= 1. n n 2 )= n n 2 σ2 = σ2


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1 SAMLE STATISTICS A rado saple of size fro a distributio f(x is a set of rado variables x 1,x,,x which are idepedetly ad idetically distributed with x i f(x for all i Thus, the joit pdf of the rado saple is f(x 1,x,,x =f(x 1 f(x f(x = f(x i A statistic is a fuctio of the rado variables of the saple, also kow as the saple poits Exaples are the saple ea x = x i / ad the saple variace s = (x i x / A rado saple ay be regarded as a icrocos of the populatio fro which it is draw Therefore, we ight attept to estiate the oets of the populatio s pdf f(x by the correspodig oets of the saple statistics To deterie the worth of such estiates, we ay deterie their expected values ad their variaces Beyod fidig these siple easures, we ight edeavour to fid distributios of the statistics, which are described as their saplig distributios We ca show, for exaple, that the ea x of a rado saple is a ubiased estiate of the populatio oet µ = E(x, sice Its variace is E( x =E ( xi = 1 E(xi = µ = µ V ( x =V ( xi = 1 V (xi = σ = σ Here, we have used the fact that the variace of a su of idepedet rado variables is the su of their variaces, sice the covariaces are all zero Observe that V ( x 0as Sice E( x =µ, this iplies that, as the saple size icreases, the estiates becoe icreasigly cocetrated aroud the true populatio paraeters Such a estiate is said to be cosistet The saple variace, however, does ot provide a ubiased estiate of σ = V (x, sice { 1 } [ 1 E(s =E (xi x { = E (xi µ+(µ x } ] [ 1 { = E (xi µ +(x i µ(µ x+(µ x }] = V (x E{( x µ } + E{( x µ } = V (x V ( x Here, we have used the result that E { 1 } (xi µ(µ x = E{(µ x } = V ( x 1
2 It follows that E(s =V (x V ( x =σ σ ( 1 = σ Therefore, s is a biased estiator of the populatio variace ad, for a ubiased estiate, we should use ˆσ = s 1 = (xi x 1 However, s is still a cosistet estiator, sice E(s σ as ad also V (s 0 The value of V (s depeds o the for of the uderlyig populatio distributio It would help us to kow exactly how the estiates are distributed For this, we eed soe assuptio about the fuctioal for of the probability distributio of the populatio The assuptio that the populatio has a oral distributio is a covetioal oe, i which case, the followig theore is of assistace: Theore Let x 1,x,,x be a rado saple fro the oral populatio N(µ, σ The, y = a i x i is orally distributed with E(y = a i E(x i = µ a i ad V (y = a i V (x i=σ a i I geeral, ay liear fuctio of a set of orally distributed variables is itself orally distributed Thus, for exaple, if x 1,x,,x is a rado saple fro the oral populatio N(µ, σ, the x N(µ, σ / The geeral result is best expressed i ters of atrices Let µ =[µ 1,µ,,µ ] = E(x deote the vector of the expected values of the eleets of x =[x 1,x,,x ] ad let Σ = [σ ij ; i, j =1,,,] deote the atrix of their variaces ad covariaces If a =[a 1,a,,a ] is a costat vector of order, the a x N(a µ, a Σa is a orally distributed rado variable with a ea of E(a x=a µ = a i x i ad a variace of V (a x=a Σa = i a i a j σ ij = j i a i σ ii + i a i a j σ ij j i A iportat case is whe the vector a =[a 1,a,,a ] becoes a vector of uits, deoted by ι =[1, 1,,1] ad described as the suatio vector The, if x =[x 1,x,,x ] is the vector of a rado saple with x i N(µ, σ for all i, there is x N(µι, σ I, where µι =[µ, µ,, µ] is a vector with µ repeated ties ad I is a idetity atrix of order Writig this explicitly, we have x = x 1 x x N µ σ 0 0 µ, 0 σ 0 µ 0 0 σ
3 The, there is x =(ι ι 1 ι x = 1 ι x N(µ, σ / ad 1 σ = ι {σ I}ι = σ ι ι = σ, where we have used repeatedly the result that ι ι = Eve if we do ot kow the for of the distributio fro which the saple has bee take, we ca still say that, uder very geeral coditios, the distributio of x teds to orality as Thus we have Theore The Cetral Liit Theore states that, if x 1,x,,x is a rado saple fro a distributio with ea µ ad variace σ, the the distributio of x teds to the oral distributio N(µ, σ / as Equivaletly, ( x µ/(σ/ teds i distributio to the stadard oral N(0, 1 distributio To describe the distributio of the saple variace, we eed to defie the chisquare distributio If x N(0, 1 is distributed as a stadard oral variable, the x χ (1 is distributed as a chisquare variate with oe degree of freedo Moreover Theore The su of two idepedet chisquare variates is a chisquare variate with degrees of freedo equal to the su of the degrees of freedo of its additive copoets I particular, if x χ ( ad y χ (, the (x + y χ ( + M It follows that, if x 1,x,,x is a rado saple fro a stadard oral N(0, 1 distributio, the x i χ ( Moreover, if x 1,x,,x is a rado saple fro a N(µ, σ distributio, the (x i µ /σ χ ( Cosider the idetity (xi µ = ({x i x} + { x µ} = {x i x} + { x µ}, which follows fro the fact that the cross product ter is { x µ} {x i x} =0 This decopositio of a su of squares features i the followig result: The Decopositio of a Chisquare statistic If x 1,x,,x is a rado saple fro a stadard oral N(µ, σ distributio, the with (x i µ σ = (1 ( (x i x ( x µ σ + σ, (x i µ χ (, σ (x i x σ χ ( 1, ( x µ (3 σ χ (1, 3
4 where the statistics uder ( ad (3 are idepedetly distributed Defiitios (1 If u χ ( ad v χ ( are idepedet chisquare variates with ad degrees of freedo respectively, the F = { / } u v F (,, which is the ratio of of the chisquares divided by their respective degrees of freedo, has a F distributio of ad degrees of freedo, deoted by F (, ( If x N(0, 1 is a stadard oral variate ad if v χ ( is a chisquare variate of degrees of freedo, ad if the two variates are distributed idepedetly, the the ratio / v t = x t( has a t distributed of degrees of freedo, deoted t( Notice that t = x v/ { χ / (1 χ } ( = F (1, 1 CONFIDENCE INTERVALS Cosider a stadard oral variate z N(0, 1 Fro the tables i the back of the book, we ca fid ubers a, b such that, for ay Q (0, 1, there is (a z b =Q The iterval [a, b] is called a Q 100% cofidece iterval for z We ca iiise the legth of the iterval by disposig it syetrically about the expected value E(z = 0, sice z N(0, 1 is syetrically distributed about its ea of zero We ca easily costruct cofidece itervals for the paraeters uderlyig our saple statistics Sice they are cocered with fixed paraeters, such cofidece stateets differ i a subtle way fro those regardig rado variables A cofidece iterval for the ea of the N(µ, σ distributio Let x 1,x,,x be a rado saple fro a oral N(µ, σ distributio The x N (µ, σ ad Therefore, we ca fid ubers ±β such that x µ σ/ ( β x µ σ/ β = Q 4 N(0, 1
5 But, the followig evets are equivalet: ( β x µ σ/ β ( β σ x µ β σ ( β σ µ x β σ ( x β σ µ x + β σ Hece ( x β σ µ x + β σ = Q This says that the probability that the rado iterval [ x βσ/, x + βσ/ ] falls over the true value µ is Q Equivaletly, give a particular saple that has a ea value of x, weareq 100% cofidet that µ lies i the resultig iterval Exaple Let (1, 34, 06, 56 be a rado saple fro a oral N(µ, σ =9 distributio The x = 7 ad x µ σ/ = 7 µ 3/ N(0, 1 Hece ( µ 3/ 196 = Q, ad it follows that (04 µ 564 is our 95% cofidece iterval A cofidece iterval for µ whe σ is ukow Usually, we have to estiate σ The ubiased estiate of σ is ˆσ = (x i x /( 1 With this estiate replacig σ, we have to replace the stadard oral distributio, which is appropriate to ( x µ/σ, by the t( 1 distributio, which is appropriate to ( x µ/ˆσ To deostrate this result, cosider writig { / (xi } x µ ˆσ/ = x µ σ/ x σ, ( 1 ad observe that we ca cacel the ukow value of σ fro the uerator ad the deoiator Now, (x i x /σ χ ( 1, so the deoiator cotais the root of a chisquare variate divided by its 1 degrees of freedo The uerator cotais a stadard oral variate That is to say, the statistic has the for of { / } χ ( 1 N(0, 1 t( 1 1 5
6 To costruct a cofidece iterval, we proceed as before, except that we replace the ubers ±β, obtaied fro the table of the N(0, 1 distributio, by the correspodig ubers ±b, obtaied fro the t( 1 table Our stateet becoes ( x b ˆσ µ x + b ˆσ = Q A cofidece iterval for the differece betwee two eas Iagie a treatet that affects the ea of a oral populatio without affectig its variace A istace of this ight be the applicatio of a fertiliser that icreases the yield of a crop without adversely affectig its hardiess We ight wish to estiate the effect of the fertiliser; ad, i that case, we would probably wat to costruct a cofidece iterval for the estiate To establish a cofidece iterval for the chage i the ea, we would take saples fro the populatio before ad after treatet Before treatet, there is x i N(µ x,σ ; i =1,, ad x N (µ x, σ, ad, after treatet, there is y j N(µ y,σ ; j =1,, ad ȳ N (µ y, σ The, o the assuptio that the two saples are utually idepedet, the differece betwee the saple eas is Hece ( x ȳ N ( x ȳ (µ x µ y σ + σ (µ x µ y, σ + σ N(0, 1 If σ were kow, the, for ay give value of Q (0, 1, we could fid a uber β fro the N(0, 1 table such that { } σ ( x ȳ β + σ σ µ x µ y ( x ȳ+β ++σ = Q This would give a cofidece iterval for µ x µ y Usually, we have to estiate σ fro the saple iforatio We have (xi x σ χ ( 1 ad (yj ȳ σ χ ( 1, 6
7 which are idepedet variates with expectatios equal to the ubers of their degrees of freedo The su of idepedet chisquares is itself a chisquare with degrees of freedo equal to the su of those of its costituet parts Therefore, (xi x + (y j ȳ σ χ ( + has a expected value of +, whece ˆσ = (xi x + (y j ȳ + is a ubiased estiate of the variace If we use the estiate i place of the ukow value of σ, we get ( x ȳ (µ x µ y ˆσ + ˆσ / (xi ( x ȳ (µ x µ y x = + (y j ȳ σ ( + σ N(0, 1 χ (+ + + σ = t( + This is the basis for deteriig a cofidece iterval that uses a estiated variace i place of the ukow value A cofidece iterval for the variace If x i N(µ, σ ; i =1,, is a rado saple, the (x i x /( 1 is a ubiased estiate of the variace ad (x i x /σ χ ( 1 Therefore, by lookig i the back of the book at the appropriate chisquare table, we ca fid ubers α ad β such that ( α (xi x σ β = Q for soe chose Q (0, 1 Fro this, it follows that ( 1 α σ (xi x 1 ( (xi x = Q β β σ (xi x = Q α ad the latter provides a cofidece iterval for σ We ought to choose α ad β so as to iiise the legth of the iterval [α 1,β 1 ] The chisquare is a asyetric distributio, so it is tedious to do so The distributio becoes icreasigly syetric as the saple size icreases, ad so, for large values of, we ay choose α ad β to dearcate equal areas withi the two tails of the distributio The cofidece iterval for the ratio of two variaces Iagie a treatet that affects the variace of a oral populatio We ight also wish to allow for the possibility that the ea is also affected Let x i N(µ x,σx; i =1,, 7
8 be a rado saple take fro the populatio before treatet ad let y j N(µ y,σy; j =1,, be a rado saple take after treatet The (xi x σ χ ( 1 ad (yj ȳ σ χ ( 1, are idepedet chisquared variates, ad hece { (xi / (yj } x ȳ F = σx( 1 σy( F ( 1, 1 1 It is possible to fid ubers α ad β such that (α F β = Q, where Q (0, 1 is soe chose probability value Give such values, we ay ake the followig probability stateet: ( (yj ȳ ( 1 α (xi x ( 1 σ y (yj ȳ ( 1 σx β (xi x = Q ( 1 8
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