Introduction to mathematical Statistics

Transcription

1 Itroducto to mthemtcl ttstcs Fl oluto. A grou of bbes ll of whom weghed romtely the sme t brth re rdomly dvded to two grous. The bbes smle were fed formul A; those smle were fed formul B. The weght gs tted from brth to ge s moths were recorded for ech bby. The results were s follows: mle : mle : (. Plese costruct 95% cofdece tervl for the me dffereces weght gs betwee the two formuls. d, K, (, d (b. Let N d Y,, ~ (, K Y N be two deedet smles. Furthermore, s ukow. Plese derve the geerl (-α% cofdece tervl for. *Plese clude detled dervtos d roofs for full credt. oluto: ( Iferece o two oulto mes. Two smll d deedet smles. Formul A (smle : 7.33, s.58, 9 Formul B (smle : 8.67, s.58, 9 [] Uder the ormlty ssumto, we frst test f the two oulto vrces re equl. Tht s, : >. The test sttstc s s.58 F, F8,8,.5, U s ce F < 3.44, we cot reject. Therefore t s resoble to ssume tht. [] The 95% C. I. for the me dfferece s : versus ± ( *.58 / 9 + ± t6,.5 s where s ( s + ( s.58 + Therefore 95% C.I. s [-.9,.4]. (b Iferece o two oulto mes, deedet smles: Pooled vrce t: Both smles re from orml oultos. Furthermore, we ssume the oulto vrces re ukow but equl, tht s: Prmeter of terest Pot estmtor for the rmeter of terest Y Y ~ N(, + ~ N(, + ere oe should derve ths dstrbuto usg the momet geertg fucto method, or other method. More ttstcs tutorl t

2 Y ( ~ N (, + 3 Z Z s ot votl qutty for ( sce s ukow. ere oe should derve ths dstrbuto usg the momet geertg fucto method, or other method. 4 ( W ~ χ deedet ( W ~ χ W W ~ + W χ + ere oe should derve ths dstrbuto usg the momet geertg fucto method, or other method. 5 W, W,, d Y re deedet. Thus, W d Z re deedet. By the defto of the T-dstrbuto, we hve: T where Z W + ~ t Y ( + + ( + ( + s clled the ooled-vrce. Y ( Therefore the votl qutty s T ~ t + 6 ( α % cofdece tervl for ( : + ( + + P t T t, /, / α α α P Y t +, α + Y + t +, α + α Thus the cofdece tervl s: Y ± t +, α + More ttstcs tutorl t

3 . Durg oe of the beer wrs the erly 98 s, tste test betwee chltz d Budweser ws the focus of TV commercl. eole greed to drk umrked mugs d dcte whch of the two beers they lked better. The results: ffty-four chose Bud whle the rest chose chltz. (. Plese costruct d terret the corresodg 95% cofdece tervl for - the roorto of beer drkers who refer Bud to chltz. (*Note: Plese derve the geerl formul for the (-α% cofdece tervl for oulto roorto bsed o lrge smle frst. (b. ow lrge does the smle sze eed to be order for the smle roorto ˆ to hve 95% chce of lyg wth.3 of? Plese clculte the smle sze for two sceros: ( we hve o estmte for ; (b we hve estmte for ˆ.54 (*Note: Plese frst derve the geerl formul for smle sze clculto bsed o mmum error of E d cofdece level of (-α%. oluto: ( d... Let ~ Beroull(,, L,, lese fd the (-α% CI for. Pot estmtor : ˆ (e., ˆ.6 Our gol: derve (-α% C.I. for E( By the cetrl lmt theorem (CLT, for lrge smle, we hve Z ~ & N (, Vr( E( E E(, ( Q (, ~ B ( Vr( Vr Vr( Z ~ & N(, ( ˆ Z ~ & N(, ( ˆ Z ~ & N (, (By lustky s theorem 垐 ( # ( α % (romte, or lrge smle C.I. for P( Zα/ Z Zα/ α ˆ > P( Zα/ Zα/ α 垐 ( 垐 垐 > ( ˆ P Z / Z / α α α More ttstcs tutorl t 3

4 垐 ( 垐 ( P( 垐 Z + Z α α α > / / 垐 ( 垐 ( P( 垐 + Z Z α α α > / / 垐 ( 垐 ( P( 垐 Z + Z α α α > / / ˆ( ˆ ˆ( ˆ > The ( α % lrge smle C.I. for s [ ˆ ˆ /, Zα + Zα/ ]. # CLT > lrge usully mes 3 # secl cse for the ferece o. lrge mes Let, lrge smle mes: ˆ 5 ( totl # of, d ( ˆ 5 (- totl # of F For the gve roblem, we hve, 54, d we wt 95% CI for α For 95% cofdece tervl, α.95, α.5,.5 54 ˆ.54 ; Z.5.96 ˆ( ˆ (.54( ˆ( ˆ Z The 95% cofdece tervl for s [.444,.636 ] (b Derve the geerl formul for P( ˆ E α P( E ˆ E α E 垐 E P( α d Z ~ & N(, 垐 ( 垐 ( 垐 ( 垐 ( E E Thus : P( Z α 垐 ( 垐 ( More ttstcs tutorl t 4

5 E c Zα 垐 ( ( Z 垐 α ( ( Zα E 4 E Plug Z.96,.5, E.3, we hve 68.5 ˆ Plug Z.96,.54, E.3, we hve 6.5 ˆ 3. To test whether the brth rtes of boys d grls re equl, rdom smle of fmles wth ectly three chldre wth the ge of 8 ws tke d the dstrbuto of the chldre s geder s rovded below. Plese test whether the brth rtes re equl or ot t the sgfcce level of α.5. 3 grls grls, boy grl, boys 3 boys No. of fmles oluto: Ths roblem c be doe two wys usg ether ( the test o oe oulto roorto or ( the Ch-squre goodess-of-ft test wth four ctegores. ( For the frst roch, ferece o oe roorto, lrge smle, we hve 3. Let be the totl umber of grls mog the 3 chldre, we hve 5. Let be the roorto of etrereeurs wth domestc crs, we hve 5 ˆ, d we re testg: versus 3 :. 5 :. 5. ˆ.5 The test sttstcs s: Z /3 ce Z.365 <.96 Z.5, we c ot reject the ull hyothess of equl brth rtes t the sgfcce level of.5. ( Altertvely, d equvletly, you c use the Ch-squre goodess-of-ft test. The bove tble s smly the followg four-ctegory tble: 3 grls grls, boy grl, boys 3 boys More ttstcs tutorl t 5

6 No. of fmles Let Y be the umber of grls 3-chldre fmly, uder the ull hyothess of equl brth rte (chce of gvg brth to 3 grl, t ech brth, s /, we hve (/ ( / 3 P,,,, 3 Let,,, 3 4be the roortos of fmles fll to these 4 ctegores resectvely, we re testg: : P( 3 / 8, P( 3/ 8, P( 3/ 8, P( / versus : s ot true. ece we hve e e4 */ 8, 5 e e3 *3/ The test sttstc s: W ( e.667 < χ3,.5, uer e Therefore we c ot reject the ull hyothess t the sgfcce level of.5. Of course you oly eed to show oe of the two roches bove to get full credt. 4. Let,, K, ~ N(, vrce s ukow. be rdom smle from the gve orml oulto, d furthermore, the ( Plese derve the test for : versus : t the sgfcce level of α usg the votl qutty roch. (Plese clude the dervto of the votl qutty, the roof of ts dstrbuto, d the dervto of the rejecto rego for full credt. (b Plese derve the lkelhood rto test for : versus : t the sgfcce level of α d show tht ths test s detcl to the test derved rt (. oluto: (. [] Frst we derve the votl qutty d ts dstrbuto. Pot Estmtor for : ~ N(, ; s NOT votl qutty sce s ukow. The we cosder Z ~ N(, ; Ths s lso NOT votl qutty sce s ukow. W Z T ~ t T ( Z d W re deedet. W ( By the: Theorem. mle from orml oulto Z ~ N(,, we kow Ad by the: Defto. T s votl qutty for ~ χ [] Net we derve the oe-smle t-test d ts rejecto rego. For -sded test of : versus :, the test sttstc s the votl qutty t, tht s, T. Itutvely, we would reject fvor of f T c. The roblem s how to determe c. By / the defto of the sgfcce level, we hve α P reject_ P T c P T c Thus ( ( ( / PT c d subsequetly we hve α ( c t, α / More ttstcs tutorl t 6

7 Tht s, t the sgfcce level α, we reject fvor of f T t., α / (b. For -sded test of : versus :, whe the oulto s orml d oulto vrce s ukow, we ow derve the lkelhood rto test. [] Wrte dow your rmeter sce uder {(, :, } ω f [] Wrte dow the urestrcted/orgl rmeter sce. {(, : R, } f [3] Wrte dow the lkelhood (of the dt L f(,, L, ; f( ; [4] Wrte dow your log-lkelhood. ( l l L l( π [5] Fd MLEs uder ω d lug to get m L ω ( dl + 4 d ( ˆ ω m L L(,, L, ;, ω ω ˆ ( ( ( π e ( π ( e [6] Fd MLEs uder d lug to get m L e π ( ( π e ( More ttstcs tutorl t 7

8 4 dl d dl d + ˆ ˆ ˆ ˆ m (,,, ;, L L L e π e π [7] Get the lkelhood rto m m L LR L ω [8] Derve the decso rule bsed o sgfcce level α (Reject s true α P? ( : More ttstcs tutorl t 8 PLR c ( P c : Recll t-test sttstc : ~ T t, t sgfcce level α, we reject fvor of f, T t α ( : P c ( : P c

9 ( + * ( c : ( P ( P + c * ( : ( f T t, α At α, we reject ( + ( ( + ( * ( : ( P c PT ( c : ** PT ( c : ** The LR test s equvlet to the t-test. 5. uose we hve two deedet rdom smles from two orml oultos:,, K, ~ N(, Y, Y, K, Y ~ N (,, d. Furthermore, s ukow. At the sgfcce level α, lese costruct test to test whether 3 + 4or ot. (*Plese clude the dervto of the votl qutty, the roof of ts dstrbuto, d the dervto of the rejecto rego for full credt. oluto: Gve tht d thus. ere s smle outle of the dervto of the test: : 3+ 4 versus : 3 + 4, whch re equvlet to: : 3 4versus : 3 4 ( We strt wth the ot estmtor for the rmeter of terest( 3 N( 3, / + 9/ usg the mgf for (, deedece roertes of the rdom smles. From ths we hve : ( 3Y. Its dstrbuto s N whch s M ( t e( t + t / ( 3Y ( 3 Z, d the ( / + 9/ ~ N,. Ufortutely, Z c ot serve s the votl qutty becuse s ukow. (b We et look for wy to get rd of the ukow followg smlr roch the costructo of the ooledvrce t-sttstc. We foud tht W ( ( / ~ χ + + usg the mgf for χ k whch k / s M ( t, d the deedece roertes of the rdom smles. t (c The we foud, from the theorem of smlg from the orml oulto, d the deedece roertes of the rdom smles, tht Z d W re deedet, d therefore, by the defto of the t-dstrbuto, we hve 3Y 3 obted our votl qutty: T ( ( + * / ( + 9/ + ~ t +. More ttstcs tutorl t 9

10 (d The rejecto rego s derved from P ( T c α, where T ( Y + ( * / ( + 9/ + sgfcce level of α, we reject fvor of ff ~ t T t +, α /. Thus c t +, /. Therefore t the α 6. Let,, K, be deedet d detclly dstrbuted wth df ( ; ( f θ θ θ, < θ <,, (. Derve the method of momet estmtor for θ (b. Derve the mmum lkelhood estmtor for θ (c. Is there effcet estmtor for θ? Plese show the etre dervto. t: Crmér-Ro Iequlty: Let ˆ θ h(,, L, be ubsed forθ, where,, L,, smle from oulto wth df f ( ; θ stsfyg ll regulrty codtos. The ( ˆ l f ; θ l f ; θ Vr θ E E θ θ s rdom oluto: P ( f( ; θ θ ( θ ;, ; (. The oulto me s θ (becuse E( * θ *( θ θ Therefore the momet estmtor of θ s ˆ θ (b. L f( ; θ θ ( θ θ ( θ l l L l θ + ( l( θ l θ θ θ + d the smle me s.. ˆ s the MLE for θ θ (c. E( ˆ θ θ More ttstcs tutorl t

11 ˆ θ ( θ vr( θ Now we derve the C-R lower boud for ubsed estmtor of θ : P ( f( ; θ θ ( θ ;, ; l f( ; θ l θ + l( θ l f( ; θ θ θ θ l f( ; θ θ θ ( θ θ θ E θ ( θ θ ( θ θ( θ C-R lower boud ˆ θ ( θ vr( θ l f E θ The MLE of θ s ubsed d ts vrce C-R lower boud. Thus t s effcet estmtor of θ. Defto. Effcet Estmtor If ˆ θ s ubsed estmtor of θ d ts vrce C-R lower boud, the ˆ θ s effcet estmtor of θ. More ttstcs tutorl t