Statistical Estimation

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1 Learig Objecives Cofidece Levels, Iervals ad T-es Kow he differece bewee poi ad ierval esimaio. Esimae a populaio mea from a sample mea f large sample sizes. Esimae a populaio mea from a sample mea f small sample sizes. Esimae a populaio propio from a sample propio. Esimae he miimum sample size ecessary o achieve give saisical goals. aisical Esimaio Poi esimae -- he sigle value of a saisic calculaed from a sample Ierval Esimae -- a rage of values calculaed from a sample saisic(s) ad sadardized saisics, such as he. elecio of he sadardized saisic is deermied by he samplig disribuio. elecio of criical values of he sadardized saisic is deermied by he desired level of cofidece. Cofidece Ierval o Esimae µ whe is Large Poi esimae Ierval Esimae X X = X ± X µ X + Disribuio of ample Meas f (1-)% Cofidece level - igificace Level 1 ces f Cofidece Iervals i Relaio o

2 Disribuio of ample Meas f (1-)% Cofidece Disribuio of ample Meas f 95% Cofidece % µ X Values f ome of he Me Commo Levels of Cofidece Probabiliy Ierpreaio of he Level of Cofidece Cofidece Level 90% 95% 98% 99% Value Pr ob[ X µ X + ] = 1 Exercise: 95% Cofidece Ierval f µ X = 4.6, = 1.1, ad = 60. X µ X µ µ µ 454. Cofidece Ierval o Esimae µ whe is Large ad is Ukow X ± X X µ +

3 Esimaig he Mea of a Nmal Populaio: mall ad Ukow The populaio has a mal disribuio. The value of he populaio sadard deviaio is ukow. The sample size is small, < 30. disribuio is o appropriae f hese codiios disribuio is appropriae A family of disribuios -- a uique disribuio f each value of is parameer, degrees of freedom (d.f.) ymmeric, Uimodal, Mea = 0, Flaer ha a fmula The Disribuio = X µ Compariso of eleced Disribuios o he adard Nmal The disribuio adard Nmal (d.f. = 5) (d.f. = 5) (d.f. = 1) f() (1) / / -c 0 c red area = rejecio regio f -sided es f() / Probabiliy aemes (1) - 0 c c P( < - c ) = P( > c ) = / / P(- c < < c ) = 1 Excel: P =TDIT( c, -1, #_of_ails) Iverse: c =TINV(a, -1) Iferece abou a populaio MEAN whe he populaio TDEV Is Ukow x µ = s / Whe he sampled populaio is mally disribued, he saisic is ude disribued wih -1 degrees of freedom. s Cofidece Ierval: x ± /, 1 where /, 1 is he / quaile of he ude -disribuio wih -1 degrees of freedom. 3

4 Checkig he required codiios Table of Criical Values of I derivig he es ad cofidece ierval, we have made wo assumpios: (i) he sample is a radom sample from he populaio; (ii) he disribuio of he populaio is mal. / f() df (1) The es is robus he resuls are sill approximaely valid as log as (i)he populaio is o exremely o-mal. (ii), if he sample size is large / / - 0 c c Excel: c =TINV(, -1) Cofidece Iervals f µ of a Nmal Populaio: mall ad Ukow X ± X µ X + df = 1 Ex: Fid 99% Cofidece Ierval X =.14, = 1.9, = 14, df = 1 = = = = ,13 X µ X µ µ µ 318. fmula Deermiig ample ize whe Esimaig µ X µ = Err of Esimaio (olerable E = X µ err) Esimaed ample ize = = E E Esimaed 1 4 rage Ex: ample ize whe Esimaig µ E = 1, = 4 90% cofidece = = = E (1.645) (4) 1 =

5 Example Compaies ha sell groceries over he Iere are called e-grocers. Cusomers eer heir ders, pay by credi card, ad receive delivery by ruck. A poeial e-grocer aalyzed he marke ad deermied ha o be profiable he average der would have o exceed $85. To deermie wheher a e-grocer would be profiable i oe large ciy, she offered he service ad recded he size of he der f a radom sample of cusomers. Ca we ifer from he daa ha he e-grocery will be profiable i his ciy a sigificace level 0.05? 5

Chapter Chapter 10 Two-Sample Tests X 1 X 2. Difference Between Two Means: Different data sources Unrelated. Learning Objectives

Chapter Chapter 10 Two-Sample Tests X 1 X 2. Difference Between Two Means: Different data sources Unrelated. Learning Objectives Chaper 0 0- Learig Objecives I his chaper, you lear how o use hypohesis esig for comparig he differece bewee: Chaper 0 Two-ample Tess The meas of wo idepede populaios The meas of wo relaed populaios The