z z ) Z x y, reject H, if, reject H, if, reject H, if Exmple 5.5. The mout of certi trce elemet i blood is kow to vry with stdrd devitio of 4. ppm (prts per millio) for mle blood doors d 9.5 ppm for femle doors. Rdom smples of 75 mle d 5 femle doors yield cocetrtio mes of 8 d 33 ppm, respectively. Wht is the likelihood tht the popultio mes of cocetrtios of the elemet re the sme for me d wome? Assume the test sigifict level is.5.

21 Level ---- Number --- Solutio: Step :.) the mes, stdrd devitios d smple sizes:?, 4., 75,?, 9.5, 5. b.) Stte the sigifict level.5. c.) the coditios for testig: 3 d 3 d, re less th % of the popultio. b.) the smple mes d stdrd error for X Y : x i x i 8, y i y i 33, SE[ X Y] Step Step 3: the ull hypothesis H : two-sided ltertive H : Step 4: Test Sttistic Step 5: x y 8 33 z.37. P-vlues d Decisios x y P vlue ( Z z ) Z ( Z.37).78 The ull hypothesis is rejected. Tht is, the differece i cocetrtio betwee me d wome is sigifict.

22 Level ---- Number Two-Smple Me t Test Testig Procedures for Two-Smple t-test for Idepedet Smples Step :.) Stte the smple sizes:,. b.) Stte the sigifict level. c.) Check the coditios for testig: smple size 5 3 d 5 3, d, re less th % of the popultio. d.) Clculte the smple mes d stdrd error for X Y : x i x i, y i y i, s xi x, s y i y i.) For the popultio vrices ukow but ssumed uequl, the stdrd error is s s SE[ X Y], the degrees of freedom k mi{, }..) For the popultio vrices ukow but ssumed to be equl, the pooled stdrd error is s pooled ( ) s ( ) s i SE[ X Y] spooled, the degrees of freedom k. Step : Stte the ull hypothesis H Step 3: Stte the possible ltertives: :.) H, oe-sided ltertive. : : b.) H, oe-sided ltertive.

23 Level ---- Number --- c.) H, two-sided ltertive. : Step 4: Test Sttistic x y t SE[ X Y ] Step 5: P-vlues d Decisios x y.) H :, P vlue k ( T z) k T SE[ X Y ], reject H, if P. vlue x y b.) H :, P vlue k ( T z) k T SE[ X Y ], reject H, if P. vlue x y c.) H :, P vlue k ( T t ) k T, reject H, if SE[ X Y ] P. vlue Exmple 5.6. At THS Stts fil exm, the me d stdrd devitio for 9 femle studets re 7. d.57, respectively; d the me d stdrd devitio for the mle studets re 6.7 d Do mle studets d femle studets differ i their test scores? Assume the sigifict level is.5 d the stdrd devitios re the sme for both mle studets d femle studets. Solutio:.) the mes, stdrd devitios d smple sizes: 9,. b.) the sigifict level.5. c.) the coditios for testig: smple size 5 3 d 5 3, d, re less th % of the popultio. d.) Clculte the smple mes d stdrd error for X Y : x i x i 7., y i y i 6.7, 3

24 Level ---- Number --- s x x i.57, s y i y i 3.63 Assume the vrices ukow but equl, the stdrd error is i s pooled ( ) s ( ) s (9 )(.57) ( )(3.63) SE[ X Y] s pooled Step Step 3: The ull hypothesis H : H : The two-sided ltertive Step 4: Test Sttistic Step 5: xy t.3953 SE[ X Y ]. k 9 37 P-vlues d Decisios x y P vlue 37( T t ) 37 T 37 T SE[ X Y ] The ull hypothesis c ot be rejected. Tht is, we c ot coclude if there is differece betwee the mes of the mle studets d femle studets. 4

25 Level ---- Number Type I d Type II Errors The Type I error hppes whe H is true while rejectig H, d Type II error hppes whe H is flse while cceptig H. The decisios d errors c be summrized i the followig tble Decisio\Result H is True H is Flse Accept H Correct Decisio ( ) Type II Error ( ) Reject H Type I Error ( ) Correct Decisio ( ) The sigifict level is the probbility of mkig Type I error: P( reject H H is true) The is the probbility of mkig Type II error: The power of test is. P( ccept H H is flse) Methods to icrese power : Icrese to mke smller, d hece bigger Use oe-sided isted of two-sided test Reduce vrice by icresig the smple size so tht the smple distributio is thier to reduce the overlppig re 5

26 Level ---- Number --- Exmple 5.7.[CollegeBordAPSttsProblem] I test of H : μ = 8 versus H : μ 8, smple of size leds to p-vlue of.34. Which of the followig must be true? (A) A 95% cofidece itervl for μ clculted from these dt will ot iclude μ = 8. (B) At the 5% level if H is rejected, the probbility of Type II error is.34. (C) The 95% cofidece itervl for μ clculted from these dt will be cetered t μ = 8. (D) The ull hypothesis should ot be rejected t the 5% level. (E) The smple size is isufficiet to drw coclusio with 95% cofidece. Solutio: The swer is A. With 95% CI, 5%. 34, the p-vlue. 6

27 Level ---- Number AP Exmples Exmple 5.8. [5APSttsFRQ, #4] A resercher coducted medicl study to ivestigte whether tkig low-dose spiri reduces the chce of developig colo ccer. As prt of the study,, dult voluteers were rdomly ssiged to oe of two groups. Hlf of the voluteers were ssiged to the experimetl group tht took lowdose spiri ech dy, d the other hlf were ssiged to the cotrol group tht took plcebo ech dy. At the ed of six yers, 5 of the people who took the low-dose spiri hd developed colo ccer d 6 of the people who took the plcebo hd developed colo ccer. At the sigificce level.5, do the dt provide covicig sttisticl evidece tht tkig low-dose spiri ech dy would reduce the chce of developig colo ccer mog ll people similr to the voluteers? Solutio: ,.5, pˆ ˆ ˆ.3, p.5, ppool s p pˆ ( ˆ pool ppool ) (.4)(.959).54 5 The ull hypothesis H : p p The oe-side ltertive H : p p pˆ ˆ p z s.54 p p ( Z.754).397 vlue So, the ull hypothesis is rejected, d the differece betwee the two results is sigifict. 7

28 Level ---- Number --- Exmple 5.8. [APSttsFRQ, #5] Solutio:.) The type II error is tht the Recretio Deprtmet ccept the fct tht 35% of dult residets pss physicl fitess test, d it is ctully less th 35% psses the test. b.) Becuse the p vlue is greter th the sigifict level of.5, so the fct tht 35% psses the test c ot be rejected. c.) It is ot SRS. 8

29 Level ---- Number --- Exmple 5.8.3[3APSttsFRQ, #5] 9

30 Level ---- Number --- Solutio:.). It is ot cuse for the reductio. There could be cofoudig fctors to reduce the blood pressure. A proper wy to sy is tht the reductio is ssocited with the medittio. 8 8 b.) Smple size is too smll. The smple pool proportio pˆ pool.86. So, 7 8 pˆ, ( pˆ ) d pˆ, ( pˆ ) eeds to stisfy to pool m m pool pool c c pool pproximte the biomil distributio by the orml distributio c.) The smple pˆ ˆ m pc.47 d P( pˆ ˆ m pc.47).5, so 7 the ull hypothesis c be rejected. Tht is, there is covicig evidece tht me i this retiremet commuity who meditte were less likely to hve high blood pressure th me i this retiremet commuity who do ot meditte. 3

31 Level ---- Number --- Quiz Hypothesis Tests.) [APStts, #7] A mufcturer clims its Brd A bttery lsts loger th its competitor s Brd B bttery. Nie btteries of ech brd re tested idepedetly, d the hours of bttery life re show i the tble below Brd A Brd B Provided tht the ssumptios for iferece re met, which of the followig tests should be coducted to determie if Brd A btteries do, i fct, lst loger th Brd B btteries? (A) A oe-sided, pired t-test (B) A oe-sided, two-smple t-test (C) A two-sided, two-smple t-test (D) A oe-sided, two-smple z-test (E) A two-sided, two-smple z-test.) [APStts, #9] A rdomized experimet ws performed to determie whether two fertilizers, A d B, give differet yields of tomtoes. A totl of 33 tomto plts were grow; 6 usig fertilizer A, d 7 usig fertilizer B. The distributios of the dt did ot show mrked skewess d there were o outliers i either dt set. The results of the experimet re show below. Which of the followig sttemets best describes the coclusio tht c be drw from this experimet? (A) There is o sttisticl evidece of differece i the yields betwee fertilizer A d fertilizer B (p >.5). (B) There is borderlie sttisticlly sigifict differece i the yields betwee fertilizer A d fertilizer B (. < p <.5). (C) There is evidece of sttisticlly sigifict differece i the yields betwee fertilizer A d fertilizer B (.5 < p <.). (D) There is evidece of sttisticlly sigifict differece i the yields betwee fertilizer A d fertilizer B (. < p <.5). (E) There is evidece of sttisticlly sigifict differece i the yields betwee fertilizer A d fertilizer B (p <.). 3

32 Level ---- Number ) [APStts, #35] Perchlorte is chemicl used i rocket fuel. People who live er former rocket-testig site re cocered tht perchlorte is preset i usfe mouts i their drikig wter. Drikig wter is cosidered sfe whe the verge level of perchlorte is 4.5 prts per billio (ppb) or less. A rdom smple of 8 wter sources i this re produces me perchlorte mesure of 5.3 ppb. Which of the followig is pproprite ltertive hypothesis tht ddresses their cocer? 4.) [APStts, #38] A physici believes tht the exercise hbits of Est Cost dults re differet from the exercise hbits of West Cost dults. To study this, she gthers iformtio o the umber of hours of exercise per week from rdom smple of Est Cost dults d rdom smple of West Cost dults. Which of the followig might be pproprite ull hypothesis for this study? (A) The verge umber of hours of exercise per week for Est Cost dults is differet from the verge umber of hours of exercise per week for West Cost dults. (B) The verge umber of hours of exercise per week for Est Cost dults is the sme s the verge umber of hours of exercise per week for West Cost dults. (C) The verge umber of hours of exercise per week for Est Cost dults is greter th the verge umber of hours of exercise per week for West Cost dults. (D) The verge umber of hours of exercise per week for Est Cost dults is less th the verge umber of hours of exercise per week for West Cost dults. (E) The probbility is.5 tht Est Cost dult d West Cost dult exercise equl umber of hours per week. 5.) [APStts, #8] A experimeter coducted two-tiled hypothesis test o set of dt d obtied p-vlue of.44. If the experimeter hd coducted oe-tiled test o the sme set of dt, which of the followig is true bout the possible p-vlue(s) tht the experimeter could hve obtied? (A) The oly possible p-vlue is.. (B) The oly possible p-vlue is.44. (C) The oly possible p-vlue is.88. (D) The possible p-vlues re. d.78. (E) The possible p-vlues re. d.88. 3

33 Level ---- Number ) [7APStts, #5] 7.) [7APStts, #3] 8.) [7APStts, #5] 33

34 Level ---- Number ) [7APStts, #7].) [7APStts, #3].) [7APStts, #39] 34

35 Level ---- Number ---.) [7APStts, #37] 3.) [997APStts, #6] 35

36 Level ---- Number ) [997APStts, #] 5.) [997APStts, #6] 6.) [997APStts, #9] 36

37 Level ---- Number ) [997APStts, #34] 8.) [APStts, #] A mufcturer of blloos clims tht p, the proportio of its blloos tht burst whe iflted to dimeter of up to iches, is o more th.5. Some customers hve complied tht the blloos re burstig more frequetly. If the customers wt to coduct experimet to test the mufcturer's clim, which of the followig hypotheses would be pproprite? (A) Ho: p.5, H: p =.5 (B) Ho: p =.5, H: p >.5 (C) Ho: p =.5, H: p.5 (D) Ho: p =.5, H: p <.5 (E) Ho: p <.5, H: p =.5 9.) [APStts, #] The mger of fctory wts to compre the me umber of uits ssembled per employee i week for two ew ssembly techiques. Two hudred employees from the fctory re rdomly selected d ech is rdomly ssiged to oe of the two techiques. After techig employees oe techique d employees the other techique, the mger records the umber of uits ech of the employees ssembles i oe week. Which of the followig would be the most pproprite iferetil sttisticl test i this situtio? (A) Oe-smple z-test (B) Two-smple t-test (C) Pired t-test (D) Chi-squre goodess-of-fit test (E) Oe-smple t-test 37

38 Level ---- Number ---.) [APStts, #4] A cosultig sttistici reported the results from lerig experimet to psychologist. The report stted tht o oe prticulr phse of the experimet sttisticl test result yielded p-vlue of.4. Bsed o this p-vlue, which of the followig coclusios should the psychologist mke? (A) The test ws sttisticlly sigifict becuse p-vlue of.4 is greter th sigificce level of.5. (B) The test ws sttisticlly sigifict becuse p =.4 =.76 d this is greter th sigificce level of.5. (C) The test ws ot sttisticlly sigifict becuse times.4 =.48 d tht is less th.5. (D) The test ws ot sttisticlly sigifict becuse, if the ull hypothesis is true, oe could expect to get test sttistic t lest s extreme s tht observed 4% of the time. (E) The test ws ot sttisticlly sigifict becuse, if the ull hypothesis is true, oe could expect to get test sttistic t lest s extreme s tht observed 76% of the time..) [APStts, #35] I test of the hypothesis H o : = versus H : >, the power of the test whe = would be gretest for which of the followig choices of smple size d sigificce level? (A) =, =.5 (B) =, =. (C) =, =.5 (D) =, =. (E) It cot be determied from the iformtio give..) [APStts, #39] As lb prters, Slly d Betty collected dt for sigificce test. Both clculted the sme z-test sttistic, but Slly foud the results were sigifict t the =.5 level while Betty foud tht the results were ot. Whe checkig their results, the wome foud tht the oly differece i their work ws tht Slly used twosided test, while Betty used oe-sided test. Which of the followig could hve bee their test sttistic? (A).98 (B).69 (C).34 (D).69 (E).78 38

39 Level ---- Number --- Aswers.) B. Smple size is smll d compre gretess of two mes (oe-side)..) D. Assume tht the stdrd re equl. For the smple size d ukow stdrd devitio, use the two-smple me t test. Use Ti-84 Stts->TESTS->-SmpTTest. Or 6, x 9.54, s 3.68, 7, x 3.39, s 4.93, The degrees of freedom is k s pooled ( ) s ( ) s (6 )(3.68) (7 )(4.93) SE[ X Y] s pooled x x t.53 SE[ X Y ].5, p T vlue ) D. 4.) B. 5.) D. The probbility o ech outside the CI is.. So depedig o H testig greter or less th, the pvlue could be. or ) C. 7.) D. The degrees of freedom re k mi{, } mi{3,3 } 39 The pvlue 39 T ) C. x is too lrge whe is smll. 9.) B. To ot iclude zero, or to reject the ull hypothesis, P vlue.56, regrdless if it is two-sided, the cofidece level should be , d hece 94% is the swer..) B. The vribility is reduced by oe-hlf : ew old. Hece, 5 the CI is lso reduced by oe-hlf: z ew, z ew z old, z old..) A. 3 39

40 Level ---- Number ---.) A. pvlue ) A 4.) D. 5.) E. 6.) A. pvlue.7.. At lest 98% cofidece tht the test is sigifict. 7.) A. x 94.3,,, 7 x 94.3 z.7 / 7 pvlue ( Z z ) ( Z.7).6. 8.) B. 9.) B. The two-smple me test whe the vrice is ukow..) D..) C. Icrese smple size d icrese.) A. 4

41 Level ---- Number Chi-Squre ( ) Test From the DeMoivre-Lpce Theorem, whe p p, m p b(, p,m) pq pq where m is the observed umber of successes i trils, the probbility of success is p, d q = p. So, the rdom vrible m p pq c be cosidered s the ormlized rdom vrible. Squrig both sides of the equtio gives m p pq p qm p pq ( ) ( m p) p c m E[ X ] m E[ X ] p m p q m q pq m q q c E[X] E[ X ] Therefore, Perso's cumultive test sttistic, which symptoticlly pproches distributio, is: Chi-Squre tests re used i three wys: observed ected exp exp ected.) The goodess-of-fit test for uivrite..) Ctegoricl homogeeity with bivrite ctegorize first, d the collect smples. 3.) Ctegoricl idepedece with bivrite collect smples first, the ctegorize. Mthemticlly, the lst two tests re the sme. The tests re oe-sided d right-tiled. 4

42 Level ---- Number --- The smple size must be lrge eough tht the followig coditios re met:.) No expected couts re less th oe..) All of expected couts should be o less th five d if they re ot, 3.) o more th % of the expected couts re less th 5. Exmple A club t THS sells douts for fudrisig. It sells pli, strwberry, blueberry, d cimo douts. The members of the club woder if there is preferece for oe of these types of douts or if ech type is preferred by the sme proportio of the studets. Suppose smple of 6 sells is give below. The tble etries re observed frequecies or couts. Assume tht the sigifict level is.5. ( The goodess-of-fit test for uivrite) Pli Strwberry Blueberry Cimo Observed Cout Solutio: Step : Ech colum hs more th 5 etries, so, the coditios re stisfied for this uivrite goodess-of-fit chi-squre test. The expected couts re p p p p 6(.5) 5: p s b c Pli Strwberry Blueberry Cimo Observed Cout Expected Cout df Step - Step 3: H : p pp ps pb pc.5 4 H : Not ll proportios equl to.5 Step 4: Test Sttistic: 4

43 Level ---- Number --- Step 4: observed exp ected exp ected I Ti-84, F upper df cdf lower upper df, (,, ) p F vlue, Sice vlue p,there is isufficiet evidece to reject the ull hypothesis. Tht is, there is ot covicig evidece tht the four types of douts re ot equlity preferred. Exmple A SRS of 9 THS studets ws collected to ctegorize themselves s liberl, moderte or coservtive. A two-wy tble with resultig dt is give below: Liberl Moderte Coservtive Totl Mles Femles 6 58 Totl Do these dt provide evidece of ssocitio betwee politicl philosophy d geder t THS? Assume the sigifict level is. d the smple ws collected SRS without ctegorized first by geder. (The test for Idepedece) Solutios: Sice there is more th oe row d the dt were collected before ctegorized, so the test is for idepedece. Step : Use row Expected cout colum tble totl totl totl 43

44 Level ---- Number --- to fid the expected couts: Liberl Moderte Coservtive Mles 3(36) Femles df (# of rows ) # of colums ( )(3 ). All dt re SRS d the expected couts re greter th five, so the coditios re stisfied. Step Step 3: H : Geder d politicl ffilitio re idepedet. H : Geder d politicl ffilitio re depedet. Step 4: Test sttistic: observed exp ected exp ected Step 5: p F (.5, ).34 vlue Sice the pvlue is greter th the sigifict level, there is ot eough evidece to reject the ull hypothesis. Tht is, o reltioship betwee geder d ffilitio c be cocluded. 44

45 Level ---- Number --- Exmple [6APSttsFRQ #] Solutio:.). Sice there re more th two thigs to determie d the smple dt were collected fter studets were ctegorized, the proper test is chi-squre test for homogeeity. Assume tht H : o differece due to ds H : there is differece due to ds Ad df (3) () Group/Type/Me Choco-Zuties Apple-Zuties Totl A 56(5) 9(5) B 56(5) 9(5) C 56(5) 9(5) Totl

46 Level ---- Number --- The sttistic is observed exp ected exp ected The p vlue is p F (.9, ).58 vlue The p vlue is less th the sigifict level give, so the ull hypothesis is rejected. Tht is, the differece betwee with ds d without ds is sigifict. b.) Cse A d C re ot much differet i choosig Choco-Zuties ( vs. ), so 5 5 ruig Choco-Zuties ds did ot improve much, but Apple-Zuties ds seems to mke impct ( 3 vs. 5 5 ). Exmple [4APSttsFRQ #] 46

47 Level ---- Number --- Solutio:.) p o d 33 p off b.) Percetge wise, more people re ot exercisig d bout the sme percetge of people who hve two or more ctivities. c.) Sice the p-vlue of.3 is greter th 5% of orml sigifict level, the ull hypothesis c ot be rejected. Tht is, we c ot sy tht there is ssocitio betwee residetil sttus d level of prticiptio i ctivities. Exmple [3APSttsFRQ #4] The Behviorl Risk Fctor Surveillce System is ogoig helth survey system tht trcks helth coditios d risk behviors i the Uited Sttes. I oe of their studies, rdom smple of 8,866 dults swered the questio Do you cosume five or more servigs of fruits d vegetbles per dy? The dt re summrized by respose d by ge-group i the frequecy tble below. 47

48 Level ---- Number --- Do the dt provide covicig sttisticl evidece tht there is ssocitio betwee ge-group d whether or ot perso cosumes five or more servigs of fruits d vegetbles per dy for dults i the Uited Sttes? Solutio: Averge Yes No (97) or older observed expected expected df ()(3 ), p F (8.983, ).. The ull hypothesis is vlue rejected. Tht is, the smple dt provide strog evidece tht there is ssocitio betwee ge group d cosumptio of fruits d vegetbles for dults i the Uited Sttes. 48

49 Level ---- Number --- Quiz - Test.). [APStts, #] The followig two-wy tble resulted from clssifyig ech idividul i rdom smple of residets of smll city ccordig to level of eductio (with ctegories "ered t lest high school diplom" d "did ot er high school diplom") d employmet sttus (with ctegories "employed full time" d "ot employed full time"). Ered t lest high school diplom Did ot er high school diplom Employed full Not employed Totl time full time Totl If the ull hypothesis of o ssocitio betwee level of eductio d employmet sttus is true, which of the followig expressios gives the expected umber who ered t lest high school diplom d who re employed full time? (A) (B) (C) (D) (E) ) [APStts, #9] A geeticist hypothesizes tht hlf of give popultio will hve brow eyes d the remiig hlf will be split evely betwee blue- d greeeyed people. I rdom smple of 6 people from this popultio, the idividuls re distributed s show i the tble below. Wht is the vlue of the sttistic for the goodess of fit test o these dt? Brow Eyes Gree Eyes Blue Eyes (A) Less th (B) At lest, but less th (C) At lest, but less th (D) At lest, but less th 5 (E) At lest

50 Level ---- Number --- Aswers rowtotl columtotl 9(8).) B. From Expected cout. tbletotl 57.) B. From the give, the expected vlues re 3 for brow eyes, 5 for gree eyes d 5 for blue eyes. observed exp ected exp ected

51 Level ---- Number --- The followig is smple R code. 5

52 Level ---- Number ANOVA Alysis of vrice (ANOVA) is method for testig the hypothesis tht there is o differece betwee two or more popultio mes (usully t lest three). Assumptios All popultios ivolved follow orml distributio. All popultios hve the sme vrice (or stdrd devitio) The smples re rdomly selected d idepedet of oe other. Exmple 5... Suppose the Ntiol Trsporttio Sfety Bord (NTSB) wts to exmie the sfety of compct crs, midsize crs, d full size crs. It collects smple of ech of the tretmets (cr types.) Usig the hypotheticl dt provided below, test whether the me pressure pplied to the driver s hed durig crsh test is equl for ech types of cr. Use. 5. Pressure Compct crs ( r ) Midsize crs ( r ) Full-size crs ( r 3 ) Smple # Smple # Smple # Me x j Stdrd Devitio Solutio: Step : Assume H : 3 : or 3 or from the rest. H 3 Step : Clculte the squres. Tht is, t lest oe of the mes is differetly Totl umber of smples: N 9 Number of rows: r 3 Number of colums: c 3 Degree of freedom: df c 3, df N c 93 6 Grd me: 5

53 Level ---- Number --- x r, c i, j N x ij Totl Sum of Squres: SST, r c i, j x ij x Tretmet Sum of Squres (Betwee Sum of Squres): r j j j [ ] [ [ SSTR r x x Error Sum of Squres (Withi Sum of Squres): rc, i, j j i, j SSE x x Note tht SST SSTR SSE Totl Me Squres (verge totl): MST SST N (9 ) Me Squre Tretmet (verge betwee): SSTR MSTR c (3 ) 53

54 Level ---- Number --- Me Squre Error (verge withi): MSE SSE N c (9 3) Note tht MST MSTR MSE. Step 3: Clculte the test sttistic -- rtio of betwee vritio to withi vritio. If the verge betwee vritio rises reltive to the verge withi vritio, the F sttistic will rise d so will our chce of rejectig the ull hypothesis. MSTR F 5.7 MSE 79 p Sice df, df 6d. 5, F,6 5.7 p., d the criticl vlue.5 F, So, the ull hypothesis is rejected. (I Ti 84: Fcdf(5.7,,,6)) ANOVA i R 54

55 Level ---- Number --- Appedices A. Q - Q PLOT A populr plot, the qutile-qutile plot or Q-Q plot, is bsed o qutile fuctios for two distributios. The plots show how cogruet the shpes (how close i probbilities) re, by exploitig how well the qutile fuctios ( the vlues of rdom vribles) re lierly relted... Costructio of the Q-Q Plot Step : Sort the two sets of dt. The size for the qutile fuctios is the miimum size of dt of both sets, if the size of either set if both sets hve the sme umbers of dt. Step : Compute the qutiles fuctios Q ( p ) d Q ( p ) (the iverse CDF fuctios i. 5 of the rdom vribles), where p i (cosider p i s the CDF fuctio), for i,,...,. If oe of the dt sets is stdrd distributio such s the orml distributio, use the iverse CDF s the qutile fuctios. Step 3: Plot the poits Q p ), Q ( p ), for i,,...,, s obtied i Step... Lierity of the QQ Plot ( i i i i For two orml distributios X N(, ), Y N(, ), if Z ~ N(, ) d ( z) P( Z z), the for the give p P p P Y ~ th p qutile, X ~ Q ( p) Q ( p) X Q ( p) P Y Q ( p) Q ( p) Q ( p) P Or Q ( p) ( p) Q ( p) ( ) p 55

56 Level ---- Number --- Now, it c esily be see tht the qutile fuctios of the two orml distributios Q ( p) d Q ( p) lierly relted: Q ( p) Q ( p) Compre to the orml distributios, the plot is cocve up if the distributio of the dt set is skewed to the left, d skewed to the right if the plot is cocve dow. A.. A Exmple of Q-Q Plot i R 56