STUDENT S t-distribution AND CONFIDENCE INTERVALS OF THE MEAN ( )

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1 STUDENT S t-distribution AND CONFIDENCE INTERVALS OF THE MEAN Suppoe that we have a ample of meaured value x1, x, x3,, x of a igle uow quatity. Aumig that the meauremet are draw from a ormal ditributio havig mea µ ad variace µσ, with the arithmetic mea σ it i reaoable to etimate thee populatio parameter ( ) ad the ample variace x = = 1 x (1) = = 1 ( x x ) 1 where tadard deviatio, σ (ample ad populatio repectively) are poitive quare-root of variace, σ. If the ample ize i large ( 30) we may etimate cofidece iterval for the mea µ uig the tatitic z, where x µ x µ x µ z = = σ σ ad z i aumed to be a value of the tadard ormal variable Z. Here we are uig a theorem of tatitical amplig theory: If all poible radom ample X1, X,, X of ize are draw with replacemet from a ormal populatio with mea µ ad tadard deviatio σ, the the amplig ditributio of the mea X will be approximately ormally ditributed with mea µ = µ ad tadard deviatio σ = σ. X X X µ X µ X µ It follow that Z = = = ha a tadard ormal ditributio. σ σ σ X Note that i equatio (3) σ (which may be uow) ca be replaced by ; which i a reaoable aumptio for 30. Cofidece iterval (CI) ca be determied from the followig probability tatemet ( ) α α 1 α P z < Z < z = (4) where P ( ) i probability, α i a igificace level ad z α the tadard ormal ditributio. For a 95% CI ( ) from table of the tadard ormal ditributio z α = () (3), z α are two tail-ed value of α = 0.05, 1 α = 0.95, α = 0.05 ad 1

2 f(z) α/ 1 α α/ z α/ z z 0 α/ Figure 1: Stadard ormal ditributio curve Subtitutig the tatitic z (equatio (3)) for the tadard ormal variable Z i equatio (4) give x µ P zα < < zα = 1 α σ ad the iequality o the left-had-ide may be re-arraged to give σ σ P x zα < µ < x + zα = 1 α Thu the lower ad upper cofidece limit the mea µ are: (5) (6) lower cofidece limit upper cofidece limit σ For a 95% CI thee limit are x ± 1.96 = x z = x + z Note that i equatio (5), (6) ad (7) the populatio tadard deviatio σ ca be replaced by the ample etimate. Thi i a reaoable aumptio for 30. α α σ σ (7)

3 If the ample ize i mall ( < 30) we may etimate cofidece iterval for the mea µ uig the tatitic t where x µ x µ t = = ad t i aumed to be a value of the radom variable T havig a Studet t ditributio with = 1 degree of freedom. The t ditributio wa itroduced by W.S. Goet who publihed uder the peudoym Studet (1908). Studet t ditributio (or jut the t ditributio) i the ditributio of the radom variable T ( x µ ) ( ) = of mall ample whoe variace may fluctuate coiderably. The t ditributio i the familiar bell-haped curve ad it ca be how that the ormal ditributio i the limitig cae of the t ditributio. (8) 0.4 t ditributio ad tadard ormal ditributio 0.3 f(t) t Figure : Stadard ormal ditributio curve (blac) ad t ditributio curve (broe) for = 1,3,5 degree of freedom. A icreae the t ditributio approache the ormal ditributio. Cofidece iterval (CI) ca be determied from the followig probability tatemet where ( ) ( ) P t < T < t = (9) α, α, 1 α P i probability, α i a igificace level ad t α,, t α, are two tail-ed value of the t ditributio with v degree of freedom. = degree of freedom, ( ) For a 95% CI with 3 α = 0.05, 1 α = 0.95, α = 0.05 ad from table of the t ditributio for = 3 degree of freedom, t α, =

4 f(t) α/ 1 α α/ 0 t t t α/ α/ Figure 3: t ditributio curve with degree of freedom. Subtitutig the tatitic t i equatio (8) for the radom variable T i equatio (9) give x µ P tα, < < tα, = 1 α ad the iequality o the left-had-ide may be re-arraged to give P x tα, < µ < x + tα, = 1 α Thu the lower ad upper cofidece limit the mea µ are: lower cofidece limit upper cofidece limit = x t = x + t α, α, For a 95% CI ad v = 3 degree of freedom thee limit are x ± (10) (11) Example 1 A ditace i meaured four time givig the followig value. What are the 95% cofidece limit of the mea? [Thi example i from Lauf (1983, 4, p.56] [1] Value; reidual v ( x x ) freedom = ; etimate of tadard deviatio ; ad degree of x (m) v ( x x ) (mm) = ( ) x = v = 0 ( ) v v = 610 4

5 etimate of variace: ( x x ) ( v ) = 1 = 1 = = 1 1 etimate of tadard deviatio: degree of freedom: = 1= 4 1= 3 [] For a 95% CI with 3 ( v ) = = = = mm = 0.09 m = degree of freedom, ( ) from table of the t ditributio, t α, = α = 0.05, 1 α = 0.95, α = 0.05 ad 0.09 [3] Cofidece limit for the mea: x ± tα, = x ± = x ± m 4 ad < µ < If we had ued the ormal ditributio itead of the t ditributio the value of z α would have bee which would have give a 95% cofidece limit of ± 0.09 m, which i oly 63% of the correct value (Lauf, 1983, p.56) Example The bearig ad ditace of a lie i determied idirectly by a group of urveyig tudet whoe reult are tabulated below. Dicard the bluder; calculate the mea, tadard deviatio ad rage of the remaiig bearig ad ditace ad the 95% cofidece limit of the mea reult for the group ' 55" ' 50" ' 40" ' 18" ' 55" ' 11" ' 8" ' 43" ' 46" ' 53" ' 46" ' 38" ' 43" ' 55".39 43' 8" ' 46" ' 56".39 43' 39".37 44' 11" ' 9" ' 05".389 5

6 [1] Bluder, mea, tadard deviatio, rage ad tem & leaf plot Ordered data Stem & Leaf Plot plot = mea t.dev rage = mea m t.dev m rage 0.03 m [] For a 95% CI: ( ) α = 0.05; 1 α = 0.95; α = For bearig; = 1 = 17 degree of freedom ad from table of the t ditributio, t α, = For ditace; = 1 = 0 degree of freedom ad from table of the t ditributio, t α, = [3] Cofidece limit for the mea: x ± tα, Limit for bearig: 61 x ±.1098 = x ± ad < µ < Limit for ditace: x ±.0860 = x ± m 1 ad < µ < m 6

7 TABLE 1: t DISTRIBUTION Probability for a give degree of freedom (the area i the right-had tail of the ditributio) α/ 0 t α/ t α For a 95% CI with 3 from above, t α, = = degree of freedom, ( ) α = 0.05, 1 α = 0.95, α = 0.05 ad 7

8 Equatio of the t ditributio f(t) 0 t The ditributio of t (or the probability deity fuctio pdf) i ( ) f t + 1 t = 1 + Γ π Γ where Γ ( ) i the gamma fuctio. The gamma fuctio wa itroduced to exted the factorial fuctio from iteger to real umber. The factorial fuctio for the poitive iteger i defied by with zero factorial defied a 0! = 1 The gamma fuctio i defied by ( ) ( ) ( )! = (1) with pecial reult ( ) 0 x Γ + 1 = x e dx for > 1 (13) 1 ( ) π ( ) Γ =, Γ 1 = 1 (14) The gamma fuctio ca be evaluated from the recurrece relatiohip ( 1) ( ) otig that if i ay poitive iteger the ( ) Γ + = Γ (15) Γ + 1 =! For example if = 9 the i equatio (1) we have (Lauf, 1983 p.55) Γ =Γ =Γ ( 5) = 4! = 1 3 4= Γ =Γ = Γ ( ) = π =

9 Hece ( ) f t If t = 0.5 the f ( t ) = t 384 t = = + π 9π 9 315π 9 Followig Lauf (1983, p.56) we ca ee that whether i poitive or egative the value of f ( t ) i the ame ad o the ditributio curve (the pdf) i ymmetrical about the axi t = 0. We may how that for large value of, For example for = ad 1 e + Γ Γ 1 ( + 1) t 1 t Γ Γ = = = ad Γ( 15) Γ ad for = 30 ad t = = = ( ) ( ) t = 1+ = ad e 1 t = e = So that ( ) 1 1 t 1 t f t e = e which i the ormal ditributio. π π Thi mea that we may alway ue the Studet t ditributio whether i mall or large (otig that the ormal ditributio i valid oly for large value of ) The probability p that a igle obervatio from the t ditributio with degree of freedom, x i that area uder the curve betwee ad t = x ad will fall i the iterval [ ) + 1 t= x Γ 1 ( + 1 ) t p = 1 dt + (16) t= Γ π For give value of p the correpodig value of t ca be evaluated from equatio (16) by umerical itegratio. 9

10 α =, ( α ) For example, for a 95% CI = 0.95 ad α = 0.05 i the area uder the right-had tail of the ditributio curve. Thi correpod to a value of t = t α ad the area uder the curve betwee ad t = t α i equal to p = ad t α obtaied from equatio (16) The value for Table 1 were obtaied by uch method uig fuctio tiv( ) from Matlab Statitic Toolbox. table = zero(30,6); for = 1:30 table(,1) = ; row = tiv([ ],); table(,:6) = row; ed table Referece Lauf, G.B., (1983), The Method of Leat Square with applicatio i urveyig, Tafe Publicatio Uit, Colligwood, Autralia. 10