Statistics Parameters

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1 Saplig Ditributio & Cofidece Iterval Etiator Statitical Iferece Etiatio Tetig Hypothei Statitic Ued to Etiate Populatio Paraeter Statitic Saple Mea, Saple Variace, Saple Proportio, Paraeter populatio ea populatio variace p populatio proportio 1 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. 3 4 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. 5 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 (, ) 6 CI - 1

2 Saplig Ditributio & Cofidece Iterval P(X > 5) =? Paraeter of the aplig ditributio of the ea: Choleterol Level ha a ea 11, d The aplig ditributio of the ea i orally ditributed. X ~ N ( 11, 4.6) P(X > 5) =? X N ( 11, 4.6) = P( X 5) P( Z 3.04) Choleterol Level ha a ea 11,.d = z 8 A Special Notatio Itroductio to Etiatio Cofidece Iterval & Saple Size z = the z core that the proportio of the tadard oral ditributio to the right of it i. Z z.05 = 1.96? z.010 =? z 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 = 11 Cofidece liit (lower) Cofidece liit (upper) Margi of Error 98 1 F 1 CI -

3 Saplig Ditributio & Cofidece Iterval Saplig Ditributio of Mea The Cofidece Iterval.05 -?.95 _ +?.05 X Cofidece Level _ 1.96 = z.05 / / = =.95 X Withi how ay tadard deviatio of the ea will have 95% of the aplig ditributio? 13 Cofidece Iterval => 95% Saple Mea The Cofidece Iterval.5% 95% Saple _.5% 15 X 95 % of iterval cotai. 5% do ot. 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. It i ot practical to aue that we kow. Margi of Error 16 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 ) 17 Studet t Table t value For a 90% C.I.: = 3 df = - 1 = =.10 / =.05 t / =? t CI - 3

4 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: How do we kow whether orality auptio i OK? 19 0 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 / (67.09, 9.91) 1 Shapiro-Wilk orality tet data: TeYearOldChild$Weight W = , p-value = Oe Saple t-tet data: TeYearOldChild$Weight t = , df = 9, p-value =.033e-07 alterative hypothei: true ea i ot equal to 0 95 percet cofidece iterval: aple etiate: ea of 80 Thikig Challege What i the average ittig pule rate for tudet i cla? Fid the 95% cofidece iterval etiate. What i your pule rate? 3 Cofidece iterval with z-core: The (1- )% cofidece iterval etiate for populatio ea: Auptio: If apled fro oral populatio with kow variace,, z / Auptio: If large aple ad if ukow variace, replace, z / 4 CI - 4

5 Saplig Ditributio & Cofidece Iterval 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 ize i large the orality auptio i iigificat.) t z a aple becoe large 5 Fidig Saple Size for Etiatig C.I. : Margi z z E of Error E Z 6 Proportio Etiatio Paraeter: Populatio Proportio p (or p) (Percetage of people ha o health iurace) Statitic: Saple Proportio i uber of uccee i aple ize Reark: If data i coded a 1 or 0, aple ea i the ae a aple proportio of Data: 1, 0, 0, 1, 0. 4 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 - ) - z, z ) (1 - ) z 8 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 / (1 -.08) z z %.7% ( 5.3%, 10.7% ) 9 Saple Size (1 - ) C.I.: z (1 - ) Margi of Error E Z z (1 - ) if pilot tudy i doe. E or z to get the larget aple to 0.5 E achieve the goal. 30 CI - 5

6 Saplig Ditributio & Cofidece Iterval Saple Size (No prior iforatio o p) Saple Size 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 ize 1068 i eeded. 31 Saple Size (With prior iforatio o p) Saple Size 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 ize 41 i eeded. How large a aple wa ued for pilot tudy? 3 Eaple: Reearcher wih to etiate the percetage of hopital eployee ifected by a certai dieae i a coutry. Out of 500 radoly choe hopital eployee, 64 were ifected. Fid the 95% cofidece iterval etiate for percetage of hopital eployee ifected by thi dieae i thi coutry. 33 CI - 6