STATISTICAL INFERENCE

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1 (STATISTICS) STATISTICAL INFERENCE COMPLEMENTARY COURSE B.Sc. MATHEMATICS III SEMESTER ( Admsso) UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION CALICUT UNIVERSITY P.O., MALAPPURAM, KERALA, INDIA

2 School of Dstace Educato UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION STUDY MATERIAL B.Sc. MATHEMATICS ( Admsso owards) III SEMESTER COMPLEMENTARY COURSE (STATISTICS) STATISTICAL INFERENCE Prepared by: Scrutsed by: Dr.K. ANEESH KUMAR Departmet of Statstcs Mahatma Gadh College, Irtty Keezhur.P.O. Kaur Dr. ANIL KUMAR.V. Reader & Head Departmet of Mathematcs Uversty of Calcut Layout & Settgs: Computer Cell, SDE Reserved STATISTICAL INFERENCE Page

3 School of Dstace Educato CONTENTS. SAMPLING DISTRIBUTIONS 5 9 Samplg dstrbutos Ch-square dstrbuto Studet s t-dstrbuto Sedecor s F-dstrbuto. THEORY OF ESTIMATION-I (POINT ESTIMATION) 3-5 Statstcal Iferece Pot estmato Desrable propertes of good estmator Methods of estmato 3. INTERVAL ESTIMATION 5 68 Iterval estmato Cofdece terval for the mea of a ormal populato Cofdece terval for the dfferece of meas Cofdece terval for the varace of a ormal populato Cofdece terval for large samples 4. TESTING OF HYPOTHESIS Statstcal hypothess Testg of hypothess Errors testg of hypothess Steps testg of hypothess Most powerful test Neymaa-Pearso theorem 5. LARGE SAMPLE TESTS 9 4 Large sample tests Testg mea of a populato Testg the equaltes of meas of two populatos Testg the populato proporto Testg the equalty of proporto of two populatos Goodess of ft test of depedece 6. SMALL SAMPLE TESTS 5 39 Small sample tests Tests based o ormally dstrbuted test statstcs Tests based o statstcs followg t dstrbuto Tests based o statstcs followg - dstrbuto Tests based o statstcs followg F - dstrbuto STATISTICAL INFERENCE Page 3

4 School of Dstace Educato STATISTICAL INFERENCE Page 4

5 School of Dstace Educato CHAPTER SAMPLING DISTRIBUTIONS.. Samplg Dstrbuto The characterstcs of a large group of dvduals are studyg ay statstcal vestgato. Ths group s referred to as the populato uder vestgato. Let us study oly oe characterstc, say X the heght of the dvduals the populato. Correspodg to each dvdual of the populato we get a umber deotg magtude of the characterstc cosdered. The set of such umbers are called the statstcal populato. Ths statstcal populato s cosdered as the set of admssble values of the varable X (here heght). The dstrbuto of the values of X s kow as the dstrbuto of ths statstcal populato (or smply sayg populato ow owards). Ay fucto of the statstcal populato values are called populato parameter. For eg., mea, varace, meda etc., of the varable cosdered. The process of makg ferece about the populato based o samples take from the populato s kow as statstcal ferece. Cosder a radom sample take from a populato the the fucto of sample values lke sample mea, sample varace sample momets etc., are kow as statstc. The dstrbuto of statstc s kow as the samplg dstrbuto of that statstc. Samplg dstrbuto of meas of radom samples take from a ormal populato N(,) : Let X, X,..., X be the radom samples take from N(,). The, X X... X Sample mea X mgf of X, M ()()() t M t M X X X... X X X... X t t t t M X ()()...()( M '.) X M X X s are d STATISTICAL INFERENCE Page 5

6 We have for X~ N(,), Therefore, School of Dstace Educato t t M () t e X t t t t t t M () t e. e... e X t t t () t e e, t s the m.g.f. of a ormal populato wth parameters ad. Ths mples that N, Problem : A radom sample of sze 5 s take from a ormal populato wth mea ad varace 9. What s the probablty that the sample mea s egatve? Soluto: Gve sample sze = 5, ad 3. We have X ~( N,). The requred probablty = P( X ) = P < = P < /, where Z = = P (Z < -.67) = P(Z >.67) = P (Z > ) P ( < Z <.67) N(, ) ( Stadard ormal curve s symmetrc about the y-as) = (from stadard ormal table) =.475 ======== Problem : A radom sample s take from a ormal populato wth mea ad varace 9. How large a sample should be take f the sample mea s to le betwee 9 ad wth probablty.95? STATISTICAL INFERENCE Page 6

7 Soluto: School of Dstace Educato To fd the mmum umber of samples so as P(9 X ).95 Let samples are take from the gve populato N (, 3), the we have, X ~(,) N 3 X e., Z ~(,) N 3 9 X P(9 X ).95().95 P For Z ~ N (,), e.,() P.95 Z 3 3 e.,() P.475 Z P() Z.475 happes whe 3 So the mmum umber of sample for the requred probablty codto s Ch-square dstrbuto ( -dstrbuto) A radom varable X wth pdf, f () e,, otherwse s sad to follow dstrbuto wth degrees of freedom, where s the parameter of the dstrbuto, deoted by X ~(). Momet geeratg fucto of X ~() : Observe that, M ()() t E e tx X STATISTICAL INFERENCE Page 7

8 Ths mples that Mea ad Varace: Observe that, School of Dstace Educato e t e d () t e d. ( = d ) t t M (t) = ( ) d E()() X M t dt X / t = ( ) t = = ( ) t = =. Hece, E(X) = Moreover, E(X ) = ( ) t = = ( ) = ( + ). Hece, V(X) = (+) = t = Addtve property of dstrbuto: Let X ad X are two depedet radom varables where X ~() ad X ~(). The X X follow ch-square dstrbuto wth degrees of freedom. STATISTICAL INFERENCE Page 8

9 School of Dstace Educato Proof: Observe that X X ~() M () t t X ad ~() M () t t X. Therefore, M ()()() t M t M t t t X X X X t Ths s the m.g.f. of a radom varable wth degrees of freedom. Hece X X follow ch-square dstrbuto wth degrees of freedom. I geeral f, X, X, Xk are depedet radom varables wth,, k degrees of freedom respectvely, the X+ X + Xk follow ch-square dstrbuto wth + + k degrees of freedom. Tables of ch-square dstrbuto: Ch square table gves the values of () for a varable wth varous degrees of freedom ad for varous values of, such that P (. () ) Theorem: If X ~ N (, ), the Y X follow ch-square dstrbuto wth oe degree of freedom. Proof: Observe that M ()()() t M t E e Y X tx t e e d STATISTICAL INFERENCE Page 9

10 School of Dstace Educato t t e d e d () e t s a eve fucto put u d du du u e () t d. u() t du M Y () t e u u() t e u du. () t But Ths mples that My(t) = ( ). ( )t Hece Y. s the m.g.f. of a ch-square radom varable wth oe degree of freedom. Theorem: If X, X,..., X are radom samples take from a stadard ormal populato, the the sum of squares of radom sample follows Proof: Sce X ' s are radom samples from N (, ), they are depedet ad X ~() for all. If X follows N (, ), we have X ~(). Here X ~() for all. Hece, by addtve property of ch-square dstrbuto, the sum of squares of,,..., X X X follow. Theorem: If X, X,..., X are radom samples take from N(,), Show that Y ~() STATISTICAL INFERENCE Page

11 Proof: School of Dstace Educato Gve X, X,..., X are depedet samples from N(,). The,,... N (, ) we get,,,..., each follow () Sce X, X,..., X are depedet, by addtve property of ch-square dstrbuto, ~(). Problem : If X, X,..., X are radom samples take from N(,), fd the dstrbuto of sample varace Soluto: S. For the radom samples X, X,..., X take from N(,), we have Y ~(). Note that Y ca be wrtte as Y X X, where X s the sample mea e., Y ()() X X ()() X()() X X X X X X () () X ()() () X () X (() ) X () X X S X X where ~(,) N STATISTICAL INFERENCE Page

12 School of Dstace Educato Observe that N(, ) ad X follow ch-square dstrbuto wth oe degree of freedom. Sce X ad S are depedetly dstrbuted, the by addtve property of ch-square dstrbuto S ~( ). Let U = S ~( ) () The, f () u u e u, u, otherwse Equato () ca be wrtte as S = S, the, f( S )= f(u) terms of u, therefore f (S ) = f(u) du ds S f () S du e ds S S e S S. Hece, S f () S e S, S Problem : If Soluto: X ~(), fd the mode of X. We have f () e STATISTICAL INFERENCE Page

13 School of Dstace Educato Mode s the pot where f() attas ts mamum. That s, the pot, where f ' () () ad f ''. ' () f e e ' f e e () e e For the value of = -, f '' (). Hece the mode s at = -. Problem 3: If X s dstrbuted as follows ch-square dstrbuto wth degrees of freedom. Soluto: Let Y log() e. Ths ca be wrtte as = Note that = f(y) = f() terms of y. d dy =.. y e = Ths mples that Y X() y e y f (),, show that log() e y f () y e y, y ; STATISTICAL INFERENCE Page 3.

14 School of Dstace Educato Problem 4: For large, show that ch-square dstrbuto appromately ormally dstrbuted. Soluto: Cosder a radom varable X followg ch-square dstrbuto wth degrees of freedom. The M () t t X. More over, E() X ad V(X) =. Cosder Z X, the t ()()() t M t M t e M Z X X t t e. Therefore, t log() M log t e Z t t t log t t t... t t t... + (may terms volvg hgher power deomator) ad ts As become very large, log() M Z t t. That s M () t e Z t. Ths s the m.g.f. of a stadard ormal radom varable. The by uqueess theorem of m.g.f., X~N(, ) for large. STATISTICAL INFERENCE Page 4

15 School of Dstace Educato Problem 5: For a radom sample of sze 6 from N(,) populato, the sample varace s 6. Fd a ad b such that P() a.6 b Soluto: P() a.6 P a b b mples ().6 P() S.6, S S ~ where S ( ). a b Puttg S =6 ad = 6 the above equato, we get That s, a (6) b P() b (5) a P() () From the table of ch-square dstrbuto, P( ad (5).37).8 P(. (5) 9.3). Therefore, P( ) () (5) Comparg () ad (), we get ad b a Hece b 4.84, ad a e.,( P ).6..3 Studet s t-dstrbuto Ths s the probablty dstrbuto whch was troduced by W.S Gossset ad kow hs pe ame studet. A cotuous radom varable t wth desty fucto () t f t, t degrees of freedom. s sad to follow studet s t-dstrbuto wth STATISTICAL INFERENCE Page 5

16 Note that f X ~(,) N ady ~() School of Dstace Educato, the t X Y follows t-dstrbuto wth degrees of freedom, where X ad Y are depedet. Eamples of statstcs followg studet s t-dstrbuto:. Let X be the mea of radom samples take from N(,). Let S be the sample varace. The we have X ~( N,) S ad ~( ). X Note that ~(,) N. Therefore, X t S ~ t(. ) ( ) Ths mples that t () X ~ t(. ) S. Let X ad X be the meas, S ad S be the stadard devatos of samples of szes ad take depedetly from two ormal populatos wth same mea ad stadard devato. The, t X X S S ~ t ( ) Proof: We have X ~( N,) ad X ~( N,) The, X X ~(,) N. X X Ths mples that ~(,) N STATISTICAL INFERENCE Page 6

17 Also we have, S ~( ) School of Dstace Educato ad S ~( ). Usg addtve (reproductve) property of ch-square dstrbuto, we get S S ~( )( S S ad.) are d So, t X X S S ~ t ( ) That s, That s, X X t t X X S S S S ~ t ~ t ( ) ( ) Tables of t-dstrbuto: Note that t-dstrbuto s symmetrc about zero ad bell shapped. Tables of t- dstrbuto gves the values of t for varous degrees of freedom ad for varous value of, such that P( t ) t. Problem : If t ~ t ( ), fd the mode of t. Soluto: We have, () t f t, t STATISTICAL INFERENCE Page 7

18 School of Dstace Educato Mode s the pot t where f(t) attas ts mamum. f ' () t () ad f '' t. That s the pot t, where ' t t f () t. 3 t () ' f t t t At '' () t t f t t. '' at t,() f. Hece Mode of s at t =. Problem : If t ~ t ( ), the as, prove that t ~(,) N. Soluto: Gve t ~ t ( ), we have () t f t, t Usg the followg results we ca prove as, t ~(,) N () as becomes very large, k k () lm e Note that as STATISTICAL INFERENCE Page 8

19 f(t) School of Dstace Educato t e whch s the probablty desty fucto of stadard ormal dstrbuto. Problem 3: If t ~ t (5), fd a such that, P() a.98 t a Soluto: The graph of t ~ t(5) s symmetrc about zero. To fd a such that the area uder the t curve betwee a ad a s.98. e., to fd a such that P() t.98 a or P() t. a From table of t-dstrbuto, for 5 d.f. we get, a = Problem 4: Prove that the rato of two depedet stadard ormal radom varables s a studet s t radom varable wth degree of freedom. Soluto: Gve X ~(,) N, X ~(,) N ad they are depedet. The, X. Ths mples that ~ () t X X X That s, t X ~ t ~ t () () That s, the rato of two depedet stadard ormal varables follows t (). Problem 5: If X ad X X of t X X. are two depedet stadard ormal varables, fd the dstrbuto STATISTICAL INFERENCE Page 9

20 Soluto: School of Dstace Educato Gve X ~(,) N, X ~(,) N ad they are depedet. So, X ad X. ~ () () ~ By addtve(reproductve) property of ch-square dstrbuto, X X. ~ () The, t X X X ~ t (). X That s, t X X follow t-dstrbuto wth degrees of freedom. Problem 6: Fd the mamum dfferece that we ca epect wth probablty.95 betwee the meas of samples of szes ad from a ormal populato, f ther stadard devatos are foud to be ad 3 respectvely. Soluto: Let ad be the meas of the samples of szes = ad = take radomly from two ormal populatos. Assume samples are take depedetly. The sample varaces S = 4 ad S = 9 respectvely. To fd the value of k such that, P k.95. For samples take from two ormal populatos depedetly, we kow that, X X S S ~ t ( ) The to fd k such that, P X X k.95 S S S S STATISTICAL INFERENCE Page

21 School of Dstace Educato That s, k P t( ) That s, k P t() () The table of t- dstrbuto for d.f., P t() () k Comparg () ad (), we get,.86. Hece.65 k That s the mamum dfferece that ca epect wth 95% probablty s Sedecor s F-dstrbuto A cotuous radom varable F wth pdf f F F F (), (,) F s sad to follow F-dstrbuto wth (,) degrees of freedom. If X ad X are depedet radom varables followg ch-square dstrbuto wth ad degrees of freedom respectvely, the, F X X ~( F,) Statstc followg Sedecor s F-dstrbuto:. Let depedet samples of szes ad are take from ormal populato wth mea ad stadard devato. Let S ad S are the respectve sample varace, the S ( ) F ~( F, ) S ( ) STATISTICAL INFERENCE Page

22 Proof: School of Dstace Educato For the set of samples take from ormal populato, we have, S ~ ( ) ad S ~ ( ) The, S F S ~( F, ) Hece S ( ) F ~( F, ) S ( ) Tables of F-dstrbuto: Tables of F-dstrbuto gves the values of F for varous values of, ad, such that P() F, F. Mode of F-dstrbuto: Mode s the pot F where f(f) attas ts mamum. That s the pot F, where f ' () F () ad f '' F or the F where log() f F F ad log() f F F. We have, f () F (,) F F Therefore, log() f F log log log( F,) log F STATISTICAL INFERENCE Page

23 School of Dstace Educato log() f F.e.,.. F F F. F F log() f F. F F F F. Remark: at ths pot t ca be verfed that Hece mode of F ~( F,) s F = The mode log() f F F.. ca be epressed as. Sce F, the mode caot be egatve. Hece should ot be less tha. So the mode ests oly whe uty.. Aga sce Problem : ad are les tha, the mode s always less tha Prove that the rato of the squares of two depedet stadard ormal radom varables s a F- radom varable wth (, ) degree of freedom. Soluto: Let X ~(,) N, X ~(,) N ad they are depedet So, X ad X ~ () () ~ the X X ~(,) F X e., ~(,) F X. STATISTICAL INFERENCE Page 3

24 School of Dstace Educato Problem : If X s a radom varable followg F dstrbuto wth, Fd the dstrbuto of Y. X Soluto: Gve Y we have f(y) = f() terms of y. d X dy degrees of freedom. Here f (), (,) Y d, so X X Y dy y so,() f y y (,) y y y f () y y y (,) y y y (,) y () y y (,) STATISTICAL INFERENCE Page 4

25 School of Dstace Educato y (,) y f () y (,) y y (,)(,) Y ~( F,) Problem 3: If X followg F dstrbuto wth, degrees of freedom Y follow F dstrbuto wth, degrees of freedom. Prove that P ()() X c P Y. c Soluto: P ()() X c P X c But, gve, X ~( F,), the, ~( F,) X Hece, Also Y s a varable followg F(,). P ()() X c P Y c Problem 4: If X followg F dstrbuto wth, degrees of freedom. If, ( such that P ()() X Soluto: P X. Show that. ) are Gve P ()() X Sce X ~( F,), P ()() X X P X also ~( F,) P X ----() P()() P X X STATISTICAL INFERENCE Page 5

26 School of Dstace Educato P()() X P X (by ()) Problem 5: If t follows studet s t-dstrbuto wth degrees of freedom, prove that dstrbuto wth, degrees of freedom. t follows F Soluto: Gve t ~ t ( ), we have () t f t, t Let Y= t ; t Note that d t Y Therefore d y, f(y) = f(t) terms of y. dt dy Y That s, Y f () Y.., Y Y.. () Y f Y Y.. Y Y Y Y, m, m m STATISTICAL INFERENCE Page 6

27 School of Dstace Educato. Y f () Y, Y, ths s the desty,. Y fucto of a radom varable followg F-dstrbuto wth (, ) degrees of freedom. Hece Y t follows F (,). Problem 7: If X followg F dstrbuto wth,, Y= X follows ch-square dstrbuto wth degrees of freedom. Soluto: degrees of freedom, prove that as Gve Y= X, d dy ; f(y) = f() terms of y. d dy X F ; ~ (,) f (), (,) f () y y (,) y y STATISTICAL INFERENCE Page 7

28 School of Dstace Educato As k k as Note that Also ote that y y y lm y y y as, lm lm e y y lm e y Hece, as,() f y y y e y y e Hece, y f () y e. y, y That s Y follow ch-square dstrbuto wth degrees of freedom. STATISTICAL INFERENCE Page 8

29 School of Dstace Educato EXERCISES. Epla what s meat by samplg dstrbuto. State the relatoshp betwee ormal ad ch-square dstrbuto.. Defe ch-square dstrbuto wth degrees of freedom. Derve ts mea ad varace. 3. State ad prove the reproductve property of ch-square dstrbuto. 4. Show that for Studets t-dstrbuto wth degrees of freedom, the mea devato s gve by. 5. Defe F-dstrbuto. Epla ts use statstcal ferece. 6. State the ter-relatoshp of t, ch-square ad F dstrbutos. A radom varable X has F- dstrbuto wth (, m) degrees of freedom. Fd the dstrbuto ofy X. 7. Derve Studet s t- dstrbuto ad establsh ts relato wth F- dstrbuto. 8. If F has F-dstrbuto wth (, m) degrees of freedom, prove that as, F teds to be dstrbuted as ch-square wth degrees of freedom. 9. If X ad Y are depedet stadard ormal varables, fd the dstrbuto of Z X Y ad wrte dow ts p.d.f.. X, X, ad X 3 are depedet N(,) varables. Fd the dstrbuto of () X X () X X ad () X X X 3 **** STATISTICAL INFERENCE Page 9

30 School of Dstace Educato CHAPTER THEORY OF ESTIMATION- POINT ESTIMATION.. Statstcal Iferece Makg fereces about the ukow aspects of the populato usg the samples draw from the populato s kow as statstcal ferece. The ukow aspects may be the form of the probablty dstrbuto of the populato or values of the parameters (e., fucto of populato values) volved, or both. Two mportat subdvso of statstcal ferece are. () () Estmato Testg of hypothess. Estmato of parameters: The theory of estmato was fouded by Prof.R.A.Fsher, who s kow as the father of moder Statstcs. Estmato deals wth fucto of sample values, the value of whch may be take as the values of the ukow parameters ( kow as pot estmato) as well as wth the determato of the tervals whch wll cota the ukow parameters wth a specfed probablty (kow as terval estmato), based o the samples take from the populato. Testg of hypothess deals wth the method of decdg whether to accept or reject the hypothess regardg the ukow aspects of the populato, based o the samples take from the populato... Pot estmato I pot estmato a umber s suggested as a value of the ukow parameter, usg the values of the sample observatos take radomly from the populato. The fucto of sample values, suggested as a good appromato for the requred parameter s kow as a estmator, ad a partcular value of the estmator s kow as the estmate. For eg., to estmate populato mea, sample mea s take as a estmator ad the value of sample mea of a partcular sample s a estmate of populato mea..3. Desrable Propertes of Good Estmator A estmator of a parameter s sad to be a good estmator f t satsfed some desrable propertes. They are () Ubasedess () Cosstecy () Effcecy (v) Suffcecy STATISTICAL INFERENCE Page 3

31 () Ubased ess: School of Dstace Educato Let,,..., are radom samples take from a populato wth ukow parameter. The statstc t t(,,...,) s sad to be a ubased estmator of, f E() t. t s a ubased estmator of a fucto of, say f (), f E()() t f. Problem : A radom sample,,..., s take from a populato wth mea. Show that the sample mea s a ubased estmator of. Soluto: Sce the samples are take from a populato wth mea, E E E ()()...() we have E ()( E...) E E () (...) = () Hece s a ubased estmator of Remark: Ubased estmator for a parameter eed ot be uque. For eg. the above case cosder the frst two observatos ad oly. The E E ()() E. That meas s also a ubased estmator of. I smlar way we ca fd may ubased estmators for. Problem : A radom sample,,..., s take from a ormal populato wth mea ad stadard devato. Show that t s a ubased estmator of Soluto: E() t E E ( )() E...() E Gve the populato varace as, ad populato mea as,. ))) E((( ad E E for all STATISTICAL INFERENCE Page 3

32 )) (( E E for all School of Dstace Educato ) E( for all Hece, E() t... E() t, e., t s a ubased estmator of. Problem 3: For the radom sample,,..., take from N(,), show that the sample varace s a based estmator of the populato varace. Soluto: Here to show that E() S, where samples,,..., take from N(,). S s the sample varace of the radom Note that E()() S S f S ds.e, S f ()() S, e S S S E()() S S e S ds S e () S ds S e () S ds STATISTICAL INFERENCE Page 3

33 School of Dstace Educato m e () d m E() S Hece That s That s, S S s ot a ubased estmator of. s a based estmator of. Here, S E() S S E() s a ubased estmator of. Problem 4: For the radom sample,,..., take from Posso populato wth parameter,. obta a ubased estmate of e Soluto: Cosder a statstc t defed as follows, t, f the frst observato of the sample f zero =, otherwse. P(the frst observato of the sample f zero) = Hece, E() t e () e e e! e Ths mples the statstc t s a ubased estmator of e. STATISTICAL INFERENCE Page 33

34 School of Dstace Educato Problem 5: For the radom sample,,..., take from B(,) p, show that ubased estmator p, wheret. Soluto: Here,,..., are from B(,) p. The by the addtve property, T follows B(,) p... Note that E() T p,() V T pq ( ) T ( T ) E E T T E T T ( )( )( ) E()()() T V T E T pq p T ( T ) E E pq p p ( )( ) ( ) E p() p p p = ( ) E p p T ( T ) ( ) s a T ( T ) E ( ) p, e., T ( T ) ( ) s a ubased estmator of p. ( ) Cosstecy: Let,,..., are radom samples take from a populato wth ukow parameter. The statstc t t(,,...,) s sad to be a cosstet estmator of, f P t as or t s a cosstet estmator of a fucto of, say f (), f P t f () as. I other words t p deoted as t. s a cosstet estmator of, f t coverges to probablty, ad s Suffcet codtos for cosstecy: Let { t } sequece of estmators of, ad f, () E() t or, as ad () V () t, STATISTICAL INFERENCE Page 34 as

35 The Proof: t s a cosstet estmator of. Cosder the statstc t, School of Dstace Educato the by Tchebycheff s equalty, Let t.() SD t c t t.() SD t P t c, the t SD() t P t c SD() t c as, f E() t or, ad V () t ; the t.() SD t c umber ad,, SD() t c becomes a small c P t e., t Hece uder the gve codtos, p t s a cosstet estmator of. Problem : For the radom sample,,..., take from N(,), show that sample mea s cosstet estmator of populato mea. Soluto:... E()() ad V as, E() ad V (). Hece, s cosstet estmator of populato mea. Problem : For the radom sample,,..., take from Posso populato wth parameter, show that s cosstet estmator. Soluto: Here,,..., are take from Posso populato wth parameter, so E()() X ad V X for all, the Hece, as,... E()() ad V STATISTICAL INFERENCE Page 35

36 School of Dstace Educato E E() E()() E, ad V V () Here satsfes the suffcet codtos to be satsfed by cosstet estmator ad hece t s a cosstet estmator of. Problem 3: For the radom sample,,..., take from B(,) p, show that T () T s a cosstet estmator of p() p, wheret. Soluto: Note that,,..., are from B(,) p. The by the addtve property, X follows B(,) p... T X pq E()() T E,()() p V T V X E()()() T V T E T pq E() T p () pq E T ()()() T E T E T p p as, ()() p p()()() p p p p p p p () p E T T p p () ()()() V T T E T T E T T STATISTICAL INFERENCE Page 36

37 School of Dstace Educato 4 3 pq E T T T p p (3) E T E E X 4 4 X 4 () ( )( )( 3) p 6( )( ) p 7( ) p p 4 as, E T p smlarly,( E T)( ) 3( ) p p p 4 as, E T p 3 3 The, as, from (3), we get, ad E T p 4 3 V T () T p p p p p (by ()) e.,() V T ; T as (4) Hece by () ad (4), T () T s a cosstet estmator of p() p Problem 4: For the radom sample,,..., take from N(,), show that the sample varace s a cosstet estmator of the populato varace. Soluto: Let The we have. S s the sample varace of the radom samples S f ()() S, e S S It s already foud a problem of last secto, E() S The, as, E() S () S E S S e () S ds,,..., take from N(,). STATISTICAL INFERENCE Page 37

38 School of Dstace Educato 3 S e () S ds S 3 e () S ds 3 3 Hece, V () S E S E S as, V () S () From () ad (), t ca fer that varace S s a cosstet estmator of the populato Problem 5: Let t be a cosstet estmator of, ad let () be a cotuous fucto of. The prove that ()t s a cosstet estmator of (). Soluto: Sce t s cosstet for, P( t ) as becomes large. If s a cotuous fucto, we have for such that, t, ()() t P( ()() t ) STATISTICAL INFERENCE Page 38

39 ( ) Effcecy: School of Dstace Educato As we already see, to estmate a partcular populato parameter there may est more tha oe ubased estmators. Based o the varace of these ubased estmators ther effcecy s defed. Cosder t ad t are two ubased estmators of the parameter. The estmator t s sad to be more effcet tha t, f Var( t ) < Var( t ). Let t be the most effcet estmator for the parameter, the effcecy of ay other ubased var() t estmator t of s defed as E() t. The effcecy of var() t most effcet estmator s ad ay other ubased estmator s less tha E var() t var() t The relatve effcecy of t wth respect to t. s deoted by E ad s defed as Problem : For the radom sample,,..., take from N(,), test whether the followg statstcs are ubased estmators of. Whch oe s more effcet? 3 4 ()()() 3 4 Soluto: () E E (),so E E, so () E E ubased estmator of. 4 4 V V V V V V 3 4 s ubased estmator of. s a ubased estmator of. 3 4, so 4 ( ' s are radom samples) 3 4 Amog these V V V Hece 4 s more effcet. also s a STATISTICAL INFERENCE Page 39

40 (v ) Suffcecy : School of Dstace Educato I may problems of statstcal ferece, a fucto of the sample observatos cotas as much formato about the ukow parameter as do all observed values. To estmate probablty of head (p) whe a co s tossed, let the co s tossed tmes ad let, f the t th toss s a success ad = otherwse. The, - the total umber of heads out of tosses s eough to estmate p. It seems uecessary to kow whch toss resulted a head. That s t s suffcet to estmate the parameter p. The result of tosses,,..., cotas o other formato about p tha that cotas t. Hece the codtoal probablty of,,..., gve s depedet of p. p () p That s P(,,..., /) t. C p () p Ct t Hece a statstc t s sad to be suffcet for the parameter, f t cotas all formato about the parameter cotaed the sample, or If the codtoal dstrbuto of ay other statstc gve t = r, s depedet of. Fsher-Neymaa Factorzato Theorem (Codto for Suffcecy): Let,,..., be a radom sample from a populato wth pmf/pdf f (,) the the jot pmf/pdf of the sample (usually called the lkelhood of the sample) s L(,,...,,)(,)( f,)...( f,) f, the statstc t s a suffcet estmator of, f ad oly f t s possble to wrte L(,,...,,)(,)( L, t,...,) L where L ( t,) s fucto of t ad aloe ad L (,,...,) s a fucto depedet of. Proof: If t s a suffcet estmator of, the the codtoal dstrbuto of,,..., gve t = r s depedet of. That s, STATISTICAL INFERENCE Page 4 P(,,..., /)( t, r,...,) h, whch s depedet of () P(,,...,)(,,..., L,) But but P(,,..., /) t r P()( t,) r P t L(,,...,,) The by (), for suffcet estmator t, P( t,) L(,,...,,)(, h,...,)(,) P t h(,,...,)

41 School of Dstace Educato Problem : For a Posso dstrbuto wth parameter, show that sample mea s the suffcet estmator of. Soluto: Let,,..., be the radom sample take from P()... L(,,...,)(,)( f,)...( f,) f where L (,) e e e....!!! e e e!!...!!!...!!!...! L (,)(, L,...,) e ad L (,,...,)!!...! Hece, by factorzato theorem, s a suffcet estmator of. Problem : Let,,..., be the radom sample take from a populato wth p.d.f. ; f (,),. Fd a suffcet estmator for. Soluto: Lkelhood fucto L(,,...,,)(,)( f,)...( f,) f where L (,) ad.... L (,)(, L,...,) STATISTICAL INFERENCE Page 4 L (,,...,), the by

42 factorzato theorem t School of Dstace Educato ca be cosdered as a suffcet estmator of. Problem 3: Obta a suffcet estmator for p, usg samples,,..., take from B (, p). Soluto: The lkelhood fucto of,,...,, L(,,...,,)(,)( f,)...( p f,) p f p C p () p.() C...() p p C p p p p C C C ().... L (,)( p L,,...,) (,)() where L p p p, ad L (,,...,) C. C... C theorem s a suffcet estmator of p., the by factorzato log L Problem 4: If t s suffcet estmator for, prove that s a fucto of t ad oly. Soluto: If t s a suffcet estmator of, the the lkely hood fucto L(,,...,,)(,)( L, t,...,) L log(,,...,,) log(,) log(,,...,) L L t L log( L,,...,,) log(,) L t Sce, (,) L t s a fucto of t ad oly, log L s also a fucto of t ad oly. Problem 5: If t s suffcet estmator for, the prove that ay - fucto of t s also suffcet for. Soluto: Let h g() t, assume h s a - fucto of t, the t g () h Sce t s suffcet for, L(,,...,,)(,)( L, t,...,) L L(,,...,,)((),)( L, g,...,) h L Hece L(,,...,,) s the product of a fucto of h ad oly ad a fucto depedet of. The by factorzato theorem h s also suffcet for. STATISTICAL INFERENCE Page 4

43 .4. Method of Estmato School of Dstace Educato () Mamum Lkelhood Estmator: Let,,..., be the sample take from the populato wth p.m.f/p.d.f f (,,,..) k, where, k,.. k are the parameters volved. The lkelhood fucto of the sample L(,,.. k,,,..) k f (,, k,..).( k f,,,..)...( k k,, f,..) k k. The method of mamum lkelhood suggests, the best estmators for estmatg the parameters,,.. are the estmators whch mamzes the lkelhood fucto. Such k k estmators are kow as Mamum Lkelhood Estmators (M.L.E) of,,... The Prcple of M.L.E says that the best estmators of the parameters based o a sample obtaed are, those values of the parameters whch make the probablty of gettg that sample a mamum. Usg the method of dfferetal calculus, the fucto of sample values for a parameter whch mamzg the lkelhood fucto- called MLE of that parameter, ca be obtaed. Let L(,,..,,,..) k be the lkelhood fucto correspods to the sample,,... The value of, as a fucto of,,...,, mamzg the lkelhood fucto ca be obtaed from L, ad f for that value of, k k L. But sce we kow the value of, whch mamzg the lkelhood fucto also mamzes logl, such value of ca also be obtaed by usg tha zero. log L f for that value of, log L s less Mamum Lkelhood Estmators possess some desrable propertes of a good estmator. () () () (v) (v) MLE s are asymptotcally ubased. MLE s are cosstet. MLE s are most effcet. MLE s are suffcet, f suffcet statstcs est. MLE s are asymptotcally ormally dstrbuted. Problem : Fd the M.L.E. of ad, usg the radom sample,,..., take from the ormal populato N(,) Soluto: Gve the radom sample,,..., from N(,) The lkelhood fucto, STATISTICAL INFERENCE Page 43

44 School of Dstace Educato L(,,.. k,,) e e... e e k log( L,,..,,) log log L ( ) log L () Hece, s the MLE of To obta the MLE of, log L ( ) 3 3 At log L, 3 4 log L 3 STATISTICAL INFERENCE Page 44

45 School of Dstace Educato The the MLE of s. Sce MLE of s, MLE of s cosdered as, whch s the sample varace. Problem : Fd the MLE of, based o radom samples take from Posso populato wth parameter. Soluto: Let,,..., are the radom sample take from P(), the L(,,...,,)(,).( f,)...( f,) f e e e e....!!!!!...! log L log log(!!...!) log L () log L log L at ; ( samples ' s from Hece, s the MLE of Problem 3: Obta the MLE of for the followg dstrbuto Soluto: Posso populato are ()) for P Let,,..., be the sample from the gve populato, the, L(,,...,,)(,).( f,)...( f,) f e. e... e e f () e,. STATISTICAL INFERENCE Page 45

46 School of Dstace Educato Therefore, log L log Here log L s mamum whe s mmum. Ths happes whe s the meda of the radom sample,,...,. So MLE of s the meda of,,...,. Problem 4: Obta the MLE of ad usg the radom samples,,..., take from the populato wth pdf f () e,,. Soluto: The lkelhood fucto L (,,...,,, ) ca be wrtte as L(,,...,,,) e. e... e log L e log L log () () log L () () () () Equato () caot mply the MLE of. But we kow log L s mamzed whe () a mmum s. Ths happes whe s a mamum. But caot be greater tha M. Hece M s the MLE of. The by (), the value of ca be wrtte as () M ad t ca be verfed that atths value of, STATISTICAL INFERENCE Page 46

47 log L School of Dstace Educato. Hece the MLE of s () M. Problem 5: Obta the MLE of a ad b usg the radom samples,,..., take from a rectagular populato over the terval ( a b,) a b. Soluto: Here radom samples,,..., take from a rectagular populato over the terval ( a b,) a b. Hece f() s gve by f (), a b a b a b a b b I ths case the lkelhood fucto L(,,...,, a,) b a b a b The method of dfferetal calculus caot be appled here. The lkelhood fucto L s mamum whe a b a b s mmum. Ths happes whe a b s takg ts mmum ad a b s takg ts mamum possble value. But a b caot be less tha the largest value of,,..., greater tha the smallest value of,,...,. Hece MLE of a b = Ma (,,..., ) ad MLE of a b = M (,,..., ) Ma(,,...,)(, M,...,) The, the MLE of a Ma(,,...,)(, M,...,) the MLE of b ad a b ad caot be Problem 6: Obta the MLE of the parameter usg the radom samples,,..., take from a populato wth pdf, f (),. The lkelhood fucto L(,,...,,) = The lkelhood fucto gettg ts mamum value whe s mmum, e., s mmum. But for the gve pdf,, so, caot be less tha the hghest observato of the sample. Hece the mmum possble value of s the hghest value of the sample. That s the MLE of = Ma (,,..., ) Problem 7: If a suffcet statstcs T ests for, the prove that MLE of s a fucto of T. Soluto: STATISTICAL INFERENCE Page 47

48 log L School of Dstace Educato If T s suffcet for estmatg, the, L(,,...,,)(,)( L, T,...,) L log( L,) T Ths mples s a fucto of T. log L ad (,) L T e Problem 8: Gve that the frequecy fucto f (,), where X ca assume oly! oegatve tegral values ad gve the followg observed values, 4,5,7,,4,,5,7,9,4. Fd the M.L.E of. Soluto: Here observatos are take from the gve Posso populato. If let,,..., are the radom sample take from P(), the L(,,...,,)(,).( f,)...( f,) f e e e e....!!!!!...! log L log log(!!...!) log L () log L log L at ; ( samples ' s from = Hece, s the MLE of Posso populato are ()) for P =.8. STATISTICAL INFERENCE Page 48

49 () Method of momets: School of Dstace Educato Let f (,,,...,) be the pdf of the populato ad,,..., be the radom sample take from t. I the method of momet, we fd the frst k momets of the populato ad equate them to the correspodg momets of the sample. The values of,,...,, obtaed as a fucto of,,...,, by solvg the equatos are cosdered as the momet estmators of,,...,. Problem : For a ormal populato N(,), fd the estmators of momets. Soluto: Let,,..., be the radom sample take from N(,), ad by the method of of. Frst raw momet of the populato E() X Frst raw momet of the sample s.. Equatg frst momet of the sample ad the populato we get as the estmator Secod raw momet of the populato E()()() X E X V X Secod raw momet of the sample s Equatg these two, we get Hece momet estmator of s ; But s the estmator of., whch s the sample varace. Problem : X s a radom varable wth probablty masses as show below. X : f () : ;, Fd the Momet estmate of, f a 5 observatos there were oes ad 4 twos. Soluto: Out of 5 samples take from the populato, t s recorded oes, 4 twos ad the remag 9 zeroes. The frst momet of the sample s Frst momet of the populato, 9 4 8, 5 5 STATISTICAL INFERENCE Page 49

50 School of Dstace Educato E() X 8 Equatg these we get,, e., Solvg ths quadratc epresso, we get =.95. Problem.3: Obta the momet estmate of, f the probablty masses are X : 3 4 ;, ad the observed frequeces are f () : ,5,7 ad 7 respectvely. Soluto: Out of samples take from the populato, t s recorded oe, 5 twos, 7 threes ad the remag 7 fours. The frst momet of the sample s = 3, Frst momet of the populato, E() X Equatg these we get, 4 3, 4 4 solvg ths quadratc epresso, we get =.5. EXERCISES. X, X, X 3 are radom samples from populato wth mea ad stadard devato. T, T, T3 are defed as T X X X 3 ; T X 3X 3 4X ; ad T3 X X X 3. Are () T ad T are ubased estmator? () Fd, such that 9 T s a ubased estmator of () Whch s the most effcet estmator? 3. Defe cosstet estmator. Obta the suffcet codtos for cosstecy. STATISTICAL INFERENCE Page 5

51 School of Dstace Educato 3. For Posso dstrbuto wth parameter, show that. s a cosstet estmator of 4. Let,,..., are radom sample take from N(,). Fd suffcet estmators of ad. 5. Defe suffcet statstc. Fd a suffcet statstc whe f (,), 6. Fd the MLE of p, based o sample take from Bomal dstrbuto wth parameters N ad p. 7. Obta the MLE of f (,)(), based o radom sample,,..., take from the populato. Also verfy whether the MLE s a suffcet estmator of. 8. Fd the MLE of, where the radom samples,,..., are take from the populato wth pdf f () e, 9. A ur cotas whte ad black balls ukow proportos, the total umber of balls beg. Four balls are draw at radom, of whch 3 are foud to be whte ad black. Fd the mamum lkelhood estmate of the umber of whte balls the ur.. Epla the method of mamum lkelhood estmato. Fd M.L.E. of, whe f (,), f, otherwse p. Fd the momet estmator of, f f (,) e, ; p s kow p p based o the radom samples,,..., are take from the populato.. Epla the method of momets. Usg ths, obta the estmators of the parameters a ad b of a uform dstrbuto over the terval [a,b] 3. Fd a estmator of, based o radom samples take from Posso populato wth parameter by the method of momets. 4. Obta the momet estmate of, f the probablty masses are X : 3 4 ;, ad the observed f () : frequeces are,5,7 ad 7 respectvely. ******************** STATISTICAL INFERENCE Page 5

52 School of Dstace Educato CHAPTER 3 INTERVAL ESTIMATION 3.. Iterval Estmato I pot estmato we are fdg a estmator to estmate the ukow parameter uder the epectato that the true value of the parameter s very close to the value of the estmator suggested. We cosder the value of the estmator as the value of the parameter. For eg., case of N(,), s suggested as a estmator of. But whe we are dealg wth terval estmato, we are estmatg a terval where the value of the ukow parameter lyg wth a pre-assged probablty. t ad That s case of terval estmato for a parameter, t s estmatg two statstcs t ()t t such that the probablty that the terval ( t,) t cotas the true value of the ukow parameter wth a specfed probablty ;. The ( t,) t s termed as % cofdece terval for the parameter, ad s the cofdece coeffcet. It s to be oted that there may be may cofdece terval for a partcular parameter wth same cofdece coeffcet. Shortess, stablty etc., are some desrable property to detfy a good terval. 3.. Cofdece terval for the mea of a ormal populato wth cofdece coeffcet : Case I: Whe s kow Let,,..., be the sample take from N(,) ad let the sample mea be. We use - the pot estmator of for ts terval estmato. The mea follows N(,), or t ~( N,). From stadard ormal table t ca observe the value STATISTICAL INFERENCE Page 5 t such that,

53 School of Dstace Educato P( t ) t P( ) t P() t t P() t t Multplyg by - P() t t P() t t Hece the cofdece terval for wth cofdece coeffcet s, t, t. Case II: Whe s u-kow Let,,..., be the sample take from N(,) ad let the sample mea be. It s already derved for radom samples take from N(,), t ~ t. s From the table of t-dstrbuto, t ca be observed a umber t such that, P( t ) t STATISTICAL INFERENCE Page 53

54 School of Dstace Educato P( ) t s s s P() t t s s P() t t s s P() t t Hece the cofdece terval for wth cofdece coeffcet s, s s t, t. Problem : Estmate a 95% cofdece terval for, based o radom samples 7,,,8,9,,,,6,9 take from N(,3) Soluto: Here s kow. Hece the % cofdece terval for s, t, t We have, , STATISTICAL INFERENCE Page 54

55 = 3 ; Gve =.95, School of Dstace Educato Hece, from stadard ormal table we get, t.96, so that, P( t ) t , Hece the cofdece terval s, = 7.44,.6 Problem : Fd the least sample sze requred f the legth of 95% cofdece terval for the mea of a ormal populato wth stadard devato 4 should be less tha 5. Soluto: Let radom samples are take from the populato N(, 4). The cofdece terval for the mea s, t, t.96. Gve the cofdece coeffcet s 95 %. Hece from the stadard ormal table t = To fd the mmum umber of samples such that, the legth of the terval of, t t t 5 STATISTICAL INFERENCE Page 55

56 School of Dstace Educato Hece the least umber of samples requred s Problem 3: A sample of sze 7 take from N(,). Mea of the sample s ad the sample varace s 4. Usg the data, fd a 9% cofdece terval for. Soluto: Here the value of s ukow. The the cofdece terval for s, s s t, t Sce the cofdece coeffcet s 9%, from the table of t dstrbuto for 6 d.f., we get t =.746, so as P( t6 ) t.9. Also gve ad s 4 The the 9% cofdece terval for s,.746,.746.7, Problem 4: For a N(,3) populato, costruct a 95% cofdece terval for 3 5, o the bass of the radom sample of sze 5. The sample mea was foud to be 3. Soluto: CosderY 3X 5; the Y follows N (3 5, 3)(3 5, N9 9) STATISTICAL INFERENCE Page 56

57 School of Dstace Educato Sample mea correspodg to Y, y ~(3N 5,) 9 Here =5 ad, y ( 3) u y (3 5) ~(,) N 9 5 Gve =.95, hece from stadard ormal table, P( t ) t.95. Hece, y (3 5) P(.96) t.96,so as Ths mples 95% cofdece terval for 3 5 as 9 9 y.96, y Sce y 95,the terval s , , Problem 5: Show that the legth of the cofdece terval for the mea of a ormal populato wth kow varace ca be made however small we please by creasg the sample sze. Soluto: The cofdece terval for the mea of a ormal populato whe s kow s gve by t, t. The legth of the terval = t t t, STATISTICAL INFERENCE Page 57

58 School of Dstace Educato Here as the umber of sample creases the legth of the terval decreases ad whch ca be adjusted to ay gve umber for a gve sgfcace level by selectg sutable umber of samples Cofdece terval for the dfferece of meas of two ormal populatos havg kow commo varace : Let ad populatos N are the umber of samples depedetly draw from to ormal ad N,, respectvely. Let ad be the meas ad S ad S be the stadard devatos of the samples draw from the frst ad secod populato respectvely. ~ N, ad ~ N, The, ~ N, t ~(,) N From stadard ormal table t ca observe the value t such that, P( t ) t P ( ) t P t t P t t STATISTICAL INFERENCE Page 58

59 Hece the () School of Dstace Educato % cofdece terval for t, t s, Problem : The average mark scored by 3 boys a eamato s 7 wth a stadard devato of 8, whle that scored by 3 grls s 7 wth a stadard devato of 6. Costruct a 99 % cofdece terval for the dfferece of meas. (Assume S.D s are equal) Soluto: The cofdece terval for dfferece mea, t, t The commo value for varace s s (sce we have large samples) = = ; 8; 3 ad 7 ; 6; 3 Cofdece coeffcet s 99%. From stadard ormal table t.57 Hece the cofdece terval s, , = 4.543, =.543, STATISTICAL INFERENCE Page 59

60 School of Dstace Educato 3.4. Cofdece terval for the varace of a ormal populato: Let,,..., be the sample take from N(,) wth sample varace S. The, S follow ch-square dstrbuto wth (-) d.f. From the table of -dstrbuto, detfy the umbers ad so as, P ( ) ad P ( ) respectvely. P S e., P P S S S P S S P S Hece the cofdece terval for wth cofdece coeffcet s, STATISTICAL INFERENCE Page 6

61 School of Dstace Educato S, S Problem : A sample of sze take from N(,). Mea of the sample s ad the sample varace s 9. Fd a 9% cofdece terval for. Soluto: Gve =, S = 9 Cofdece terval for s gve by S, S For 9% cofdece terval, e., for., from table of ch-square dstrbuto for d.f, ad P. P ad Hece the 9% cofdece terval for s,, , 3.58 Problem : A optcal frm purchases glass for makg leses. Assume that the refractve de of 4 peces of glass have varace of. X. Costruct a 95% cofdece terval for the populato varace. Soluto: Cofdece terval for wth cofdece coeffcet s, S, S. STATISTICAL INFERENCE Page 6

62 School of Dstace Educato Here samples are draw ad the sample varace s. X coeffcet 95%, ad for ( ) = 9 d.f., ch-square table mples, 4. For cofdece = ad = Hece the 95% cofdece terval for s,.., = , Problem 3: Costruct a 95% cofdece terval for the varace wth ukow mea usg the followg sample: of the ormal populato 4.5,.,.5, 9.8, 3., 9., 5.5, 3.3,.8 ad 6.4 Soluto: Gve = samples from the ormal dstrbuto. ()% Cofdece terval for s gve by S, S For ()% =95%,.5 ; from table of ch-square dstrbuto for - = 9, d.f, ad 9 P 9.5 P ad 9.8 To fd the sample varace of the populato, usg the gve samples The calculatos follow: STATISTICAL INFERENCE Page 6

63 School of Dstace Educato s Hece the 9% cofdece terval for s, , , STATISTICAL INFERENCE Page 63

64 3.5. Cofdece terval for large samples: School of Dstace Educato. % Cofdece terval for the proporto of a bomal populato: Cosder a bomal populato wth parameters N ad p. Assume N s kow. Repeat the Beroull tral for umber of tmes. The the bomal varable X, follow B(,p). Whe becomes very large, X follows ormal dstrbuto N( p,) pq. X p The, t ~(,) N pq t X p pq ~(,) N As a appromato, t p p ~(,) N p q ; q p (where, p X s the sample proporto) Hece from stadard ormal table, obta P( t ) t t such that, p p p p P( ) () t P t t p q p q STATISTICAL INFERENCE Page 64

65 School of Dstace Educato p q p q P() p t p p t p q p q P() p t p p t Hece the p q p q e.,() P p t p p t % cofdece terval for p for large s, p q p t, p t p q where, q p Problem : Radom samples of workers of a factory 4 are dssatsfed wth ther workg codtos. Form a 95% cofdece terval for the proporto of dssatsfed workers of the factory. Soluto: Gve the sample proporto of dssatsfed workers p = 4. Also gve the 3 cofdece coeffcet = 95% The cofdece terval for proporto of dssatsfed workers s gve by, p q p t, p t p q For 95% cofdece coeffcet, from table of stadard ormal dstrbuto, t.96 Hece the cofdece terval s, , , Problem : Each computer chp produced by a certa maufacturer s ether acceptable or uacceptable. A large batch of such chps produced ad t s supposed that each chp ths batch wll be depedetly acceptable wth some ukow probablty p. To obta a 99% cofdece STATISTICAL INFERENCE Page 65

66 School of Dstace Educato terval for p, whch s to be of legth appromately.5, a sample of sze 3 s tally take. If 4 of the 3 chps are deemed acceptable, fd the appromate sample sze. (=49) Soluto: The cofdece terval for proporto of mortalty rate s gve by, p q p t, p t p q The gve cofdece coeffcet s 99%. From stadard ormal table From the 3 samples take, 4 4 p. 3 5 t.57 The legth of the cofdece terval = t p q. To fd the appromate sample sze so as legth of 99% cofdece terval p q t =.5. That s to get, such that % Cofdece terval for Posso populato: Cosder radom samples from Posso populato ad assume s large. As become very large, the Posso radom varable X follows ormal dstrbuto N (,). The sample mea ~( N,) for large. t ~(,),(()) N appromately E STATISTICAL INFERENCE Page 66

67 School of Dstace Educato From stadard ormal table, obta P( t ) t t such that, P() t t P() t t P() t t Hece the cofdece terval wth cofdece coeffcet s, t, t STATISTICAL INFERENCE Page 67

68 School of Dstace Educato EXERCISES. The mea of a sample of sze 3 draw from a ormal wth mea ad varace 36 was recorded as Fd a 95% cofdece terval for.. Costruct a % cofdece terval for the dfferece of meas of two ormal populatos havg commo varace. 3. Obta % cofdece terval for varace of a ormal populato N(,), where the mea s kow. 4. Derve 95% cofdece terval for the mea of a ormal populato whe () varace s kow () varace s ukow. 5. Obta 95% cofdece terval for the parameter of Posso dstrbuto o the bass of a radom sample of sze. 6. A radom sample of sze s draw from a ormal populato N(,). Sample mea ad the sample varace are respectvely ad 6. Fd a 9% cofdece terval for. 7. The mea of a sample of sze 4 draw from N(,) s 5.5. The sample varace s 4. Costruct a 95% cofdece terval for. 8. Usg a radom sample of sze 4 draw from a Posso populato, costruct a 9% cofdece terval for the parameter. 9. I a radom sample of 4 artcles 4 are foud to be defectve. Obta 95% cofdece terval for the true proporto of defectves the populato of artcles.. Obta a large sample () % cofdece terval for the parameter, radom samplg from the populato wth pdf f () e,,. ********************* STATISTICAL INFERENCE Page 68

69 School of Dstace Educato CHAPTER 4 TESTING OF HYPOTHESIS 4.. Statstcal Hypothess: Statstcal ferece maly deals wth estmato ad testg of hypothess. As we already see, the area of estmato cossts of makg a estmate of a approprate value of the ukow parameter or a terval for the ukow value of the parameter wth a specfed probablty. A statstcal hypothess s some statemet or asserto about the populato parameters or about the form of probablty dstrbuto or the populato. testg of hypothess was tated by J.Neymaa ad E.S.Pearso. Theory o Smple ad composte hypothess: A statstcal hypothess whch completely specfes the populato, e., t specfes the values of all parameters volved the probablty dstrbuto of the populato, the t s called smple hypothess, otherwse t s called composte. For eg., assume,,..., are radom samples take from a ormal populato N(,). The the hypothess H :, s a smple hypothess. But the hypotheses () H : ; () H : ; () H :, (v) H :, (v) H :, etc., are ot specfyg o the eact values of all the parameters volved, hece they are cosdered as composte hypothess. 4.. Testg of Hypothess: Testg of hypothess s a decso makg whether to accept or reject the proposed hypothess about the populato based o the radom samples draw from the populato. STATISTICAL INFERENCE Page 69

70 School of Dstace Educato Cosder the stuato where a lght bulb maufacturg frm troducg a ew type of lght bulb. They are clamg that the ew product s superor to the estg stadard type terms of the lfe legth. Suppose to test the clam of the frm. Assume t s kow that the average lfe legth of the bulb of stadard type as 5 hrs. Here we propose a hypothess regardg the average lfe legth, of the ew type of lght bulb as H : 5. Ths hypothess s to be tested, agast the alteratves, () s greater tha 5 or () s less tha 5 or () ether >5 or <5; e., 5. I a statstcal testg of hypothess, the hypothess s to be tested s termed as Null hypothess. The ull hypothess s deoted by H. Here the ull hypothess s H : 5. Aother hypothess our md whch we wll accept or reject accordg as we reject or accept H s termed as Alteratve hypothess, deoted by H. I the above llustrato, H ca be () H : 5 or () H : 5 or () H : 5. I ths stuato we are takg a radom sample of bulbs of ew type ad fd the average lfe legth of the sample tem. Based o the sample mea (whch s a good estmator of populato mea) we decde whether to accept or reject the ull hypothess. Roughly speakg, f the alterate hypothess cosdered s H : 5, ad the sample mea s much hgher tha 5, the hypothess H : 5 s rejected. If H : 5, ad the sample mea s much lesser tha 5, H s rejected ad f H : 5, ad the sample mea s reasoably dstat from 5, H s rejected. Here we are makg decso based o the sample mea, because we had to make a decso o populato mea ad sample mea s a good statstc to say somethg about the populato mea. As ths, ay statstcal test we have to fd a approprate statstc to make decso based o ts value. The value of statstc ca be calculated by the value of the samples selected. Such a statstc used testg of hypothess s termed as test statstc. I a statstcal test, accordg to the alteratve hypothess selected, we dvde the rage of varato of the test statstc to two. Oe s acceptace rego ad the other s STATISTICAL INFERENCE Page 7

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