Confidence Intervals for One Mean with Tolerance Probability


 Randolf Norris
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1 Chapter 421 Cofidece Itervals for Oe Mea with Tolerace Probability Itroductio This procedure calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) with a give tolerace probability at a stated cofidece level for a cofidece iterval about a sigle mea whe the uderlyig data distributio is ormal. Techical Details For a sigle mea from a ormal distributio with ukow variace, a twosided, 100(1 α)% cofidece iterval is calculated by t X ± / 2, 1 A oesided 100(1 α)% upper cofidece limit is calculated by t X + ˆ σ ˆ σ, 1 Similarly, the oesided 100(1 α)% lower cofidece limit is t X ˆ σ, 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 twosided cofidece itervals, this distace is sometimes called the precisio, margi of error, or halfwidth. We will label this distace, D. The basic equatio for determiig sample size whe D has bee specified is D t σ α = 1 / 2,
2 Solvig for, we obtai = t / 2, 1 This equatio ca be solved for ay of the ukow quatities i terms of the others. The value α/2 is replaced by α whe a oesided iterval is used. There is a additioal subtlety that arises whe the stadard deviatio is to be chose for estimatig sample size. The sample sizes determied from the formula above produce cofidece itervals with the specified widths oly whe the future sample has a sample stadard deviatio that is o greater tha the value specified. As a example, suppose that 15 idividuals are sampled i a pilot study, ad a stadard deviatio estimate of 3.5 is obtaied from the sample. The purpose of a later study is to estimate the mea withi 10 uits. Suppose further that the sample size eeded is calculated to be 57 usig the formula above with 3.5 as the estimate for the stadard deviatio. The sample of size 57 is the obtaied from the populatio, but the stadard deviatio of the 57 idividuals turs out to be 3.9 rather tha 3.5. The cofidece iterval is computed ad the distace from the mea to the cofidece limits is greater tha 10 uits. This example illustrates the eed for a adjustmet to adjust the sample size such that the distace from the mea to the cofidece limits will be below the specified value with kow probability. Such a adjustmet for situatios where a previous sample is used to estimate the stadard deviatio is derived by Harris, Horvitz, ad Mood (1948) ad discussed i Zar (1984) ad Hah ad Meeker (1991). The adjustmet is D ˆ σ 2 2 t ˆ / 2, 1σ = F1 γ ; 1, m 1 D where 1 γ is the probability that the distace from the mea to the cofidece limit(s) will be below the specified value, ad m is the sample size i the previous sample that was used to estimate the stadard deviatio. The correspodig adjustmet whe o previous sample is available is discussed i Kupper ad Hafer (1989) ad Hah ad Meeker (1991). The adjustmet i this case is t = 2 ˆ / 2, 1σ γ, 1 D 2 χ1 1 where, agai, 1 γ is the probability that the distace from the mea to the cofidece limit(s) will be below the specified value. Each of these adjustmets accouts for the variability i a future estimate of the stadard deviatio. I the first adjustmet formula (Harris, Horvitz, ad Mood, 1948), the distributio of the stadard deviatio is based o the estimate from a previous sample. I the secod adjustmet formula, the distributio of the stadard deviatio is based o a specified value that is assumed to be the populatio stadard deviatio
3 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. OeSided or TwoSided Iterval Iterval Type Specify whether the iterval to be used will be a oesided or a twosided 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
4 Cofidece ad Tolerace Cofidece Level (1 Alpha) 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 Tolerace Probability This is the probability that a future iterval with sample size N ad the specified cofidece level will have a distace from the mea to the limit(s) that is less tha or equal to the distace specified. If a tolerace probability is ot used, as i the 'Cofidece Itervals for Oe Mea' procedure, the sample size is calculated for the expected distace from the mea to the limit(s), which assumes that the future stadard deviatio will also be the oe specified. Usig a tolerace probability implies that the stadard deviatio of the future sample will ot be kow i advace, ad therefore, a adjustmet is made to the sample size formula to accout for the variability i the stadard deviatio. Use of a tolerace probability is similar to usig a upper boud for the stadard deviatio i the 'Cofidece Itervals for Oe Mea' procedure. Values betwee 0 ad 1 ca be etered. The choice of the tolerace probability depeds upo how importat it is that the distace from the iterval limit(s) to the mea is at most the value specified. You ca eter a rage of values such as or 0.70 to 0.95 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 twosided itervals, it is also kow as the precisio, halfwidth, or margi of error. You ca eter a sigle value or a list of values. The value(s) must be greater tha zero
5 Stadard Deviatio Stadard Deviatio Source This procedure permits two sources for estimates of the stadard deviatio: S is a Populatio Stadard Deviatio This optio should be selected if there is o previous sample that ca be used to obtai a estimate of the stadard deviatio. I this case, the algorithm assumes that future sample obtaied will be from a populatio with stadard deviatio S. S from a Previous Sample This optio should be selected if the estimate of the stadard deviatio is obtaied from a previous radom sample from the same distributio as the oe to be sampled. The sample size of the previous sample must also be etered uder 'Sample Size of Previous Sample'. Stadard Deviatio Populatio Stadard Deviatio S (Stadard Deviatio) Eter a estimate of the stadard deviatio (must be positive). I this case, the algorithm assumes that future samples obtaied will be from a populatio with stadard deviatio S. Oe commo method for estimatig the stadard deviatio is the rage divided by 4, 5, or 6. You ca eter a rage of values such as or 1 to 10 by 1. Press the Stadard Deviatio Estimator butto to load the Stadard Deviatio Estimator widow. Stadard Deviatio Stadard Deviatio from Previous Sample S (SD Estimated from a Previous Sample) Eter a estimate of the stadard deviatio from a previous (or pilot) study. This value must be positive. A rage of values may be etered. Press the Stadard Deviatio Estimator butto to load the Stadard Deviatio Estimator widow. Sample Size of Previous Sample Eter the sample size that was used to estimate the stadard deviatio etered i S (SD Estimated from a Previous Sample). This value is etered oly whe 'Stadard Deviatio Source:' is set to 'S from a Previous Sample'
6 Example 1 Calculatig Sample Size A researcher would like to estimate the mea weight of a populatio with 95% cofidece. It is very importat that the mea weight is estimated withi 15 grams. Data available from a previous study are used to provide a estimate of the stadard deviatio. The estimate of the stadard deviatio is 45.1 grams, from a sample of size 14. The goal is to determie the sample size ecessary to obtai a twosided cofidece iterval such that the mea weight is estimated withi 15 grams. Tolerace probabilities of 0.70 to 0.95 will be examied. Setup This sectio presets the values of each of the parameters eeded to ru this example. First, from the PASS Home widow, load the procedure widow by expadig Meas, the Oe Mea, the clickig o Cofidece Iterval, ad the clickig o Cofidece Itervals for Oe Mea with Tolerace Probability. 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... TwoSided Populatio Size... Ifiite Cofidece Level Tolerace Probability to 0.95 by 0.05 Distace from Mea to Limit(s) Stadard Deviatio Source... S from a Previous Sample S Sample Size of Previous Sample Aotated Output Click the Calculate butto to perform the calculatios ad geerate the followig output. Numeric Results Numeric Results for TwoSided Cofidece Itervals Target Actual Sample Distace Distace Stadard Cofidece Size from Mea from Mea Deviatio Tolerace Level (N) to Limits to Limits (S) Probability Sample size for estimate of S from previous sample = 14. Refereces Hah, G. J. ad Meeker, W.Q Statistical Itervals. Joh Wiley & Sos. New York. Zar, J. H Biostatistical Aalysis. Secod Editio. PreticeHall. Eglewood Cliffs, New Jersey. Harris, M., Horvitz, D. J., ad Mood, A. M 'O the Determiatio of Sample Sizes i Desigig Experimets', Joural of the America Statistical Associatio, Volume 43, No. 243, pp
7 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 twosided itervals, it is also kow as the precisio, halfwidth, 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. Tolerace Probability is the probability that a future iterval with sample size N ad correspodig cofidece level will have a distace from the mea to the limit(s) that is less tha or equal to the specified distace. Summary Statemets The probability is that a sample size of 49 will produce a twosided 95% cofidece iterval with a distace from the mea to the limits that is less tha or equal to if the populatio stadard deviatio is estimated to be by a previous sample of size 14. This report shows the calculated sample size for each of the scearios. Plots Sectio This plot shows the sample size versus the tolerace probability
8 Example 2 Validatio usig Hah ad Meeker Hah ad Meeker (1991) page 139 give a example of a sample size calculatio for a twosided cofidece iterval o the mea whe the cofidece level is 95%, the populatio stadard deviatio is assumed to be 2500, the distace from the mea to the limit is 1500, ad the tolerace probability is The ecessary sample size is 19. Setup This sectio presets the values of each of the parameters eeded to ru this example. First, from the PASS Home widow, load the procedure widow by expadig Meas, the Oe Mea, the clickig o Cofidece Iterval, ad the clickig o Cofidece Itervals for Oe Mea with Tolerace Probability. 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... TwoSided Populatio Size... Ifiite Cofidece Level Tolerace Probability Distace from Mea to Limit(s) Stadard Deviatio Source... S is a Populatio Stadard Deviatio S Output Click the Calculate butto to perform the calculatios ad geerate the followig output. Numeric Results Numeric Results for TwoSided Cofidece Itervals Target Actual Sample Distace Distace Stadard Cofidece Size from Mea from Mea Deviatio Tolerace Level (N) to Limits to Limits (S) Probability PASS also calculated the ecessary sample size to be
9 Example 3 Validatio usig Zar Zar (1984) pages give a example of a sample size calculatio for a twosided cofidece iterval o the mea whe the cofidece level is 95%, the stadard deviatio is estimated to be by a previous sample of size 25, the distace from the mea to the limit is 1.5, ad the tolerace probability is The ecessary sample size is 53. Setup This sectio presets the values of each of the parameters eeded to ru this example. First, from the PASS Home widow, load the procedure widow by expadig Meas, the Oe Mea, the clickig o Cofidece Iterval, ad the clickig o Cofidece Itervals for Oe Mea with Tolerace Probability. 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... TwoSided Populatio Size... Ifiite Cofidece Level Tolerace Probability Distace from Mea to Limit(s) Stadard Deviatio Source... S from a Previous Sample S Sample Size of Previous Sample Output Click the Calculate butto to perform the calculatios ad geerate the followig output. Numeric Results Numeric Results for TwoSided Cofidece Itervals Target Actual Sample Distace Distace Stadard Cofidece Size from Mea from Mea Deviatio Tolerace Level (N) to Limits to Limits (S) Probability PASS also calculated the ecessary sample size to be
10 Example 4 Validatio usig Harris, Horvitz, ad Mood Harris, Horvitz, ad Mood (1948) pages give a example of a sample size calculatio for a twosided cofidece iterval o the mea whe the cofidece level is 99%, the stadard deviatio is estimated to be 3 by a previous sample of size 9, the distace from the mea to the limit is 2, ad the tolerace probability is The ecessary sample size is 49. Setup This sectio presets the values of each of the parameters eeded to ru this example. First, from the PASS Home widow, load the procedure widow by expadig Meas, the Oe Mea, the clickig o Cofidece Iterval, ad the clickig o Cofidece Itervals for Oe Mea with Tolerace Probability. You may the make the appropriate etries as listed below, or ope Example 4 by goig to the File meu ad choosig Ope Example Template. Optio Value Desig Tab Solve For... Sample Size Iterval Type... TwoSided Populatio Size... Ifiite Cofidece Level Tolerace Probability Distace from Mea to Limit(s)... 2 Stadard Deviatio Source... S from a Previous Sample S... 3 Sample Size of Previous Sample... 9 Output Click the Calculate butto to perform the calculatios ad geerate the followig output. Numeric Results Numeric Results for TwoSided Cofidece Itervals Target Actual Sample Distace Distace Stadard Cofidece Size from Mea from Mea Deviatio Tolerace Level (N) to Limits to Limits (S) Probability PASS also calculated the ecessary sample size to be
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