Confidence Intervals for One Mean

Transcription

1 Chapter 420 Cofidece Itervals for Oe Mea Itroductio This routie calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) at a stated cofidece level for a cofidece iterval about the mea whe the uderlyig data distributio is ormal. Cautio: This procedure assumes that the stadard deviatio of the future sample will be the same as the stadard deviatio that is specified. If the stadard deviatio to be used i the procedure is estimated from a previous sample or represets the populatio stadard deviatio, the Cofidece Itervals for Oe Mea with Tolerace Probability procedure should be cosidered. That procedure cotrols the probability that the distace from the mea to the cofidece limits will be less tha or equal to the value specified. Techical Details For a sigle mea from a ormal distributio with kow variace, a two-sided, 100(1 α)% cofidece iterval is calculated by z X ± 1 α / 2σ A oe-sided 100(1 α)% upper cofidece limit is calculated by z X + 1 α σ Similarly, the oe-sided 100(1 α)% lower cofidece limit is z X 1 α σ For a sigle mea from a ormal distributio with ukow variace, a two-sided, 100(1 α)% cofidece iterval is calculated by t X ± 1 α / 2, 1 A oe-sided 100(1 α)% upper cofidece limit is calculated by t X + ˆ σ, 1 ˆ σ 1 α 420-1

2 Cofidece Itervals for Oe Mea Similarly, the oe-sided 100(1 α)% lower cofidece limit is t X 1 α, 1 Each cofidece iterval is calculated usig a estimate of the mea plus ad/or mius a quatity that represets the distace from the mea to the edge of the iterval. For two-sided cofidece itervals, this distace is sometimes called the precisio, margi of error, or half-width. We will label this distace, D. The basic equatio for determiig sample size whe D has bee specified is ˆ σ D = z1 α / σ 2 whe the stadard deviatio is kow, ad D t σ α = 1 / 2, 1 whe the stadard deviatio is ukow. These equatios ca be solved for ay of the ukow quatities i terms of the others. The value α / 2 is replaced by α whe a oe-sided iterval is used. Fiite Populatio Size The above calculatios assume that samples are beig draw from a large (ifiite) populatio. Whe the populatio is of fiite size (N), a adjustmet must be made. The adjustmet reduces the stadard deviatio as follows: σ fiite = σ 1 N This ew stadard deviatio replaces the regular stadard deviatio i the above formulas. Cofidece Level The cofidece level, 1 α, has the followig iterpretatio. If thousads of samples of items are draw from a populatio usig simple radom samplig ad a cofidece iterval is calculated for each sample, the proportio of those itervals that will iclude the true populatio mea is 1 - α. Procedure Optios This sectio describes the optios that are specific to this procedure. These are located o the Desig tab. For more iformatio about the optios of other tabs, go to the Procedure Widow chapter. Desig Tab The Desig tab cotais most of the parameters ad optios that you will be cocered with. Solve For Solve For This optio specifies the parameter to be solved for from the other parameters

3 Cofidece Itervals for Oe Mea Oe-Sided or Two-Sided Iterval Iterval Type Specify whether the iterval to be used will be a oe-sided or a two-sided cofidece iterval. Populatio Populatio Size This is the umber of idividuals i the populatio. Usually, you assume that samples are draw from a very large (ifiite) populatio. Occasioally, however, situatios arise i which the populatio of iterest is of limited size. I these cases, appropriate adjustmets must be made. This optio sets the populatio size. Cofidece Cofidece Level The cofidece level, 1 α, has the followig iterpretatio. If thousads of samples of items are draw from a populatio usig simple radom samplig ad a cofidece iterval is calculated for each sample, the proportio of those itervals that will iclude the true populatio mea is 1 α. Ofte, the values 0.95 or 0.99 are used. You ca eter sigle values or a rage of values such as 0.90, 0.95 or 0.90 to 0.99 by Sample Size N (Sample Size) Eter oe or more values for the sample size. This is the umber of idividuals selected at radom from the populatio to be i the study. You ca eter a sigle value or a rage of values. Precisio Distace from Mea to Limit(s) This is the distace from the cofidece limit(s) to the mea. For two-sided itervals, it is also kow as the precisio, half-width, or margi of error. You ca eter a sigle value or a list of values. The value(s) must be greater tha zero. Stadard Deviatio S (Stadard Deviatio) Eter a value (or rage of values) for the stadard deviatio. Roughly speakig, this value estimates the average absolute differece betwee each idividual ad every other idividual. You ca use the results of a pilot study, a previous study, or a ball park estimate based o the rage (e.g., Rage/4) to estimate this parameter. Kow Stadard Deviatio Check this box whe you wat to base your results o the ormal distributio. Whe the box is ot checked, calculatios are based o the t-distributio. The differece betwee the two distributios is egligible whe the sample sizes are large (>50)

4 Cofidece Itervals for Oe Mea Example 1 Calculatig Sample Size Suppose a study is plaed i which the researcher wishes to costruct a two-sided 95% cofidece iterval for the mea such that the width of the iterval is o wider tha 14 uits. The cofidece level is set at 0.95, but 0.99 is icluded for comparative purposes. The stadard deviatio estimate, based o the rage of data values, is 28. Istead of examiig oly the iterval half-width of 7, a series of half-widths from 5 to 9 will also be cosidered. The goal is to determie the ecessary sample size. Setup This sectio presets the values of each of the parameters eeded to ru this example. First, from the PASS Home widow, load the Cofidece Itervals for Oe Mea procedure widow by expadig Meas, the Oe Mea, the clickig o Cofidece Iterval, ad the clickig o Cofidece Itervals for Oe Mea. You may the make the appropriate etries as listed below, or ope Example 1 by goig to the File meu ad choosig Ope Example Template. Optio Value Desig Tab Solve For... Sample Size Iterval Type... Two-Sided Populatio Size... Ifiite Cofidece Level Distace from Mea to Limit(s)... 5 to 9 by 1 S (Stadard Deviatio) Kow Stadard Deviatio... Not Checked Aotated Output Click the Calculate butto to perform the calculatios ad geerate the followig output. Numeric Results Numeric Results for Two-Sided Cofidece Itervals with Ukow Stadard Deviatio Target Actual Sample Distace Distace Stadard Cofidece Size from Mea from Mea Deviatio Level (N) to Limits to Limits (S) Refereces Hah, G. J. ad Meeker, W.Q Statistical Itervals. Joh Wiley & Sos. New York

5 Cofidece Itervals for Oe Mea Report Defiitios Cofidece level is the proportio of cofidece itervals (costructed with this same cofidece level, sample size, etc.) that would cotai the populatio mea. N is the size of the sample draw from the populatio. Distace from Mea to Limit is the distace from the cofidece limit(s) to the mea. For two-sided itervals, it is also kow as the precisio, half-width, or margi of error. Target Distace from Mea to Limit is the value of the distace that is etered ito the procedure. Actual Distace from Mea to Limit is the value of the distace that is obtaied from the procedure. The stadard deviatio of the populatio measures the variability i the populatio. Summary Statemets A sample size of 123 produces a two-sided 95% cofidece iterval with a distace from the mea to the limits that is equal to whe the estimated stadard deviatio is This report shows the calculated sample size for each of the scearios. Chart Sectio These plots show the sample size versus the distace from the mea to the limits (precisio) for the two cofidece levels

6 Cofidece Itervals for Oe Mea Example 2 Validatio usig Moore ad McCabe Moore ad McCabe (1999) page 443 give a example of a sample size calculatio for a cofidece iterval o the mea whe the cofidece coefficiet is 95%, the stadard deviatio is kow to be 3, ad the margi of error is 2. The ecessary sample size is 9. Setup This sectio presets the values of each of the parameters eeded to ru this example. First, from the PASS Home widow, load the Cofidece Itervals for Oe Mea procedure widow by expadig Meas, the Oe Mea, the clickig o Cofidece Iterval, ad the clickig o Cofidece Itervals for Oe Mea. You may the make the appropriate etries as listed below, or ope Example 2 by goig to the File meu ad choosig Ope Example Template. Optio Value Desig Tab Solve For... Sample Size Iterval Type... Two-Sided Populatio Size... Ifiite Cofidece Level Distace from Mea to Limit(s)... 2 S (Stadard Deviatio)... 3 Kow Stadard Deviatio... Checked Output Click the Calculate butto to perform the calculatios ad geerate the followig output. Numeric Results Target Actual Sample Distace Distace Stadard Cofidece Size from Mea from Mea Deviatio Level (N) to Limits to Limits (S) PASS also calculated the ecessary sample size to be

7 Cofidece Itervals for Oe Mea Example 3 Validatio usig Ostle ad Maloe Ostle ad Maloe (1988) page 536 give a example of a sample size calculatio for a cofidece iterval o the mea whe the cofidece coefficiet is 95%, the stadard deviatio is kow to be 7, ad the margi of error is 5. The ecessary sample size is 8. Setup This sectio presets the values of each of the parameters eeded to ru this example. First, from the PASS Home widow, load the Cofidece Itervals for Oe Mea procedure widow by expadig Meas, the Oe Mea, the clickig o Cofidece Iterval, ad the clickig o Cofidece Itervals for Oe Mea. You may the make the appropriate etries as listed below, or ope Example 3 by goig to the File meu ad choosig Ope Example Template. Optio Value Desig Tab Solve For... Sample Size Iterval Type... Two-Sided Populatio Size... Ifiite Cofidece Level Distace from Mea to Limit(s)... 5 S (Stadard Deviatio)... 7 Kow Stadard Deviatio... Checked Output Click the Calculate butto to perform the calculatios ad geerate the followig output. Numeric Results Target Actual Sample Distace Distace Stadard Cofidece Size from Mea from Mea Deviatio Level (N) to Limits to Limits (S) PASS also calculated the ecessary sample size to be