Stat 200 Testing Summary Page 1

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1 Stat 00 Testig Summary Page 1 Mathematicias are like Frechme; whatever you say to them, they traslate it ito their ow laguage ad forthwith it is somethig etirely differet Goethe 1 Large Sample Cofidece Itervals 1001 α% largesample cofidece iterval for a populatio mea µ X z α σ, X + z α σ, where z α is the costat such that P Z z α = 1 α Notes a The values of z α for various cofidece coefficiets are readily obtaied from the table of ormal probabilities I particular, 1001 α% z α 90% % % 58 b Whe σ is ukow, ad is large say, bigger tha 30, the it is usual to replace σ i the cofidece iterval formula by its estimator s c The cofidece iterval result holds as log as X i have fiite variace, ot just whe the X i s are ormal If, however, X i is ormal, i = 1,,, the the cofidece iterval above is exact ie has exact cofidece coefficiet 1001 α% 1001 α% large sample cofidece iterval for a differece i meas µ X µ Y X Y z α σ X + σ Y, X Y + z α σ X + σ Y Whe the true variaces σ X ad σ Y are ukow ad ad are large, the s X ad s Y are used i place of σx ad σ Y, respectively, i the above formula 1001 α% large sample cofidece iterval for a proportio p ˆp z α ˆp1 ˆp, ˆp + z α ˆp1 ˆp 1001 α% large sample cofidece iterval for a differece i proportios p 1 p ˆp 1 ˆp z α ˆp 1 1 ˆp 1 + ˆp 1 ˆp, ˆp 1 ˆp + z α ˆp 1 1 ˆp 1 + ˆp 1 ˆp Large sample hypothesis tests
2 Stat 00 Testig Summary Page Note: oesided tests are give i paretheses ad square brackets [ ] These tests are valid for large say, > 30, ad populatio variaces ca be replaced by sample variaces wherever appropriate Large sample hypothesis test for a populatio mea µ We wish to test H 0 : µ = µ 0 agaist the alterative H A : µ µ 0 H A : µ > µ 0 [H A : µ < µ 0 ] The test statistic is Z = X µ 0 σ/ or if the observed value of Z exceeds z α [if the observed value of Z is smaller tha z α ] Large sample hypothesis test for a differece i meas µ X µ Y We wish to test H 0 : µ X µ Y = D 0 agaist the alterative H A : µ X µ Y D 0 H A : µ X µ Y > D 0 [H A : µ X µ Y < D 0 ] The test statistic is Z = X Y D 0 σ X + σ Y or if the observed value of Z exceeds z α [if the observed value of Z is smaller tha z α ] Large sample hypothesis test for a proportio p We wish to test H 0 : p = p 0 agaist the alterative H A : p p 0 H A : p > p 0 [H A : p < p 0 ] The test statistic is Z = ˆp p 0 p0 1 p 0 or if the observed value of Z exceeds z α [if the observed value of Z is smaller tha z α ] Large sample hypothesis test for a differece i proportios p 1 p We wish to test H 0 : p 1 = p agaist the alterative H A : p 1 p H A : p 1 > p [H A : p 1 < p ] Let ˆp = X + Y / + deote the overall proportio of successes for the two populatios The test statistic is ˆp 1 ˆp 0 Z = 1 ˆp1 ˆp + 1 or if the observed value of Z exceeds z α [if the observed value of Z is smaller tha z α ]
3 Stat 00 Testig Summary Page 3 3 Smallsample cofidece itervals 1001 α% small sample cofidece iterval for a populatio mea µ X t 1 α/ s, X + t 1 α/ s, where t 1 α/ is the critical poit of a t 1 distributio such that P T t 1 α/ = 1 α, ie 1001 α% of the area uder the t 1 desity lies betwee t 1 α/ ad t 1 α/ 1001 α% small sample cofidece iterval for a differece i meas µ X µ Y from idepedet populatios To compute this iterval we eed to make the additioal assumptio that the variaces of the two populatios are equal ie that σx = σ Y The, we estimate this commo variace usig the pooled variace estimator, The cofidece iterval is the s pooled = 1s X + 1s Y + X Y t 1+ α/s pooled , X Y + t 1+ α/s pooled Note that the degrees of freedom used for the t statistic is α% small sample cofidece iterval for a differece i meas µ X µ Y from paired data The data is X 1, Y 1,, X, Y Form the differeces D 1 = X 1 Y 1,, D = X Y, ad let D ad s D deote the sample mea ad sample stadard deviatio of the differeces D i, i = 1,, The the iterval is D t 1 α/ sd, D + t 1 α/ sd 4 Small sample hypothesis tests Note: Oesided tests are give i paretheses ad square brackets [ ] Small sample hyptohesis test for a mea µ We wish to test H 0 : µ = µ 0 agaist H A : µ µ 0 H A : µ > µ 0 [H A : µ < µ 0 ] The test statistic is T = X µ 0 s/ We reject H 0 i favor of H A at sigificace level α if the observed value of T either exceeds t 1 α/ or is smaller tha t 1 α/ if the observed value of T exceeds t 1 α [if the observed value of T is smaller tha t 1 α] Note that the α/ th critical poit is used for twosided tests while the α th critical poit is used for oesided tests
4 Stat 00 Testig Summary Page 4 Small sample hypothesis test for a differece i meas µ X µ Y from idepedet populatios We wish to test H 0 : µ X µ Y = D 0 agaist H A : µ X µ Y D 0 H A : µ X µ Y > D 0 [H A : µ X µ Y < D 0 ] The test statistic is T = X Y D 0 s pooled 1 + 1, where s pooled is the pooled variace estimator s pooled = 1s X + 1s Y / + We reject H 0 i favor of H A at sigificace level α if the observed value of T e ither exceeds t 1 + α/ or is smaller tha t 1 + α/ if the observed value of T exceeds t 1 + α [if the observed value of T is smaller tha t 1 + α] freedom for this test is + ot 1 Note the degrees of Small sample hypothesis test for a differece i meas µ X µ Y from paired data We are iterested i testig H 0 : µ X µ Y = 0 agaist H A : µ X µ Y 0 H A : µ X µ Y > 0 [H A : µ X µ Y < 0] The test is based o the differeces D i = X i Y i, i = 1,,, so is equivalet to testig H 0 : µ D = 0 agaist H A : µ D 0 H A : µ D > 0 [H A : µ D < 0], where µ D is the mea of the populatio of differeces This is just a oesample test for the mea µ D, so the test statistic is T = D 0 s D / We reject H 0 i favor of H A at sigificace level α if the observed value of T e ither exceeds t 1 α/ or is smaller tha t 1 α/ if the observed value of T exceeds t 1 α [if the observed value of T is smaller tha t 1 α] 5 Tests ad cofidece itervals for the variace 1001 α% cofidece iterval for the populatio variace σ 1s χ 1 α/, 1s χ, 11 α/ where χ 1 α/ is the critical poit of the χ 1 distributio such that if X χ 1 the P X χ 1α/ = 1 α/ Hypothesis test for the variace σ We wish to test the hypothesis H 0 : σ = σ0 agaist H A : σ σ 0 H A : σ > σ 0 [H A : σ < σ 0] The test statistic is χ = 1s σ0 We reject H 0 i favor of H A if the observed χ value either exceeds χ 1 α/ or is smaller tha χ 1 1 α/ if the observed value of χ exceeds χ 1 α [if the observed value of χ is smaller
5 Stat 00 Testig Summary Page 5 tha χ 1 1 α] Note that the α/ th critical poit of the χ 1 distributio is used for the twotailed test, while the α th critical poit is used for the oesided test Hypothesis test for the equality of variaces Recall that i order to carry out a test of the hypothesis that µ X = µ Y from idepedet populatios, it was ecessary to assume σx = σ Y We ca formally test this assumptio as follows: we wish to test H 0 : σx = σ Y H A : σx > σ Y [H A : σx > σ Y ] The test statistic is agaist H A : σ X σ Y F = s X s Y We reject H 0 i favor of H A at sigificace level α if the observed F value exceeds F 1 1, 1α/, the upper α/ th critical poit of a F 1 1, 1 distributio, is falls below F 1 1, 11 α/ if the observed value of F exceeds F 1 1, 1α [if the observed value of F is smaller tha F 1 1, 11 α Tables for both the F ad χ distributios for various degrees of freedom are give at the back of the book Note that i order to coduct some of the tests above we eed use the special idetity for F distributios: F,m 1 α = 1 F m, α
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