Post Graduate Diploma in Applied Statistics (PGDAST)

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2 FORMULAE AD STATISTICAL TABLES BOOKLET for Post Graduate Dploma Appled Statstcs (PGDAST) IMPORTAT The Formulae ad Statstcal Tables Boolet cotas the ma formulae of the courses of the PGDAST programme ad Statstcal Tables. The Formulae ad Statstcal Tables Boolet wll be avalable at the eamato cetres the Term-Ed Eamatos for the courses of the PGDAST programme. School of Sceces Idra Gadh atoal Ope Uversty ew Delh 008 Copyrght, Idra Gadh atoal Ope Uversty

3 COTETS FORMULAE AD STATISTICAL TABLES Page o. MST-00 : Foudato Mathematcs ad Statstcs - MST-00 : Descrptve Statstcs -9 MST-00 : Probablty Theory 0- MST-00 : Statstcal Iferece -8 MST-00 : Statstcal Techques 9- MSTE-00 : Idustral Statstcs-I - MSTE-00 : Idustral Statstcs-II 8-9 Table : Logarthms - Table : Atlogarthms - Table : Values of e Table : Commoaly Used Values of Stadard ormal Varate Z Table : Stadard ormal Dstrbuto (Z table) 7 Table : Studet s t Dstrbuto (t table) 8 Table 7 : Ch square Dstrbuto ( table) 9 Table 8 : F Dstrbuto (F table) 0- Table 9 : Crtcal Values of Wlcoo Test Table 0 : Crtcal Values of Rus Test Table : Crtcal Values of Kolmogorov-Smrov Test for Oe Sample 7 Table : Crtcal Values of Kolmogorov-Smrov Test for Two Samples of Equal Sze 8 Table : Crtcal Values of Kolmogorov-Smrov Test for Two Samples of Uequal Szes 9 Table : Crtcal Values of Ma-Whtey U Test 0- Table : Crtcal Values of Krusal-Walls Test Table : Crtcal Values of Fredma Test Table 7 : Posso Probablty -9 Table 8 : Costats/Factors for Varable Cotrol Charts 0 Table 9 : Cumulatve Bomal Probablty Dstrbuto -7 Table 0 : Cumulatve Posso Probablty Dstrbuto 7-79 Table : Radom umber Table 80-8 Copyrght, Idra Gadh atoal Ope Uversty

4 MST-00 FOUDATIO I MATHEMATICS AD STATISTICS S. o. FORMULAE Dstrbutve Laws De-Morga s Laws A (BC) A B A C A (BC) A B A C (A B)' A' B' (A B)' A' B' Importat Relatos Betwee Dfferet Sets Laws of Logarthm (A B) = (A) + (B) (A B) (A B) = (A B) + (A B) + (B A) (A B) = (A) (A B) (B A) = (B) (A B) (A BC) (A) (B) (C) (A B) (A C) (BC) (A BC) loga m loga m loga m loga loga m loga log m a log m log b loga b log a a a loga m m loga a loga b log a Arthmetc Progresso (A.P.) Geometrc Progresso (G.P.) Sum of Specal Sequeces For a A.P. havg frst term a ad commo dfferece d th term = a T a ( )d Sum of frst terms = S [a ( )d] or S (a l), l last term ( )! = (whe at-cloc wse ad cloc wse order of arragemets does ot gve dstct permutatos) For a G.P. havg frst term a ad commo rato r th term = a T ar a( r ) Sumof frst terms S, r r a Sum of fte G.P. = S, r r b ( ) ; ( )( ) ( ) Permutato of thgs ot all dstct Combato The total umber of combatos of thgs tae r ( r ) at a, where p p... p p p p... p! tme Cr ad p, ( r)! r! where p are the th d of thg out of thgs. Result I: Total umber of permutatos of dstct thgs tae r at Permutato whe repetto s allowed a tme such that Total umber of permutatos of thgs tae r ) s (0 < s < r) partcular thgs are always cluded = s P r rs Ps at a tme aythg ca repeat ay umber of tmes ) s (0 < s < r) partcular thgs are always ecluded = s P r r =. Crcular permutato Result II: Total umber of combato of dstct thgs tae r at umber of permutatos of dstct thgs a tme such that = ( )! (whe at-cloc wse ad cloc wse ) s (0 < s < r) partcular thgs are always cluded = s Cr s order of arragemets maes dfferet permutatos) ) s (0 < s < r) partcular thgs are always ecluded = s Cr umber of permutatos of dstct thgs Bomal theorem for postve tegral de (a b) C a C a b C a b C a b... + C ab C b 0 Bomal theorem for ay de ( ) ( )( ) ( )!! ( )( )...( (r )) r! r Copyrght, Idra Gadh atoal Ope Uversty

5 Some Stadard Results o Lmt a lm a a a s lm lm 0 0s ta lm lm 0 0 ta lm cos 0 lm 0 a loge a, lm e / lm( ) e 0 e lm log ee 0 log( ) lm 0 lm 0, where 0 7 Dervatves of Some Fuctos d () 0, s costat d d ( ) d d (a b) a(a b) d d d (cu) c (u) d d where c s costat ad u s a fucto of If y = f(u), u = g(w), w = h() the dy dy du dw (Cha Rule) d du dw d 8 Itegrato of Some Fuctos d c, ; d log c; (a b) (a b) d c, a( ) 9 Itegrato by Parts Epoetal fucto d (a b ) ba b loga d d (e b ) be b d Logarthmc fucto d (loga ) logae d d (log e ) d Parametrc fuctos = f(t), y = g(t) dy dy d d dt dt d log a b c; a b a m m a a d c; mlog a m m a d c; e a mlog a d u() v() d u() v() d u() v() d d d 0 Elemetary Propertes of Defte Itegral b f ()d f ()d a b a a a b b f ()d f (a b )d a a f ()d f (a )d 0 0 Matrces ad Determats b c c c a a c c c Copyrght, Idra Gadh atoal Ope Uversty b d du dv (u v) d d d d dv du (uv) u v (Product Rule) d d d du dv v u d u d d (Quotet Rule) d v v where u, v are fuctos of e a a a e d c; e a ab a b e d c f ()d f ()d f ()d... f ()d f ()d, where a < c < c <...< c < b a a a 0 a f ()d, f f ()s a eve fucto f ()d f ()d 0 A I, where I s the detty matr of the same order as A 0 0, f f ()s a odd fucto a f ()d, f f (a ) f () 0 0, f f (a ) f () (A ) A (A) A, (A B) A B tr (A + B) = tr (A) + tr (B) tr (A) = tr (A)

6 (A B) A AB BA B (A B) A AB B f ad oly f AB = BA (A B) A B (AB) BA tr (AB) = tr (BA) tr (AB) tr (A) tr (B) A A A... A A A A... A A A A... A A A A... A A A A... A A A A... A AdjA A A A... A A A A... A th wherea j represetscofactor of (, j) elemet of thematr A A A, If A 0 the (A ) (A ) ; (AB) A (adja) A B A Graphcal Presetato Hstogram umber of classes K = +. Log, where K = the appromate umber of classes, = total umber of observatos Log = commo logarthm (logarthm to the base 0) Frequeces of the uequal class-tervals Gve frequecy The least wdth Wdth of tsclass-terval Stem-ad-leaf dsplay j m ths quatle Qj/ m where s that value of the varable below whch j m ths j observato le ad m. Bo Plot Lower hg = frst quartle, Upper hg = thrd quartle H-spread = Upper hg Lower hg, Step (H-Spread). Lower er fece = Lower hg Oe step Upper er fece = Upper hg + Oe step Lower outer fece = Lower hg Two steps Upper outer fece = Upper hg + Two steps Lower adjacet = Smallest observato Upper adjacet = Largest observato Copyrght, Idra Gadh atoal Ope Uversty

7 MST-00 DESCRIPTIVE STATISTICS S. o. Measures of Cetral Tedecy X Mea for Ugrouped Data XA d Whe s odd X FORMULAE Mea for Grouped Data Weghted Mea Combed Mea f f fd X A, Meda for Ugrouped/Dscrete Data Meda (M d ) Whe s eve M d th th observato observato observato ; th f X W w w X X X I Geeral X X Meda for Cotuous Data C Meda L h f Mode for Ugrouped/Dscrete Data Meda for Cotuous Data Emprcal Formula Mode = value of the varable correspodg to the mamum frequecy Geometrc Mea for Ugrouped Data GM... (... ) f f0 M0 L h Mode = Meda Mea f f f f 0 GM... Geometrc Mea for Grouped Data f f f f HM Harmoc Mea for Ugrouped Data Harmoc Mea for Grouped Data HM Quartles Decles Percetles f C Q L h f for,, C 0 D L h f for,,...,9 C 00 P L h f for,,...,99 Measures of Dsperso Rage Quartle Devato Mea Devato About ay Arbtrary Pot A R X X Ma M Coffcet of rage X X Ma Ma X X M M QD Q Q Q Q Coffcet of QD Q Q M D A Copyrght, Idra Gadh atoal Ope Uversty

8 σ = Varace σ = f Stadard devato SD Varace MD 7 Copyrght, Idra Gadh atoal Ope Uversty f f A Combed Varace for Several Groups d d... d... where d ad Root mea square devato (RMSD) for a arbtrary pot A RMSD A ad RMSD f A f Coeffcet of varato CV = 00 Raw Momets (Momets about Arbtrary Pot A) Cetral Momets (Momets about Mea) r r ( A) r ; f ( A) Recurrece relato r for r =,, ; r =,, r r r r r r r r r C r C r C r... ( ) r f ( ) ; for r = 0,,, r ; for r = 0,, Coeffcets of Sewess Coeffcets of Kurtoss ; Q Q Q S = Q Q S D D D 9 D Curve Fttg D 9 ; S P P P P P 90 0 ormal least square equatos for straght le Y = a + bx : y a b y a b = ormal least square equatos for epoetal curve Y = ab X : u A B u A B

9 ormal least square equatos for secod degree parabola Y = a + bx + cx : y a b c y a b c y a b c ormal least square equatos for power curve Y = ax b : u A b v uv A v b v where u = log y, v = log ad A = log a Smple, Multple ad Partal Correlato Aalyss Correlato coeffcet betwee X ad Y ( )(y y) r ( ) (y y) Correlato coeffcet betwee X ad Y by short cut method r ddy ddy y y d d d d Correlato Coeffcet betwee X ad Y for bvarate data r fd fydy fyddy fd fydy f d fd y y Ra correlato coeffcet rs ( ) d where log y = u, log a = A ad log b = B ormal least square equatos for epoetal curve Y = ae bx : u A B u A B where log y = u, log a = A ad blog e = B log - logarthm to the base 0. Correlato rato for cotuous data m f (y y) m T T y or m y f (y y) j j m where T f y ad T f y j j j j j j y Itra-class correlato coeffcet r c ( ) (j ) j ( )( j ) j m rc Multple correlato coeffcet R R r r r r r. r r r r r r. r Ra correlato coeffcet for ted ras m(m ) d... rs ( ) r T ( ) d T T s c c y T Ty p q (m m ) ad Ty y (m m ) Coeffcet of cocurret devato (c ) rc Correlato rato for dscrete data y r r r r r. r 8 Copyrght, Idra Gadh atoal Ope Uversty R Multple correlato coeffcet terms of partal correlato coeffcet.. R ( r )( r ).. R ( r )( r ).. R ( r )( r ) Partal correlato coeffcet r r r r. ( r )( r )

10 m y m j (y y) j (y y) r r.. r r r ( r )( r ) r r r ( r )( r ) 7 Regresso Aalyss Regresso le of Y o X (y y) b y ( ) ry where b y = Regresso le of X o Y ( ) b y (y y) where r b y = y Agle betwee two les of regresso ( r ) y ta r y 8 Assocato of Attrbutes Yule s coeffcet of assocato (AB)( ) (A )( B) Q (AB)( ) (A )( B) Coeffcet of collgato (A )( B) (AB)( ) (A )( B) (AB)( ) ad Q Partal Regresso Coeffcets b b. r r r r. r r r r r rr b. r Varace of resdual. r ( r r r r r r ) Ch-square statstc r s j O j Ej E Coeffcet of cotgecy C j 9 Copyrght, Idra Gadh atoal Ope Uversty

11 MST-00 PROBABILITY THEORY S. o. Probablty Laws of probablty P(A B) = P(A) P(A B) P(A B) = P(B) P(A B) P(A B) P(A) P(B) P(A B) P(A BC) P(A) P(B) P(C) P(A B) P(A C) P(BC) P(A BC) If A, B, C are mutually eclusve evets, the P(A BC) P(A) P(B) P(C) FORMULAE Codtoal probablty P(A B) = P(A) P(B A), P(A) > 0 = P(B) P(A B), P(B) > 0 Happeg of at least oe of the evets A,A,A,...,A s P(A A... A ) P(A )P(A )... P(A ) Uvarate radom varable X Dstrbuto fucto or cumulatve dstrbuto fucto F( ) = P[X ] P[X ], the case of dscrete radom varable 0 F() P[X ] f ()d, the case of cotuous radom varable Bvarate Radom Varable (X, Y) Law of total probablty P(A) = P(E ) P(A E ) + P(E ) P(A E ) + + P(E ) P(A E ) P E Bayes theorem PA E P(E )P(A E ) P(E A),,,..., P(A) where P(A) P(E )P(A E ) For Dscrete Radom Varable (X, Y) For Cotuous Radom Varable (X, Y) Jot probablty mass fucto p(, y j) PX,Y y j or p(, y j) P X Y y j where p(, y ) 0 ad p(, y ) j j j Margal probablty mass fucto of X p p(, y ) j j Margal probablty mass fucto of Y py j p(, y j) Codtoal probablty mass fucto of X gve Y = y P X Y y = PY y p( y) P[X Y y] provded P[Y = y] 0 Codtoal probablty mass fucto of Y gve X = P Y y X = PX p(y ) P[Y y X ] provded P[X = ] 0 Jot probablty desty fucto F, y y f (, y)dyd where f (, y) 0 ad f (, y)dyd Margal dstrbuto fucto of X F() P X f (, y)dyd Margal dstrbuto fucto of Y F(y) P[Y y] y f (, y)ddy Margal probablty desty fucto of X d f () f (, y)dy or f () F() d Margal probablty desty fucto of Y d f (y) f (, y)d or f (y) F(y) dy Codtoal probablty desty fucto of X gve Y = y f (, y) f ( y), where f(y) > 0 s the margal desty of Y f (y) 0 Copyrght, Idra Gadh atoal Ope Uversty

12 Margal dstrbuto fucto of X F() P X P X, Y yj Margal dstrbuto fucto of Y j PX,Y y F(y) P Y y Codto for depedece P[X Y y j] P[X ]P[Y y j] Epectato Propertes of mathematcal epectato E() = E(X) = E(X) E(aX + b) = ae(x) + b where, a ad b are costat r th order momet about ay pot A r E(X A) r r r r p ( A), f X s a dscrete r.v. ( A) f ()d, f X s a cotous r.v r th order momet about mea () r r E(X ) E XE(X) r r r p ( ), f Xsa dscrete r.v. ( ) f ()d, f Xsa cotous r.v. r Codtoal probablty desty fucto of Y gve X = f (, y) f (y ), where f() > 0 s the margal desty of X f () Codtoal dstrbuto fucto of X gve Y = y F( y) P X Y y f ( y)d, for all y such that f (y) 0 Codtoal dstrbuto fucto of Y gve X = F(y ) P Y y X y f (y )dy, for all such that f () 0 Codto for depedece f (y ) f (y) ad f ( y) f () or f(, y) = f() f(y) Addto theorem of epectato E(X Y) E(X) E(Y) Multplcato theorem of epectato E(XY)= E(X) E(Y) where X ad Y are depedet radom varable (r.v.) Propertes of varace V(aX b) a V(X) V(aX by) a V(X) b V(Y) abcov(x, Y) Cov(X, Y) = p j( )(y y), f (X, Y) s dscrete r.v. ( )(y y)f (, y)dyd, f (X, Y)s cotuous r.v. = E(X )(Y y) M.D. = E X Mea = E X E(X) p Mea for dscrete r.v. Mea f ()d for cotuous r.v. Dscrete ad Cotuous Probablty Dstrbutos Probablty Dstrbuto Dscrete Probablty Dstrbutos Beroull (wth parameter p) Bomal (wth parameters ad p) Probablty Fucto p p ; 0, PX 0; elsewhere Cp q ; 0,,,..., P[X ] 0; elsewhere Recurrece relato p p..p q The epected frequeces, = 0,,,, Mea ad Varace Mea p ad Varace p( p) Mea p ad Varace pq Other Propertes p(p )(p ) p p p p pq q p pq pq q p pq pq pq Copyrght, Idra Gadh atoal Ope Uversty

13 f () P X. C p q ; 0,,,..., p, pq ad pq pq Posso (wth parameter 0 ) e ; 0,,,,... P[X ] 0; elsewhere Recurrece relato p( ).p(), 0,,,,... Mea ad Varace =,,,, Dscrete Uform (Rectagular) (wth parameter ) Epected frequeces e f ().P X. ; 0,,,... for,,..., P[X ] 0, otherwse Epected frequecy Mea ad Varace f ().P[X ]. ;,,,...,. Hypergeometrc (wth parameters, M ad ) M M C. C for 0,,,...,m{,M} P[X ] C 0, otherwse where, M, are postve tegers such that, M M Mea ad Varace = M M Geometrc (wth parameter p) egatve Bomal (wth parameters r ad p) P X q p for 0,,,... 0, otherwse r r Crp q for 0,,,,... P[X ] 0, otherwse where r s a postve teger ad 0 < p < q Mea ad p q Varace p rq Mea ad p Varace rq p Cotuous Probablty Dstrbutos ormal (wth parameters ad ) f () e, where ad > 0 P[ X ] P[ Z ] 8.7% P[ X ] P[ Z ] 9.% P[ X ] P[ Z ] 99.7% Sum of depedet ormal varables s also a ormal varable. Stadard ormal dstrbuto z f (z) e, z Mea ad Varace 0, 0, β = 0,, Q.D., M.D. 0, 0 Copyrght, Idra Gadh atoal Ope Uversty

14 Cotuous Uform (Rectagular) (wth parameters a ad b) for a b f () b a 0, otherwse Cumulatve dstrbuto fucto 0 for a a F() for a b b a for b a b Mea ad Varace b a Epoetal (wth parameter ) e for 0 f () 0, elsewhere where > 0 Cumulatve dstrbuto fucto Mea ad Varace Memory less (Lac of memory) Property P[X + a X a] = P[X ] e for 0 F() 0, elsewhere Gamma (wth parameters r ad ) Gamma fucto wth parameter r Mea ad e d r 0 Varace Gamma probablty dstrbuto wth parameters r > 0 ad > 0 r r e, 0 f () r Mea r ad 0, elsewhere Varace r Gamma dstrbuto wth sgle parameter r > 0 r e f (), 0,r 0 r I the case of sgle parameter r > 0 If >, ( ) ( ) If s a postve teger, the Addtve property of Gamma Dstrbuto If X are depedet ad X G(,r ) the X G(, r ) If X are depedet ad X G(r )the X G( r ) Beta Dstrbuto of frst d (wth parameters m ad ) Beta fucto β(m, ) = m ( ) d 0 where m > 0, > 0 Beta dstrbuto of frst d m Mea ad m Varace = m (m ) (m ) Beta fucto s symmetrc fucto,.e., β(m, ) = β(, m) (p, q ) (p, q) q p (p,q) (p q,q) (p,q ) m ( ),0 f () (m,) 0, otherwse where m > 0, > 0 (m, ) 0 ( ) m m d Relatoshp betwee beta ad gamma fuctos If m > 0, > 0, the (m,) m m Copyrght, Idra Gadh atoal Ope Uversty

15 m Beta Dstrbuto, 0 m f () (m,)( ) of secod d 0, elsewhere (wth parameters m where m > 0, > 0. ad ) m Mea, ad Varace = m(m ), ( ) ( ) MST-00 STATISTICAL IFERECE S. o. Samplg Dstrbuto Sample mea (X) ad sample varace (S ) X X ad S X X E X Var X Samplg Dstrbuto of Sample Mea SDX SE X Samplg Dstrbuto of Dfferece of two Sample Meas (Idepedet Populatos) Y E X Var X Y SE X Y FORMULAE Samplg Dstrbuto of Sample Meda E X Var X SE X πσ Samplg Dstrbuto of Dfferece of two Sample Proportos E(p p ) = P P P Q PQ Var(p p ) p SE p P Q P Q E(p) P, Samplg Dstrbuto of Sample Proporto For fte populato E(p) P, SE(p) PQ Var(p) ad SE(p) Var p PQ PQ ad Cetral Lmt Theorem ad Law of Large umbers PQ If X,X,..., X s a radom sample of sze tae from a populato wth mea µ ad varace σ the X ~, for 0. Law of large umbers PX where 0 ad 0. Ch-Square Dstrbuto t-dstrbuto F-Dstrbuto Copyrght, Idra Gadh atoal Ope Uversty

16 Ch-square statstc S X ~ t-statstc ~t S/ t-dstrbuto wth df / ( / ) f ( ) e ( ) / ft / ;0 t B, r ; t r r / r 0; for all r 0,,,... r r r r ; r Relato betwee t, F ad ch-square dstrbuto t ~F(, ν); F whe S/σ F-statstc ~F, S/σ F / / ~ F, ν /ν F ν / ν / f (F) ν ν ν / ν B, ν F ν ;0 F r r r ; for r r where ad. F,, F,, Copyrght, Idra Gadh atoal Ope Uversty

17 Pot Estmato f,,...,, P X,X,...,X Ubasedess E(T) for all values of Cosstecy p T as for every where 0 lmp T P T ; m, 0 ad 0 Suffcet codtos for cosstecy E T as ad Var T 0 as Effcecy T s sad to be more effcet tha T f Var T Var T Mamum lelhood estmato L f,,...,, For dscrete case L P X.P X... P X For cotuous case L f,.f,... f, For mamum lelhood estmato log L 0 provded log L 0; for all,,..., provded, the matr of dervatves log L 0 ˆ Absolute effcecy of estmator T * Var(T ) e Var(T) Mmum varace ubased estmator (T) ) E(T) for all values of ) Var T Suffcecy Var T for all values of f,,..., / T t g,,..., Factorzato theorem of suffcecy f,,...,, g t(),.h,,..., For momet estmato M, r r M; r,,..., r th r sample momet about org = Mr X r th sample momet about mea = Mr X X r th populato momet about org = EX r r th populato momet about mea = EX r r r r j ˆ ˆ & j j log L 0; for all j,,..., Cofdece Iterval Legth of cofdece terval ( L) = T T where T lower cofdece lmt, ad T upper cofdece lmt. Cofdece terval for populato mea Whe populato varace s ow X z /, X z/ Whe populato varace s uow S S X t, X t, /, / S S X z /, X z / ; 0 For o-ormal populato X EX P z z / Var X / Cofdece terval for populato proporto p p p p p z /, p z/ Cofdece terval for populato varace Whe populato mea s ow X X,, /, / Whe populato mea s uow X X X X, or, /, / S S,, /, / Copyrght, Idra Gadh atoal Ope Uversty

18 Cofdece terval for dfferece of two populato meas (depedet populatos) X Y z /, X Y z/ X Y t S +, / p, X Y t + S, / p where Sp X X Y Y or S p S S ad ( ), S Y Y S X X Short-cut approach X a d, Y b d ( ) ad d d p S d d where d = (X a ) ad d = (Y b), a ad b are assumed arbtrary values. Sample Sze / z E where E samplg error or marg of error / z P( P) E Parametrc Tests Type of error = P [reject H 0 whe H 0 s true] = P[do ot reject H 0 whe H 0 s false] Power of a test - = P[Reject H 0 whe H s true] p-values p-value = P[Test Statstc (T) observed value of the test statstc] (for rght-taled test) p-value = P[Test Statstc (T) observed value of the test statstc] (for left-taled test) p-value = P T observed value of the test statstc (for two-taled test) Whe, > 0 S S S S X Y z /, X Y z/ Cofdece terval for dfferece of two populato meas (depedet populatos) S D t, D t D, /, / where D X Y ; D SD D ad D SD D D D Cofdece terval for dfferece of two populato proportos (depedet populatos) pq pq pq pq p p z /, p p z/ Cofdece terval for two populato varaces (depedet populatos) F S /S, F,., /, S, / For fte populato of sze z/ / E z / / z P( P) E z P( P) Test statstc for populato mea X μ Z 0 ~ 0, ; σ / X 0 X t ~ t ; Z 0 ~ 0, ; > 0 S/ S/ Test statstc for populato proporto Z p P0 PQ 0 0 ~ 0, S Test statstc for populato varace S σ0 Z ~ 0, ; 0 σ0 χ X X ( )S ~ χ σ0 σ0 7 Copyrght, Idra Gadh atoal Ope Uversty

19 Test statstc for dfferece of two populato meas (depedet populatos) X Y Z ~ 0, σ X Y Z ~ (0, ) σ σ t S p X Y ~ t ( ) where Sp X X Y Y or S p S S X Y Z ~ (0, ) ;, 0 S S Test statstc for dfferece of two populato meas (depedet populatos) D t S / D ~ t D D ad S D D D Test statstc for populato correlato coeffcet () r t r ~ t o-parametrc Tests Sg test/pared sg test S for rght tal test Test statstc(s) S for left tal test m{s,s } for two tals test where S umber of mus sgs ad S umber of plus sgs. p-value = PS test statstc S Test statstc (for > 0) Z ~ 0, Ru test Test statstc = R = umber of rus Test statstc (for > 0) R Z 0, Test statstc for dfferece of two populato proportos (depedet populatos) p p Z (whe P s ow) PQ where Q = -P. p p Z ˆ ˆPQ where p p X X ˆ (whe P s uow) Pˆ ad Q Pˆ Test statstc for two populato varaces (depedet populatos) S S Z ~ 0, ; ad 0 σ S S Z ~ 0, ; ad 0 S S S F S F ~ F,, (, ) F,, Wlcoo sged-ra test/wlcoo matched-par sged-ra test T for rght tal test Test statstc(t) T for left tal test m{t,t } for two tal test where T sum of ras of mus sgs ad T sum of ras of plus sgs. Test statstc (for > ) T Z ~ 0, Kolmogorov-Smrov goodess of ft test Test statstc(d ) sup S() F () 0 umber of sample observatos less tha or equal to S() Total umber of observatos 8 Copyrght, Idra Gadh atoal Ope Uversty

20 Ma-Whtey U test S ; f s small Test statstc(u) S ; f s small where S sum of the ras assged to the sample observatos of smaller szed sample. For large or > 0 Test statstc Krusal-Walls test U Z ~ 0, Test statstc(h) R where R sum of ras of th sample. If te occurs, the adjustmet factor s r C t t where r umber of groups of dfferet ted ras, ad t umber of ted values wth th group that are ted at a partcular value. Test statstc(h ) H/ C C Ch-square test for goodess of ft O E E Test statstc ( ) ~ E p where O observed frequecy of th class, ad E epected frequecy of th class. Kolmogorov-Smrov two-sample test Test statstc(d) sup S () S () S S Fredma test umber of frst sample observatos less tha or equal to umber of secod sample observatos less tha or equal to Test statstc(f) R j ( ) ( ) j where R sum of all ras for th treatmet. If te occurs, the adjustmet factor s C t t where t umber of ted observatos th bloc (sample). Test statstc(f ) F/ C C Ch-square test for depedece of attrbutes r c O E Test statstc ( ) ~ RCj Ej where R sum of th row, ad C sum of j th colum. j j r c j Ej 9 Copyrght, Idra Gadh atoal Ope Uversty

21 MST-00 STATISTICAL TECHIQUES S. o. Smple Radom Samplg Sample FORMULAE Populato Mea Mea Square Mea Mea Square Varace s X X S X X X X Smple Radom Samplg wthout Replacemet (SRSWOR) umber of possble samples = C E Var X S Smple Radom Samplg for Attrbutes Sample Smple Radom Samplg wth Replacemet (SRSWR) umber of possble samples = E Var X S Populato Proporto Mea Square Varace Proporto Mea Square a pq p s Sample Sze Smple Radom Samplg S z d S z (for large populato) Stratfed Radom Samplg Sample Var (p) = ( ). S t d S t A S (for small populato) Populato ( ) Mea Mea Square/Varace Proporto Mea Mea Square j j st W s ( ) j j Var st S W S p W p st E(p st ) Var p st X Xj X X j j X j X S (Xj X ) j Allocato of Sample Sze Equal Proporto eyma/optmum WS Var st PROP S ; 0 Copyrght, Idra Gadh atoal Ope Uversty Var st S S c S c W S W S EY

22 WS Systematc Radom Samplg Sample Mea Varace Populato Mea Populato Mea Square j j sys Cluster Samplg Var sys S S sys X X X j.. j j X j X S Xj X ( ) j S sys ( ) X j X Cluster Mea Sample Mea Populato Mea Populato Mea Square M j M j Varace X M X M j j M X X X M f f Var Sb S b M where Two Stage Samplg j j S S M w S M X j X j j (wth cluster) S (X X) b j M j M S X X M M X j XX X j j M M S Sample Mea Populato Mea Populato Mea Square m j m j Varace Var X X M X M j S M m S b M m j M X j M j Oe Way AOVA / Completely Radomsed Desg (CRD) SST T CF. SSE TSS SST TSS RSS CF w S (X X) b M w j M j (betwee samples) S X X (wth sample) (betwee clusters) Sum of Squares (SS) Mea Sum of Squares (MSS) Varace Rato SST MSST = SSE MSSE = MSST F = MSSE G where CF, RSS j y, G grad total or sum of all observatos, ad j Copyrght, Idra Gadh atoal Ope Uversty

23 T. j T j sum of the observatos of th sample or th level. Crtcal Dfferece CRD/ Oe Way AOVA For Equal Sze of Samples For Uequal Szes of Samples CD = t (for error df) MSSE CD = t α/ (for error df) MSSE j / 7 Two Way AOVA / Radomsed Bloc Desg (RBD) p SSA y CF q p j. q SSB y CF.j SSE TSS SSA SSB TSS RSS CF Sum of Squares Mea Sum of Squares Varace Rato MSSA SSA p MSSB SSB q SSE MSSE p q MSSA F A = MSSE MSSB F B = MSSE where G CF, p q RSS y, = pq j j Crtcal Dfferece RBD/ Two Way AOVA CD = t (for error df) MSSE q 8 Two Way AOVA wth m Observatos per Cell Sum of Squares Mea Sum of Squares Varace Rato p SSA y.. CF qm MSSA SSA p q SSB y.j. CF pm MSSB SSB q j p q SSAB y CF SSA SSB m j j. SSE TSS SSA SSB SSAB TSS RSS CF SSAB MSSAB p q MSSE SSE pq m F MSSA MSSE F MSSB MSSE MSSAB F MSSE where G CF, RSS p q m j y j, = pqm 9 Lat Square Desg (LSD) Sum of Squares Mea Sum of Squares Varace Rato m SSR y CF m m j.. m SSC y CF.j. SST y CF m.. m SSE TSS SSR SSC SST TSS RSS CF MSSR SSR MSSR F m R MSSE MSSC SSC MSSC m FC MSSE MSST SST MSST m FT MSSE SSE MSSE m m Copyrght, Idra Gadh atoal Ope Uversty

24 G where CF, m m m RSS y, = m, j j y.. m m j y j sum of the observatos of th row y.j. m m y j sum of the observatos of j th colum, Relatve Effcecy LSD over RBD Whe rows are tae as blocs MSSC (m )MSSE Relatve effcecy = m MSSE Whe colums are tae as blocs MSSR (m )MSSE Relatve effcecy = m MSSE Mssg Value RBD pt qbj G Ŷ (p )(q ) y.. m m j y j sum of the observatos of th treatmet Relatve Effcecy LSD over CRD MSSR Relatve effcecy = MSSC (m )MSSE (m )MSSE Mssg Value LSD m(r Cj T ) G Ŷ (m )(m ) 0 Factoral Epermets Ma ad teracto effects for [A] = [ab] + [a] [b ] [] [B] = [ab] +[ b] [a] [] [AB] = [ab] + [] [a] [ b] Sum of squares due to A A / r B B / r AB AB / r Ma ad teracto effects for [A] [abc] [bc] [ac] [c] [ab] [b] [a] [] [B] [abc] [bc] [ac] [c] [ab] [b] [a] [] [AB] [abc] [bc] [ac] [c] [ab] [b] [a] [] [C] [abc] [bc] [ac] [c] [ab] [b] [a] [] [AC] [abc] [bc] [ac] [c] [ab] [b] [a] [] [BC] [abc] [bc] [ac] [c] [ab] [b] [a] [] [ABC] [abc] [bc] [ac] [c] [ab] [b] [a] [] Sum of squares due to Stadard errors for testg the dfferece betwee meas at dfferet levels MSSE SE of dfferece betwee ma effect meas = r. SE of dfferece betwee A meas at same level of B = SE of dfferece betwee B meas at same level of A = A A / 8r ; B B / 8r ; AB AB / 8r ; AC AC / 8r ; BC BC /8r ; MSSE r. SE for testg the dfferece betwee meas case of r-factor teracto = r MSSE r. C C / 8r ABC ABC / 8r Copyrght, Idra Gadh atoal Ope Uversty

25 MSTE-00 IDUSTRIAL STATISTICS-I S. o. Process Cotrol Cotrol Chart FORMULAE Cotrol Le (CL) Lower Cotrol Le (LCL) Upper Cotrol Le (UCL) X A A X X AR X A R X ew Xew AR ew Xew AR ew X X AS X X ew Xew ASew Xew ASew A S R S p p-chart for varable sample sze where where X X ; X d j X X ew ; R d j ew R d j d R j ; S d D D R DR DR R ew DR ew DR ew c B B S BS BS S ew B S ew B S ew P p P p P( P) p( p) p ( p ) ew P p S d j d P( P) p( p) p ew ew ew ew ew pew pew d p ; p p or P p p ew P p p d ; p P p p ew P p P( P) p p ew p p ew P( P) p( p) ew p d j p ( p ) d p j or p ew P p ew d S p ( p ) d j p P p d P( P) p( p) ew d p ( p ) P( P) p( p) p ew ew ew ew ew pew pew j j ew p ( p ) Copyrght, Idra Gadh atoal Ope Uversty

26 where d d d p ; p ; ; p d d j ew d j d j j p P P P( P) P P( P) p p p( p) p p( p) p ew ew ew ew p p ( p ) pew p ew ( p ew ) c c c c c c c ew cew cew cew cew where c c ; c ew c d j d c j u u u u u u u ew u ew u ew u ew u ew u u u u u u ew ew uew u u ew u ew where c d c u ; u ; u u ; ; u c d j ew d j c j j Product Cotrol Lot qualty or proporto defectve umber of defectve uts a lot p lot sze Probablty of acceptg a lot a P p P P X c P X or or,..., or c a Sgle Samplg Plas Probablty of acceptg a lot Pa p PX c c 0 Bomal appromato a p C p C C c whe 0 0 P p C p p Posso appromato Producer s rs p P p P P rejectg a lot of acceptace qualty level Cosumer s rs c p P p P acceptg a lot of qualty = LTPD Acceptace outgog qualty (AOQ) umber of defectve uts the lot after specto AOQ Total umber of uts the lot Cosumer s rs c c p p C C Pc Pa p PX 0 0 C Bomal appromato c c 0 P C p p p P Acceptace outgog qualty (AOQ) ; a Copyrght, Idra Gadh atoal Ope Uversty

27 P a p c e whe p s fte 0 Producer s rs C c p p p a 0 C P P p Bomal appromato c p 0 P C p p Double Samplg Plas C AOQ p Pa pp a whe 0 Average sample umber (AS) = P Average total specto (ATI) Desg of sgle samplg plas p p (where p = AQL); (where p = LTPD) p p p p R ; p p ad p p ; p p ad p p a Probablty of acceptg a lot c p p a a a 0 C P (p) P P C c p p p p c C C Cy C y c y0 C C Bomal appromato c P a(p) C p p 0 c c y y C p p Cy p p c y0 Producer s rs P p C C c p p 0 C. c p p p p c C C Cy C y c y0 C C Bomal appromato c p 0 P C p p Game Theory C. c c y y Cp p Cyp p c y0 Algebrac Method Every two-perso zero-sum game wthout a saddle pot: Payoff matr for player A Player A Player B B B A a b A c d Cosumer s rs P c C C c p p 0 C c p p p p c C C Cy C y c y0 C C Bomal appromato c c 0 P C p p. c c y y C p p Cy p p c y0 AOQ p P P P a a a l AOQ ppa Pa whe 0ad 0 AS P I ( )( P I) P I = P[lot s accepted o the frst sample] + P[lot s rejected o the frst sample] ATI P P P a a a Desg of double samplg plas p (where p = AQL, = or = ); p p (where p = LTPD, = or = ) p p R p Med strateges (p,p ) for player A d c a b p, p (a b) (d c) (a b) (d c) Med strateges (q,q ) for player B d b a c q, q (a b) (d c) (a b) (d c) The value (v) of the game Copyrght, Idra Gadh atoal Ope Uversty

28 ad bc v (a b) (d c) Relablty Theory Relatos Betwee R(t), F(t), f(t) ad (t) s(t) f (t) R(t), F(t), F(t) = R( t) 0 0 where s(t) umber of compoets that are operatg at tme t, ad f (t) umber of compoets that have faled at tme t. t R(t) F(t) f (t) dt ep (t) dt t 0 t t F(t) R(t) f (t) dt ep (t) dt 0 0 t d d f (t) R(t) F(t) (t) ep (t) dt dt dt 0 d F(t) d (t) lr(t) dt dt F(t) 7 Mea Tme To Falure (MTTF) Relablty of Seres ad Parallel Systems MTTF E(T) t f (t)dt MTTF R(t)dt 0 0 If hazard rate follows epoetal dstrbuto the MTTF or MTTF 8 Stadby system wth perfect swtchg Let Q deote the urelablty of the compoet, gve that compoets to( ) have faled. Further, f R ad Q deote the relablty ad urelablty of the stadby system, the Q Q Q Q...Q ad R = Q Stadby system wth mperfect swtchg Q Ps QAQB Ps Q A, where, P s probablty of successful chageover, P P probablty of usuccessful chageover, s Q A s urelablty of compoet A, ad Q B urelablty of compoet B gve that compoet A has faled. t 0 f (t) f (t)dt Relablty (R) of a seres system = R R Relablty (R) of a parallel system = R R R R... R R where R the relablty of the th compoet. Decomposto method or codtoal probablty approach Relablty of the system (R s) P systemsuccess compoet K sgood P(compoet K sgood) P(systemsuccess compoet K s bad) P(compoet K s bad) Cut set method Urelablty of the system(q ) P C C C... C where C,C,...,C are mmal cut sets Te set method s Relablty of the system (R ) P T T T... T where T,T,...,T are mmal te sets s 7 Copyrght, Idra Gadh atoal Ope Uversty

29 MSTE-00 IDUSTRIAL STATISTICS-II S. o. Operatg Characterstcs for M/M/ Queueg Model P (t) = P[(t) = ] P, 0 P L s q P c FORMULAE L c P L s = λw s ; L s L s L q L q = λw q Ws Wq Ls Ws Lq Wq Wq Ws EOQ Models for Ivetory Cotrol Ecoomc order quatty (EOQ) model wth uform demad D Orderg cost = C O Q Ecoomc order quatty Q* DC Mmum total yearly vetory cost TC EOQ Model whe shortages are allowed M Carryg cost per cycle Ch t M t Carryg cost C h Q M Carryg cost per ut tme Ch Q S t Shortage cost C S Q S Shortage cost per ut tme CS Q Mamum vetory level M C DC C Q C C C C C * S * O S h S h h S C h O * DC C Ecoomc order quatty * DC O C S C Q ( h ) Ch Cs Total cycle tme * * Q CO CS Ch t D DCh CS Mmum total yearly vetory cost C TC* S DCOCh C h C S O h EOQ model wth dfferet rates of demad dfferet cycles QChT The carryg cost for tme T Ecoomc order quatty Q* D C CT Mmum total yearly vetory cost TC DCOChT EOQ Model wth uform repleshmet Q r d Average vetory r p Q r d Carryg cost = C h r p Q* DC Legth of each lot sze producto ru t O rp Chrp rp rd Optmum umber of producto rus per year D DCh rp rd * Q* Co rp DC p Ecoomc order quatty O r Q* Ch r p rd Mmum total yearly vetory cost * r d TC DCOCh r p EOQ model wth prce (or quatty) dscouts Ecoomc order quatty * C O D Q p Ecoomc order quatty * C O D Q p Mmum total yearly vetory cost h O * 8 Copyrght, Idra Gadh atoal Ope Uversty

30 Smple Lear Regresso Modellg Estmates of parameters ˆ â Y bx Y bˆ X SSXY ˆb SS X Sum of squares X SSX X Y X Y X SSXY r th resdual = r Y ˆ Y ;,,..., th r stadard resdual = d ;,,..., SS (SS ) ˆ SS SS XY Res Y SSX or ˆ r r r Varace of estmates ˆ Var a ˆ Var b X SS SS Var(Y) X Res Multple Lear Regresso Modellg j th ormal equato Y X B X X B X X j j 0 j j j j For matr method ˆB XX XY... B X X Yˆ X Bˆ X X X XY Test statstcs ˆB j t j ; j 0,,...p S.E.(B ˆ ) j p pj j Probabltes for ormal probablty plot Probablty for th ordered resdual Res O C D TC Qj Dp j ChQj Qj Test statstc t (aˆ a 0) X SSRes SS t (bˆ b) SSRes t() SS () 9 Copyrght, Idra Gadh atoal Ope Uversty X X t Lower ad upper cofdece lmts for b b bˆ t L / b bˆ t U / SS SS SS SS Res X Res X Lower ad upper lmts ˆ ˆ ˆ ˆ (X0 X) YL Y t/ V(Y) Y t/ SSRes SSX ˆ ˆ ˆ ˆ (X0 X) YU Y t/ V(Y) Y t/ SSRes SSX Lower ad upper lmts for ew value of Y=Y 0 ˆ ˆ ˆ ˆ 0 YL Y0 t/ V(Y 0) Y0 t / SSRes SS X ˆ ˆ ˆ ˆ (X X) 0 YU Y0 t/ V(Y 0) Y0 t/ SSRes SS X Coeffcet of determato R SS SS XY X SS R ( R ) Sum of squares SS r r YY YXBˆ Res Y XY X SS SS SS /SS For matr method T SS Y Y Y YY SS Bˆ Y 'X Y Reg j j SSRes SST SSReg Adjusted R Y Varace of estmates 0 ˆ (X X) V Y ˆ SSX ˆ 0 V Y0 SS X SS ˆ Res ( p ) V(B ˆ ) j jj ˆ j jj ˆ ˆ SE B (X X) j (X X) V B X X ( ); j, 0,,...p Cov(B ˆ,B ˆ ) ; j j j

31 p,,..., Percetage probablty for th ordered resdual P 00;,,..., Coeffcet of determato R p j0 ˆB Y'X j Y'Y Y j Y Tme Seres Modellg Tme seres values SS SS Weghted movg average y Reg T q w t t q Estmated tme seres values y w y w y t t t Estmated tme seres values lear form y t p b jt j 0 j Estmated tme seres values o lear form t yt 0 Auto-covarace ad autocorrelato coeffcet Auto-covarace coeffcet at lag c y y y y t t t Auto-covarace coeffcet at lag 0 0 t t Y t c y y y y, for all Auto-correlato coeffcet r Autoregressve processes Varace of X t AR() a Varace of X t AR() a c c Autocorrelato fucto of AR () process Partal auto-covarace fucto of AR() pacf () Partal auto-covarace fucto of AR() 0 R Adj ( )( R ) ( p ) Etra sum of squares SSB B0 SS Res (B 0) SS Res (B 0,B ) SS(B 0,B ) SS(B 0) SS B B SS (B ) SS (B,B ) SS(B,B ) SS(B ) 0 Res 0 Res SS B B,B SS (B,B ) SS (B,B,B ) 0 Res 0 Res 0 SS(B,B,B ) SS(B,B ) 0 0 SS B B,B SS (B,B ) SS (B,B,B ) 0 Res 0 Res 0 SS(B 0,B ) SS(B 0,B,B ) SS B,B B SS (B ) SS (B,B,B ) 0 Res 0 Res 0 SS(B,B,B ) SS(B ) 0 0 Seasoal dces ad seasoal relatves Seasoal de y 00, for,,...,. y 00 Seasoal de Meda Totalof Idces for,,,..., Deseasoalsed tme seres value y Z T C I t t t t t St Seasoal relatve = S I 00 = (y t /MA() ) 00 Movg average processes Varace of X t MA(q) : VX Autocorrelato fucto for MA(q)... q q q q t a Varace of X t MA() : VX t a Varace of X t MA() : VX t a Autoregressve movg average processes Auto-covarace fucto for ARMA at lag 0 a 0 Auto-covarace fucto for ARMA at lag 0 a Auto-covarace fucto for ARMA at lag, Auto-correlato fucto for ARMA at lag 0 Auto-correlato fucto for ARMA at lag, 0 Copyrght, Idra Gadh atoal Ope Uversty

32 pacf () Lmts = Statstcal Tables Copyrght, Idra Gadh atoal Ope Uversty

33 TABLE LOGARITHMS Mea Dfferece Copyrght, Idra Gadh atoal Ope Uversty

34 Mea Dfferece Copyrght, Idra Gadh atoal Ope Uversty

35 TABLE ATILOGARITHMS Mea Dfferece Copyrght, Idra Gadh atoal Ope Uversty

36 Mea Dfferece Copyrght, Idra Gadh atoal Ope Uversty

37 TABLE VALUES OF e (FOR COMPUTIG POISSO PROBABILITIES) (0 < < ) (=,,,...,0) e ote: To obta values of e for other values of, use the laws of epoets,.e., ab a b. 0. e e.e e.g. e e.e (0.)(0.7788) 0.0 Copyrght, Idra Gadh atoal Ope Uversty

38 TABLE COMMOLY USED VALUES OF STADARD ORMAL VARIATE Z Cofdece Iterval 99% 98% 9% 90% Level of Sgfcace (α) 0.0 (%) 0.0 (%) 0.0 (%) 0.0 (0%) Two-Taled ± z α/ = ±.7 ± z α/ = ±. ± z α/ = ±.90 ± z α/ = ±. Oe (rght)-taled z α =. z α =.0 z α =. z α =.8 Oe ( left)-taled z α =. z α =.0 z α =. z α =.8 7 Copyrght, Idra Gadh atoal Ope Uversty

39 TABLE STADARD ORMAL DISTRIBUTIO (Z TABLE) The frst colum ad frst row of the table dcate the values of stadard ormal varate Z at frst ad secod place of decmal. The etry represets the upper tal area uder the curve or probablty,.e., P[0 Z z] for dfferet values of Z. Z ote: The area for egatve values of Z s tae as the same as that for postve values, sce the curve s symmetrcal. 8 Copyrght, Idra Gadh atoal Ope Uversty

40 TABLE STUDET S t DISTRIBUTIO (t TABLE) The frst colum of ths table dcates the degrees of freedom ad frst row dcates a specfed upper tal area (α). The etry represets the value of the t-statstc such that area uder the curve of the t-dstrbuto to ts upper tal s equal to α. Oe-Taled Test α = Two-Taled Test α = ν = ote: The area for egatve values of Z s tae as the same as that for postve values, sce the curve s symmetrcal. 9 Copyrght, Idra Gadh atoal Ope Uversty

41 TABLE 7 CHI SQUARE DISTRIBUTIO ( TABLE) The frst colum of ths table dcates the degrees of freedom ad frst dcates row a specfed upper tal area (α). The etry represets the value of ch-square statstc such that the area uder the curve of the ch square dstrbuto to ts upper tal s equal to α. α = ν = Copyrght, Idra Gadh atoal Ope Uversty

42 0 TABLE 8 F DISTRIBUTIO (F TABLE) F-table cotas the values of the F-statstc for dfferet set of degrees of freedom (, ) of umerator ad deomator such that the area uder the curve of the F-dstrbuto to ts rght (upper tal) s equal to α. F values for α = 0. Degrees of Freedom for Deomator (ν ) Degrees of Freedom for umerator(ν ) Copyrght, Idra Gadh atoal Ope Uversty

43 F values for α = 0.0 Degrees of Freedom for Deomator (ν ) Degrees of Freedom for umerator(ν ) Copyrght, Idra Gadh atoal Ope Uversty