m = Statistical Inference Estimators Sampling Distribution of Mean (Parameters) Sampling Distribution s = Sampling Distribution & Confidence Interval

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1 Saplig Ditributio & Cofidece Iterval Uivariate Aalyi for a Nueric Variable (or a Nueric Populatio) Statitical Iferece Etiatio Tetig Hypothei Weight N ( =?, =?) 1 Uivariate Aalyi for a Categorical Variable (or a Categorical Populatio) Etiator Statitic Ued to Etiate Populatio Paraeter 1 - p p Soke Do ot Soke Sokig Habit Statitic Saple Mea, Saple Variace, Saple Proportio, Paraeter populatio ea populatio variace p populatio proportio Populatio proportio: p (or p ) 3 4 Saplig Ditributio Saplig ditributio i probability ditributio of the aple Statitic. Saplig Ditributio of Mea (Paraeter) What i the aplig ditributio of ea? Shape: Noral Paraeter: Mea, Stadard Deviatio I ay ituatio, ea ad tadard deviatio ca copletely deterie the ditributio of a pecific hape. = = 5 6 CI - 1

2 Saplig Ditributio & Cofidece Iterval Saplig Ditributio of Mea (Ditributio hape) Noral ditributio theore: If a rado aple i take fro a orally ditributed populatio, the the aplig ditributio of ea would be oral. Cetral Liit Theore: Whe a relative large rado aple i take fro ay populatio, regardle of the ditributio of the populatio, the aplig ditributio of ea would be approiately oral. 7 Saplig Ditributio of Mea = = _ Ditributio of the aple ea i oral if apled populatio i orally ditributed, or the aple ie i relatively large. 8 X Probability Related to Mea Eaple: Coider the ditributio of eru choleterol level for 40- to 70-year-old ale livig i couity A ha a ea of 11 g/100 l, ad the tadard deviatio of 46 g/100 l. If a rado aple of 100 idividual i take fro thi populatio, what i the probability that the average eru choleterol level of thee 100 idividual i higher tha 5? X N ( =, = ) 9 P(X > 5) =? Paraeter of the aplig ditributio of the ea: Choleterol Level ha a ea 11, d. 46. = = 11 = = The aplig ditributio of the ea i orally ditributed. X ~ N ( = 11, = 4.6) = = P(X > 5) =? Choleterol Level ha a ea 11,.d. 46. X N ( = 11, = 4.6) = 100 Itroductio to Etiatio 11 5 P( X 5) = P( Z 3.04) = = 3.04 Cofidece Iterval & Saple Sie CI -

3 Saplig Ditributio & Cofidece Iterval Saplig Error Saple tatitic (poit etiate) Key Eleet of Iterval Etiatio Cofidece Level: A probability that the populatio paraeter fall oewhere withi the iterval. Cofidece iterval Saple tatitic (poit etiate) Saplig Error = 13 Cofidece liit (lower) Cofidece liit (upper) Margi of Error 98 1 F 14 Saplig Ditributio of Mea (the ie of cofidece iterval) _ -? +? Withi how ay tadard deviatio of the ea will have 95% of the aplig ditributio?.05 X A Special Notatio = the core that the proportio of the tadard oral ditributio to the right of it i. Z = 1.96? =? * Margi of error i half of thi iterval The Cofidece Iterval The Cofidece Iterval Cofidece Level _ 1.96 =.05 / / = =.95 X Cofidece Iterval => 95% Saple Mea % 95% Saple _.5% 18 X 95 % of iterval cotai. 5% do ot. CI - 3

4 Saplig Ditributio & Cofidece Iterval Cofidece Iterval for Mea ( Kow) (1-) 100% Cofidece Iterval Etiate for ea of a oral populatio ( X - Z /, X Z / ) or X Z / Kow ay ea that we have very good etiate of. Margi of Error Cofidece Iterval of Mea ( ukow ad 30) (1-) 100% Cofidece Iterval Etiate for ea of a populatio whe aple ie i relative large ( X - Z /, X Z / ) It i ot practical to aue that we kow or X Z / Cofidece Iterval Mea ( Ukow & < 30) 1. Auptio Populatio Stadard Deviatio I Ukow Populatio Mut Be Norally Ditributed. Ue Studet t Ditributio 3. Cofidece Iterval Etiate S S ( X - t /, -1, X t /, -1 X t, -1 ) 1 Bell-Shaped Syetric Fatter Tail Studet t Ditributio Stadard Noral (Z) 0 t (df = 13) - t = t (df = 5) Z t Studet t Table For a 90% C.I.: = 3 df = - 1 = =.10 / =.05 t / =?.05 t value t 4 CI - 4

5 Saplig Ditributio & Cofidece Iterval Average Weight for Feale Te Year Childre I US Ifo. fro a rado aple: = 10, = 80 lb, = lb, aue weight i orally ditributed, fid the 95% cofidece iterval etiate for average weight. Data: Average Weight for Feale Te Year Childre I US Data: = 10, = 80 lb, = lb t / = t.05/ = t 0.05, d.f. = 10 1 = 9, t 0.05, 9 =.6 t / How do we kow whether orality auptio i OK? (67.09, 9.91) 6 Weight for Te Year Old ht (poud) articipat What i your e? f eale ale Mea Decriptive 95% Cofidece Iterv al for Mea 5% Tried Mea Media Variace Std. Dev iatio Miiu Maiu Rage Iterquartile Rage Skewe Kurtoi Mea 95% Cofidece Iterv al for Mea Lower Boud Upper Boud Lower Boud Upper Boud Statitic Std. Error Cofidece iterval with -core: The (1- )% cofidece iterval etiate for populatio ea: Auptio: If apled fro oral populatio with kow variace,, / Auptio: If large aple ad if ukow variace, replace, / 8 Cofidece iterval with t-core: The (1- )% cofidece iterval etiate for populatio ea: Auptio: If apled fro oral populatio with ukow variace,, t /, df = -1 (If aple ie i large the orality auptio i iigificat.) t a aple becoe large 9 Thikig Challege What i the average ittig pule rate for tudet i cla? Fid the 95% cofidece iterval etiate. What i your pule rate? 30 CI - 5

6 Saplig Ditributio & Cofidece Iterval Eaple A reearcher wihe to fid out whether a ew diet progra ca help a particular populatio i reducig Body Ma Ide (BMI). The BMI core fro a rado aple of ubject were recorded both before ad after the diet progra. Ue the followig data to fid a 95% cofidece iterval for etiatig the average reductio i BMI. BMI ID Before After Diff Fidig Saple Sie for Etiatig C.I. : Margi = E of Error = E = Z 3 Thikig Challege You pla to urvey reidet i your couty to fid the average health iurace preiu that they are payig. You wat to be 95% cofidet that the aple ea i withi ± $50. A pilot tudy howed that wa about $400. What aple ie hould you ue? Proportio Etiatio Paraeter: Populatio Proportio p (or p) (Percetage of people ha o health iurace) Statitic: Saple Proportio p ˆ = i uber of uccee i aple ie 33 Reark: If data i coded a 1 or 0, aple ea i the ae a aple proportio of Data: 1, 0, 0, 1, 0 = = =. = p Cofidece Iterval Proportio 1. Auptio Two Categorical Outcoe Noral Approiatio Ca Be Ued If p ad (1 p) are both greater tha 5.. Cofidece Iterval Etiate (for large aple) (1 - ) (1 - ) ( -, ) (1 - ) Etiatio Eaple Proportio A rado aple of 400 fro a large couity howed that 3 have diabete. Set up a 95% cofidece iterval etiate for p, the percetage of people that have diabete. = 400 (1 - ) Z / 3 = = (1 -.08) = = %.7% ( 5.3%, 10.7% ) CI - 6

7 Saplig Ditributio & Cofidece Iterval Saple Sie C.I.: or = E 0.5 (1 - ) Margi of Error = E = Z = (1 - ) E (1 - ) if pilot tudy i doe. to get the larget aple to achieve the goal. 37 Saple Sie (No prior iforatio o p) Saple Sie Eaple: If oe wihe to do a urvey to etiate the populatio proportio with 95% cofidece ad a argi of error of 3%, how large a aple i eeded? Z / = 1.96; E =.03 = (1.96 /.03 ).5 = A aple of ie 1068 i eeded. 38 Saple Sie (With prior iforatio o p) Saple Sie Eaple: If oe wihe to to etiate the percetage of people ifected with Wet Nile i a populatio with 95% cofidece ad a argi of error of 3%, how large a aple i eeded? (A pilot tudy ha bee doe, ad the aple proportio wa 6%.) Z / = 1.96; E =.03 = (1.96 /.03 ).06 (1.06) = 40.7 A aple of ie 41 i eeded. How large a aple wa ued for pilot tudy? 39 Eaple Oe claied that the percetage of iddle chool tudet i a populatio that had adequate eercie i ore tha 30%. A rado aple of 300 iddle chool tudet fro thi populatio i urveyed, ad 110 of the had adequate eercie. Fid the 95% cofidece iterval for etiatig the percetage of the iddle chool tudet i populatio A that had adequate eercie. 40 CI - 7