Confidence Intervals for Paired Means

Transcription

1 Chaper 496 Cofidece Iervals for Paired Meas Iroducio This rouie calculaes he sample size ecessary o achieve a specified disace from he paired sample mea erece o he cofidece limi(s) a a saed cofidece level for a cofidece ierval abou he mea erece whe he uderlyig daa disribuio is ormal. Cauio: This procedure assumes ha he sadard deviaio of he fuure sample will be he same as he sadard deviaio ha is specified. If he sadard deviaio o be used i he procedure is esimaed from a previous paired sample or represes he populaio sadard deviaio, he Cofidece Iervals for Paired Meas wih Tolerace Probabiliy procedure should be cosidered. Tha procedure corols he probabiliy ha he disace from he mea paired erece o he cofidece limis will be less ha or equal o he value specified. Techical Deails For a paired sample mea erece from a ormal disribuio wih kow variace, a wo-sided, 100(1 α)% cofidece ierval is calculaed by z ± 1 / α 2σ where is he mea of he paired ereces of he sample, ad σ is he kow sadard deviaio of paired sample ereces. A oe-sided 100(1 α)% upper cofidece limi is calculaed by z σ + Similarly, he oe-sided 100(1 α)% lower cofidece limi is z σ For a paired sample mea erece from a ormal disribuio wih ukow variace, a wo-sided, 100(1 α)% cofidece ierval is calculaed by ± ˆ σ / 2,

2 Cofidece Iervals for Paired Meas where is he mea of he paired ereces of he sample, ad σˆ is he esimaed sadard deviaio of paired sample ereces. A oe-sided 100(1 α)% upper cofidece limi is calculaed by +, 1 Similarly, he oe-sided 100(1 α)% lower cofidece limi is ˆ σ, 1 Each cofidece ierval is calculaed usig a esimae of he mea erece plus ad/or mius a quaiy ha represes he disace from he mea erece o he edge of he ierval. For wo-sided cofidece iervals, his disace is someimes called he precisio, margi of error, or half-widh. We will label his disace, D. The basic equaio for deermiig sample size whe D has bee specified is whe he sadard deviaio is kow, ad D = D = z ˆ σ 1 α / 2σ ˆ / 2, 1σ whe he sadard deviaio is ukow. These equaios ca be solved for ay of he ukow quaiies i erms of he ohers. The value α / 2 is replaced by α whe a oe-sided ierval is used. Fiie Populaio Size The above calculaios assume ha samples are beig draw from a large (ifiie) populaio. Whe he populaio is of fiie size (N), a adjusme mus be made. The adjusme reduces he sadard deviaio as follows: σ fiie = σ 1 N This ew sadard deviaio replaces he regular sadard deviaio i he above formulas. Cofidece Level The cofidece level, 1 α, has he followig ierpreaio. If housads of samples of iems are draw from a populaio usig simple radom samplig ad a cofidece ierval is calculaed for each sample, he proporio of hose iervals ha will iclude he rue populaio mea erece is 1 α

3 Procedure Opios Cofidece Iervals for Paired Meas This secio describes he opios ha are specific o his procedure. These are locaed o he Desig ab. For more iformaio abou he opios of oher abs, go o he Procedure Widow chaper. Desig Tab The Desig ab coais mos of he parameers ad opios ha you will be cocered wih. Solve For Solve For This opio specifies he parameer o be solved for from he oher parameers. Oe-Sided or Two-Sided Ierval Ierval Type Specify wheher he ierval o be used will be a oe-sided or a wo-sided cofidece ierval. Populaio Populaio Size This is he umber of pairs i he populaio. Usually, you assume ha samples are draw from a very large (ifiie) populaio. Occasioally, however, siuaios arise i which he populaio of ieres is of limied size. I hese cases, appropriae adjusmes mus be made. This opio ses he populaio size. Cofidece Cofidece Level The cofidece level, 1 α, has he followig ierpreaio. If housads of samples of iems are draw from a populaio usig simple radom samplig ad a cofidece ierval is calculaed for each sample, he proporio of hose iervals ha will iclude he rue populaio mea erece is 1 α. Ofe, he values 0.95 or 0.99 are used. You ca eer sigle values or a rage of values such as 0.90, 0.95 or 0.90 o 0.99 by Sample Size (Number of Pairs) N (Sample Size) Eer oe or more values for he sample size. This is he umber of pairs seleced a radom from he populaio o be i he sudy. You ca eer a sigle value or a rage of values. Precisio Disace from Mea erece o Limi(s) This is he disace from he cofidece limi(s) o he mea paired erece. For wo-sided iervals, i is also kow as he precisio, half-widh, or margi of error. You ca eer a sigle value or a lis of values. The value(s) mus be greaer ha zero

4 Cofidece Iervals for Paired Meas Sadard Deviaio of Paired ereces S (Sadard Deviaio) Eer a value (or rage of values) for he sadard deviaio. You ca use he resuls of a pilo sudy, a previous sudy, or a ball park esimae based o he rage (e.g., Rage/4) o esimae his parameer. Kow Sadard Deviaio Check his box whe you wa o base your resuls o he ormal disribuio. Whe he box is o checked, calculaios are based o he -disribuio. The erece bewee he wo disribuios is egligible whe he sample sizes are large (>50). Example 1 Calculaig Sample Size A researcher would like o esimae he mea erece i weigh followig a specific die usig a wo-sided 95% cofidece ierval. The cofidece level is se a 0.95, bu 0.99 is icluded for comparaive purposes. The sadard deviaio esimae, based o he rage of paired ereces, is 9.6 lbs. The researcher would like he ierval o be o wider ha 10 lbs. (half-widh = 5 lbs.), bu will examie half-widhs of 3, 4, 5, 6, ad 7 lbs. The goal is o deermie he ecessary sample size. Seup This secio preses he values of each of he parameers eeded o ru his example. Firs, from he PASS Home widow, load he Cofidece Iervals for Paired Meas procedure widow by expadig Meas, he expadig Paired Meas, ad he clickig o Cofidece Ierval, ad he clickig o Cofidece Iervals for Paired Meas. You may he make he appropriae eries as lised below, or ope Example 1 by goig o he File meu ad choosig Ope Example Templae. Opio Value Desig Tab Solve For... Sample Size Ierval Type... Two-Sided Populaio Size... Ifiie Cofidece Level Disace from Mea o Limi(s)... 3 o 7 by 1 S (Sadard Deviaio) Aoaed Oupu Click he Calculae buo o perform he calculaios ad geerae he followig oupu. Numeric Resuls Numeric Resuls for Two-Sided Cofidece Iervals wih Ukow Sadard Deviaio Targe Acual Sample Dis from Dis from Sadard Cofidece Size Mea Mea Deviaio Level (N) o Limis o Limis (S) (repor coiues) 496-4

5 Cofidece Iervals for Paired Meas Refereces Hah, G. J. ad Meeker, W.Q Saisical Iervals. Joh Wiley & Sos. New York. Repor Defiiios Cofidece level is he proporio of cofidece iervals (cosruced wih his same cofidece level, sample size, ec.) ha would coai he populaio mea erece. N is he size of he sample (or umber of pairs) draw from he populaio. Dis from Mea o Limi is he disace from he cofidece limi(s) o he mea paired erece. For wo-sided iervals, i is also kow as he precisio, half-widh, or margi of error. Targe Dis from Mea o Limi is he value of he disace ha is eered io he procedure. Acual Dis from Mea o Limi is he value of he disace ha is obaied from he procedure. The sadard deviaio (S) is he sadard deviaio of he paired ereces. Summary Saemes A sample size of 42 produces a wo-sided 95% cofidece ierval wih a disace from he mea paired erece o he limis ha is equal o whe he esimaed sadard deviaio of he paired ereces is This repor shows he calculaed sample size for each of he scearios. Plos Secio These plos show he sample size versus he precisio for he wo cofidece limis

6 Cofidece Iervals for Paired Meas Example 2 Validaio This procedure uses he same mechaics as he Cofidece Iervals for Oe Mea procedure. The validaio of his procedure is give i Examples 2 ad 3 of he Cofidece Iervals for Oe Mea procedure