STAT431 Review. X = n. n )
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1 STAT43 Review I. Results related to ormal distributio Expected value ad variace. (a) E(aXbY) = aex bey, Var(aXbY) = a VarX b VarY provided X ad Y are idepedet. Normal distributios: (a) Z N(, ) (b) X N(µ, σ ), the X µ N(, ) σ (c) if X N(µ x, σx), Y N(µ y, σy), ad they are idepedet, the I particular, if X i N(µ, σ ), i=,,...,, ax by N(aµ x bµ y, a σ x b σ y) X = Xi N(µ, σ ) 3 The cetral limit theorem: If X i f(x), i =,...,, µ = EX, < σ = VarX <, the whe is approx large, X i N(µ, σ ) or X approx N(µ, σ ). i= 4 Biomial Distributio (ad sample): (a) Proportio of successes i idepedet (Beroulli) trials, each with probability p of success. Has probability fuctio p(k) = ( k ) p k q k where q = p, k =,...,. (populatio) mea = p, populatio SD = (pq) (b) Sample estimate of p is ˆp = k /. SD of this estimate is σˆp = pq ˆpˆq ˆσˆp = = StadardError (c) Whe is large ˆpis approximately N(p, σˆp ) 4 χ distributio. (a) Let Z i N(, ), i =,...,, the. This is estimated by U = Zi χ. (b) If U χ, the i. U has Gamma(, ). EU=, VarU=. ii. X Gamma(α, λ), f(x) = λα Γ(α) xα e λx, x>. (c) If U i N(µi, σi ), the i= ( U i µ i σ i ) χ. -
2 5 t distributio (a) If Z N(, ), idepedet of U χ, the Z U/ t (b) The desity fuctio of t is symmetric about. (c) Whe the degrees of freedom, the f T (t) Stadard Normal 6 F m, distributio. If U χ m idepedet of V χ, the U/m V/ F m,. 7 The sample mea ad the sample variace. Suppose X i f with mea µ ad variace σ. (a) Sample mea X = X i / i= Sample variace s i= = (X i X) X = i (X). E(X) = µ, Var(X) = σ /, E(s ) = σ (b) If X i N(µ, σ ), the ( )s i) σ χ ii) X µ s/ t II. Cofidece Itervals ad Hypothesis testig. Let X, X,..., X be a radom sample from f(x, θ). (T (X, X,..., X ) < θ < T (X, X,..., X )) is a ( α)% CI of θ if P(T (X, X,..., X ) < θ < T (X, X,..., X )) = α. Hypotheses: The ull hypothesis, H, statemets we are agaist The alterative hypothesis, H a, or (H, H A, H α ), statemets we are i favor of. 3 A sigificat test is a decisio rule based o the data which decides whether to reject H. 4 Errors. Type I error: Type II error: α=p(reject H, whe H is true) β=p(fail to reject H, whe H a is true) 5 α level sigificace test: A test whose type I error is α. 6 The p- value: The smallest α at which H ca be rejected. -
3 Facts ad Formulas Chapter 7: Cofidece Itervals Oe Sample Mea: The basic form of a (-α)% cofidece iterval for a ukow populatio mea, µ, is * σ * * σ* * σ* x± C, which is the same as x C, x C, where C* comes from the appropriate table ad σ* deotes the appropriate populatio SD or estimate thereof. The meaig of such a iterval is that Prob (µ Iterv) = -α, at least approximately. (The term σ * correspods to (a estimate of) the SD of X, ad hece is sometimes referred to as the SE.) The importat situatios are: σ kow ad populatio ormal or large: σ* = σ ad C* = z α /, from stadard ormal tables σ ukow ad large: σ* = S ad C* = z α /, from stadard ormal tables. σ ukow ad populatio ormal: σ* = S ad C* = t α /, df=- from the t-table. The sample size eeded to get a total iterval width of w is (approximately) σ = z α w. If σ is ot kow, use a estimate from previous experiece or from the correspodig value of S i a pilot experimet (as described i the problem). Oe populatio proportio: Whe is large ad ˆp is ot too ear or you ca use the classical largesample formula pˆ ± zα pq ˆˆ. (Large here is approximately pq ˆˆ.) Otherwise use the score (= Wilso s ) formula pq ˆˆ z pˆ z ± z 4 with z= z α. z z The sample size eeded to get a total iterval width of w is (approximately) pq = z α where p is a prior guess for the true value of p. w
4 If o good iformatio is available to guess the true value of p the use p = ½. This gives a coservative (possibly too large) choice of. (The formula for is based o the largesample formula. If the turs out small, this will ot be a very accurate aswer, but use it ayway.) Predictio Itervals: Whe the populatio is ormal a predictio iterval for a sigle future observatio * takes the form x± C σ * with C* ad σ* as above. If we deote that future observatio by Y such a iterval has the property that Prob(Y PredIterv) -α. Chapter 8: Hypothesis Tests Geeral Theory Tests ivolve a ull hypothesis, H, ad a alterative hypothesis, H a. A typical case is H :µ = µ ad either H a :µ µ, ( two-sided ) or H a :µ < µ, ( oe-sided ). They ivolve a test statistic, call it T for ow, ad a rejectio regio such as T >K or ϒ <K. Suppose we are testig a ukow mea, µ. The Pµ H (Reject H ) = Pr µ ("Type I Error"), ad Pµ Ha( DoNotReject H ) = Pr µ ("Type II Error") = β at this µ. The sigificace level, α, is the probability of type I error at the boudary value, µ, that divides H ad H a. Oe related term is the Power at µ = P µ (Reject H ) = - β. If oe is testig H as above ad observes a value ϒ = τ the the P-value correspodig to τ is the value of α for which H is just barely rejected (or just barely otrejected). If you kow P ad α the you kow the outcome of the test sice α > P Reject H ad α P DoNotReject H. For the usual two-sided tests a level α test DoesotReject exactly whe µ is i the -α cofidece iterval. Particular Tests Oe sample mea: A test of H :µ = µ vs H a :µ µ has the form: Reject if ϒ > C* where X µ ϒ = σ *
5 ad σ* ad C* are as i the above discussio of two-sided cofidece itervals. The P-value correspodig to ϒ = τ ca be foud from a ormal or t-table (depedig whether is large or small) by lookig up P( Z > τ) i the ormal case, ad P( t df=- > τ) i the t case. NOTE that because of the sig these ivolve multiplyig the tabled values times. The sample size for which a two sided level α test of µ has Type II error = β at the alterative µ is σ( zα zβ) =. (approximately) µ µ I the oe sided case put z α i place of z α/. (The aswer should be a large sice this formula ivolves the ormal tables.) A test of the oe sided alterative H a :µ < µ would reject whe ϒ < C** where C** is as i the case of a oe sided lower cofidece boud. P-values ca be foud aalogously. NOTE they do ot ivolve multiplyig tabled etries times. Oe populatio proportio: For testig H :p = p use ˆp p ϒ = pq with rejectio regios determied i the usual maer from the stadard ormal table (as i the case of oe sample mea). P-values are also determied from this ϒ i a aalogous maer. ( should ot be too small here p q > 5 should suffice. There is a procedure based o the biomial distributio that ca be used for smaller.) The formula for the for which a two-sided test of p also satisfies Type II error = β whe p = p is zα pq z p β q = p p For a oe-sided test substitute z α for z α/. Differece of Two meas: (approximately). Chapter 8: Ifereces based o Two-Samples The ull hypothesis will be of the form H : µ - µ =. (Usually =.) If the samples from the two meas, X ad Y, are idepedet the use 3
6 ϒ = X Y ( σ ) ( ) * * σ with σ* ad C* as i the oe sample procedures, above. I the case where or are small ad the values of σ are ot kow i advace you eed to assume ormality, ad the treat ϒ accordig to a t-distributio with ν df, where s s ν =. s s Note that mi{ -, -} ν. A (-α) % cofidece iterval for µ - µ takes the form X Y ± C* SE where SE is the deomiator of the test statistic, ϒ. P-values ca be foud i the usual way from the value of ϒ ad the correspodig table. Formulas for β ca be derived from the structure of the test. Sample size calculatios are complicated, except for some special cases, ad we do ot give geeral formulas here, or for two proportios, below. If the additioal assumptio that σ = σ seems teable the use s p as the commo estimate for σ ad - as the degrees of freedom. Meas of Paired Samples: Whe the data are paired i ay possibly meaigful way the ull hypothesis H : µ - µ = should be tested by a matched pairs test. This test computes the differeces D i = X i - Y i ad the uses these for a oe-sample test of H : µ D =. Differece of two proportios: For testig H : p = p the situatio is etirely similar to the precedig, except that we ow use pˆ pˆ ϒ = alog with the ormal table, where pq ˆˆ ( ) ˆp is the COMBINED SAMPLE estimate of the proportio, defied by X Y pˆ = = pˆ ˆ p. 4
7 (For validity of this test the ad should be ot too small; similar to what is eeded i the case of testig oe proportio.) P-values ca be computed i the usual fashio. NOTE that for a Cofidece Iterval for the differece of two proportios you should ot use the SE based o the pooled estimate, ˆp, above. Istead you should use pˆˆ q ˆˆ pq as the SE. Equality of variaces: Suppose S ad S are the sample variaces from idepedet samples of size ad. Assume the populatios are (approximately) ormally distributed with respective populatio SDs σ ad σ. To test H : σ = σ use the statistic S F =. S Refer the values of this statistic to a F-table with ν = - df i the umerator ad ν = df i the deomiator. Our F table like most others is very icomplete, ad a little awkward to use. We will usually get the F-values from JMP. (O a exam suitable values from JMP ad maybe some usuitable oes will be separately supplied.) For a -sided test of this H reject if F is either too small or too large. 5
Common Large/Small Sample Tests 1/55
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