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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

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 chi-square 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 chi-squares is itself a chi-square 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 chi-square 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 chi-square 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 chi-squared 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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