S T A T R a c h e l L. W e b b, P o r t l a n d S t a t e U n i v e r s i t y P a g e 1. = Population Variance

Transcription

1 S T A T 4 - R a c h e l L. W e b b, P o r t l a d S t a t e U i v e r i t y P a g e Commo Symbol = Sample Size x = Sample Mea = Sample Stadard Deviatio = Sample Variace pˆ = Sample Proportio r = Sample Correlatio Coefficiet N = Populatio Size μ = Populatio Mea σ = Populatio Stadard Deviatio = Populatio Variace p = Populatio Proportio = Populatio Correlatio Coefficiet Decriptive Statitic x x x x I Excel ue Data > Data Aalyi > Decriptive Statiitc x x x x z Rage = Max Mi % x Upper = Q3 + (.5*IQR) IQR=iterquartile rage = Q3 Q Coefficiet of Variatio = Fidig Outlier Limit: Lower = Q (.5*IQR) Probability Uio Rule: P(A B) = P(A) + P(B) P(A B) Complemet Rule: P(A C ) = P(A) P(A B) Coditioal Probability: P(A B) P(B) Mutually Excluive Evet: P(A B) Depedet Evet: P(A B) P(B) P(A B) Idepedet Evet: P(A B) P(B) P(A) Coutig Rule Sum of Two Dice Secod Die Firt Die Stadard Deck of Card Face Card are Jack (J), Quee (Q) ad Kig (K)

2 S T A T 4 - R a c h e l L. W e b b, P o r t l a d S t a t e U i v e r i t y P a g e Dicrete Probability Ditributio E(x) x P( x) P( x) Biomial Ditributio: P x x P( x) x x ( x)cxp q p p q pq Sig Are Importat For Dicrete Ditributio! P(X=x) P(X x) P(X x) I I le tha or equal to I greater tha or equal to I equal to I at mot I at leat I exactly the ame a I ot greater tha I ot le tha a ot chaged from Withi I the ame a =BINOM.DIST(x,,p,fale) =BINOM.DIST(x,,p,true) =-BINOM.DIST(x,,p,true) TI: biompdf(,p,x) TI: biomcdf(,p,x) TI: biomcdf(,p,x ) Where: P(X>x) P(X<x) x i the value i the More tha Le tha quetio that you are Greater tha Below fidig the probability for. Above Lower tha p i the proportio of a igher tha Shorter tha ucce expreed a a Loger tha Smaller tha decimal betwee ad Bigger tha Decreaed i the ample ize Icreaed Smaller Excel: = BINOM.DIST(x,,p,true) =BINOM.DIST(x,,p,true) TI-Calculator: biomcdf(,p,x) biomcdf(,p,x ) Poio Ditributio: P( x) x e Chage mea to fit the uit i the quetio: ew uit ew old x! old uit P(X=x) P(X x) P(X x) I I le tha or equal to I greater tha or equal to I equal to I at mot I at leat I exactly the ame a I ot greater tha I ot le tha a ot chaged from Withi I the ame a =POISSON.DIST(x,mea,fale) =POISSON.DIST(x,mea,true) = POISSON.DIST(x,mea,true) TI: poiopdf(mea,x) TI: poiocdf(mea,x) TI: poiocdf(mea,x ) Where: P(X>x) P(X<x) x i the value i the More tha Le tha quetio that you are Greater tha Below fidig the probability for. Above Lower tha igher tha Shorter tha The mea ha bee recaled to Loger tha Smaller tha the uit of the quetio. Bigger tha Decreaed Icreaed Smaller Excel: = POISSON.DIST(x,mea,true) =POISSON.DIST(x,mea,true) TI-Calculator: poiocdf(mea,x) poiocdf(mea,x )

3 S T A T 4 - R a c h e l L. W e b b, P o r t l a d S t a t e U i v e r i t y P a g e 3 Cotiuou Ditributio z x x z Cetral Limit Theorem: Note that for a cotiuou ditributio there i o area at a lie uder the curve, o ad > will have the ame probability ad ue the ame Excel commad. Normal Ditributio x z Normal Ditributio Fidig a Probability P(X x) or P(X<x) P(x<X<x) or P(x X x) P(X x) or P(X>x) I le tha or equal to Betwee I greater tha or equal to I at mot I at leat I ot greater tha I ot le tha Withi More tha Le tha Greater tha Below Above Lower tha igher tha Shorter tha Loger tha Smaller tha Bigger tha Decreaed Icreaed =NORM.DIST (x,µ,,true) =NORM.DIST (x,µ,,true)- NORM.DIST (x,µ,,true) =- NORM.DIST (x,µ,,true) Note that the NORM.S.DIST fuctio i for a tadard ormal whe µ= ad =. Ivere Normal Ditributio Normal Ditributio Fidig a X-value Give a Area or Probability P(X x) or P(X<x) P(x<X<x) or P(x X x) P(X x) or P(X>x) Lower Betwee Upper Bottom Top Below Above More tha Le tha Greater tha Lower tha Shorter tha igher tha Smaller tha Loger tha Decreaed Bigger tha Icreaed =NORM.INV(area,µ,) x =NORM.INV(-area/,µ,) x = x =NORM.INV(-area, µ,)

4 S T A T 4 - R a c h e l L. W e b b, P o r t l a d S t a t e U i v e r i t y P a g e 4 Cofidece Iterval Whe < 3 the variable mut be approximately ormally ditributed. t-iterval The ( - )% cofidece iterval for, i ukow, i x t, /. O the TI-83 you ca fid a cofidece iterval uig the tatitic meu. Pre the [STAT] key, arrow over to the [TESTS] meu, arrow dow to the [8:TIterval] optio ad pre the [ENTER] key. Arrow over to the [Stat] meu ad pre the [ENTER] key. The type i the mea, ample tadard deviatio, ample ize ad cofidece level, arrow dow to [Calculate] ad pre the [ENTER] key. The calculator retur the awer i iterval otatio. Be careful, if you accidetally ue the [7:ZIterval] optio you would get the wrog awer. Or (If you have raw data i lit oe) Arrow over to the [Data] meu ad pre the [ENTER] key. The type i the lit ame, L, leave Freq: aloe, eter the cofidece level, arrow dow to [Calculate] ad pre the [ENTER] key. O the TI-89 go to the [App] Stat/Lit Editor, the elect d the F7 [It], the elect :TIterval. Chooe the iput method, data i whe you have etered data ito a lit previouly or tat whe you are give the mea ad tadard deviatio already. Type i the mea, tadard deviatio, ample ize (or lit ame (lit), ad Freq: ) ad cofidece level, ad pre the [ENTER] key. The calculator retur the awer i iterval otatio. Be careful, if you accidetally ue the [:ZIterval] optio you would get the wrog awer. pˆ pˆ Proportio z-iterval pˆ z / O the TI-84 pre the [STAT] key, arrow over to the [TESTS] meu, arrow dow to the [A:-PropZIterval] optio ad pre the [ENTER] key. The type i the value for X, ample ize ad cofidece level, arrow dow to [Calculate] ad pre the [ENTER] key. The calculator retur the awer i iterval otatio. Note: ometime you are ot give the x value but a percetage itead. To fid the x to ue i the calculator, multiply pˆ by the ample ize ad roud off to the earet iteger. The calculator will give you a error meage if you put i a decimal for x or. For example if pˆ =. ad = 4 the.*4 = 7.8, o ue x = 7. O the TI-89 go to the [App] Stat/Lit Editor, the elect d the F7 [It], the elect 5: -PropZIt. Type i the value for X, ample ize ad cofidece level, ad pre the [ENTER] key. The calculator retur the awer i iterval otatio. Note: ometime you are ot give the x value but a percetage itead. To fid the x value to ue i the calculator, multiply p by the ample ize ad roud off to the earet iteger. The calculator will give you a error meage if you put i a decimal for x or. For example if p =. ad = 4 the.*4 = 7.8, o ue x = 7. Sample Size z E z pˆ pˆ E / Alway roud up to the ext iteger.

5 S T A T 4 - R a c h e l L. W e b b, P o r t l a d S t a t e U i v e r i t y P a g e 5 ypothei Tetig α = P(Type I error) Ye I σ kow? No Ue the z α/ value ad σ i the formula.* Ue the t α/ value ad i the formula.* *If < 3, the variable mut be ormally Look for thee key word to help et up your hypothee: Two-tailed Tet Right-tailed Tet Left-tailed Tet : : : : : : Claim i i the Null ypothei = I equal to I le tha or equal to I greater tha or equal to I exactly the ame a I at mot I at leat a ot chaged from I ot more tha I ot le tha I the ame a Withi Claim i i the Alterative ypothei > < I ot More tha Le tha I ot equal to Greater tha Below I differet from Above Lower tha a chaged from igher tha Shorter tha I ot the ame a Loger tha Smaller tha Bigger tha Icreaed Decreaed The rejectio rule: p-value method: reject whe the p-value α. Critical value method: reject whe the tet tatitic i i the critical tail(). Cofidece Iterval method, reject whe the hypotheized value foud i i outide the boud of the cofidece iterval.

6 S T A T 4 - R a c h e l L. W e b b, P o r t l a d S t a t e U i v e r i t y P a g e 6 Claim i i Reject? Ye There i eough evidece to reject the claim that No There i ot eough evidece to reject the claim that Claim i i Reject? Ye No There i eough evidece to upport the claim that There i ot eough evidece to upport the claim that Fiih cocluio with cotext ad uit from quetio. Oe Sample Tet: -Sample Mea t-tet: : : Tet tatitic whe σ i ukow: x t with df= / : p p p p -Sample Proportio z-tet: Tet tatitic i z ˆ x where pˆ : p p p q Two Sample Tet: -Sample Mea t-tet Idepedet Populatio : : : or where uually you have : : : Note that i mot cae Aumig Equal Variace Tet tatitic whe ad are ukow: Aumig Uequal Variace Tet tatitic whe ad are ukow: t t -Sample Mea t-tet Depedet Populatio Fid the differece (d) betwee each matched pair. x x x x : D : D with df Tet tatitic: t d / d

7 S T A T 4 - R a c h e l L. W e b b, P o r t l a d S t a t e U i v e r i t y P a g e 7 Proportio : p p pˆ pˆ p p x x Tet tatitic z, (uually p p ) where pˆ ˆ ˆ : p p pq ˆ ˆ Correlatio ad Regreio : Tet tatitic for correlatio: t r : r with df= yˆ a bx r = ample correlatio coefficiet ρ = populatio correlatio coefficiet a = y-itercept b = lope = et = tadard error of etimate R = coefficiet of determiatio, q p Oe-Factor ANOVA table k=#of group, N=total of all : µ=µ=µ3= =µk : At leat oe mea i differet CV: Alway a right-tailed F, ue Excel =F.INV.RT(α, dfb, dfw) ANOVA Table Source SS df MS F (Tet Statitic) Betwee (Treatmet or Factor) SSB=( x x GM) k MSB = SSB / dfb F = MSB / MSW Withi (Error) SSW=( ) N k MSW = SSW / dfw Total SST N Whe you reject for a oe-factor ANOVA the you hould do a multiple compario. For example for 3 group you would have the followig 3 compario. (4 group would have 4C=6 compario) :µ=µ :µ=µ3 :µ=µ3 :µ µ :µ µ3 :µ µ3 Boferroi Tet t x x i MSE i j j with df = N k ad to get the p-value you would multiply the tail area by kc group. For example if you ue the tcdf i your calculator to fid the area i both the tail ad you have 4 group you would multiply the tail area by 4C=6.