Confidence Intervals for One Variance with Tolerance Probability

Transcription

1 Chapter 65 Confidence Interval for One Variance with Tolerance Probability Introduction Thi procedure calculate the ample ize neceary to achieve a pecified width (or in the cae of one-ided interval, the ditance from the variance to the confidence limit) with a given tolerance probability at a tated confidence level for a confidence interval about a ingle variance when the underlying data ditribution i normal. Technical Detail For a ingle variance from a normal ditribution with unknown mean, a two-ided, 100(1 - α)% confidence interval i calculated by χ1 /, n, α 1 χ α /, n 1 A one-ided 100(1 α)% upper confidence limit i calculated by χ α, n 1 Similarly, the one-ided 100(1 α)% lower confidence limit i χ 1 α, n 1 For two-ided interval, the ditance from the variance to each of the limit i different. Thu, intead of pecifying the ditance to the limit we pecify the width of the interval, W. The baic equation for determining ample ize for a two-ided interval when W ha been pecified i /, n 1 W = χ α χ 1 α /, n 1 For one-ided interval, the ditance from the variance to limit, D, i pecified. 65-1

2 Confidence Interval for One Variance with Tolerance Probability The baic equation for determining ample ize for a one-ided upper limit when D ha been pecified i D = χ α, n 1 The baic equation for determining ample ize for a one-ided lower limit when D ha been pecified i D = χ1 α, n 1 Thee equation can be olved for any of the unknown quantitie in term of the other. There i an additional ubtlety that arie when the variance i to be choen for etimating ample ize. The ample ize determined from the formula above produce confidence interval with the pecified width only when the future ample ha a ample variance that i no greater than the value pecified. A an example, uppoe that 15 individual are ampled in a pilot tudy, and a variance etimate of 6.5 i obtained from the ample. The purpoe of a later tudy i to etimate the variance with a confidence interval with width no greater than 3 unit. Suppoe further that the ample ize needed i calculated to be 105 uing the formula above with 6.5 a the etimate for the variance. The ample of ize 105 i then obtained from the population, but the variance of the 105 individual turn out to be 7. rather than 6.5. The confidence interval i computed and the width of the interval i greater than 3 unit. Thi example illutrate the need for an adjutment to adjut the ample ize uch that the width or ditance from the variance to the confidence limit will be below the pecified value with known probability. Such an adjutment for ituation where a previou ample i ued to etimate the variance i derived for the cae of confidence interval for a mean by Harri, Horvitz, and Mood (1948) and dicued in Zar (1984) and Hahn and Meeker (1991). The adjutment i made by replacing with = σ F1 γ ; n 1, m 1 where 1 γ i the probability that the width or ditance from the variance to the confidence limit will be below the pecified value, and m i the ample ize in the previou ample that wa ued to etimate the variance. The correponding adjutment when no previou ample i available i dicued in Kupper and Hafner (1989) and Hahn and Meeker (1991). The adjutment in thi cae i χ1 γ, n 1 = σ n 1 where, again, 1 γ i the probability that the width or ditance from the variance to the confidence limit will be below the pecified value. Each of thee adjutment account for the variability in a future etimate of the variance. In the firt adjutment formula (Harri, Horvitz, and Mood, 1948), the ditribution of the variance i baed on the etimate from a previou ample. In the econd adjutment formula, the ditribution of the variance i baed on a pecified value that i aumed to be the population variance. For thi procedure, both adjutment are adapted from the cae of a one-ample confidence interval for a ingle mean to the cae of a one-ample confidence interval for a ingle variance. Confidence Level The confidence level, 1 α, ha the following interpretation. If thouand of ample of n item are drawn from a population uing imple random ampling and a confidence interval i calculated for each ample, the proportion of thoe interval that will include the true population variance i 1 α. 65-

3 Confidence Interval for One Variance with Tolerance Probability Procedure Option Thi ection decribe the option that are pecific to thi procedure. Thee are located on the Deign tab. For more information about the option of other tab, go to the Procedure Window chapter. Deign Tab The Deign tab contain mot of the parameter and option that you will be concerned with. Solve For Solve For Thi option pecifie the parameter to be olved for from the other parameter. One-Sided or Two-Sided Interval Interval Type Specify whether the interval to be ued will be a two-ided confidence interval, an interval that ha only an upper limit, or an interval that ha only a lower limit. Confidence and Tolerance Confidence Level (1 Alpha) The confidence level, 1 α, ha the following interpretation. If thouand of ample of n item are drawn from a population uing imple random ampling and a confidence interval i calculated for each ample, the proportion of thoe interval that will include the true population variance i 1 - α. Often, the value 0.95 or 0.99 are ued. You can enter ingle value or a range of value uch a 0.90, 0.95 or 0.90 to 0.99 by Tolerance Probability Thi i the probability that a future interval with ample ize N and the pecified confidence level will have a width or ditance from the variance to the limit that i le than or equal to the width or ditance pecified. If a tolerance probability i not ued, a in the 'Confidence Interval for One Variance uing Variance' procedure, the ample ize i calculated for the expected width or ditance from the variance to the limit, which aume that the future variance will alo be the one pecified. Uing a tolerance probability implie that the variance of the future ample will not be known in advance, and therefore, an adjutment i made to the ample ize formula to account for the variability in the variance. Ue of a tolerance probability i imilar to uing an upper bound for the variance in the 'Confidence Interval for One Variance uing Variance' procedure. Value between 0 and 1 can be entered. The choice of the tolerance probability depend upon how important it i that the width or ditance from the interval limit to the variance i at mot the value pecified. You can enter a range of value uch a or 0.70 to 0.95 by

4 Confidence Interval for One Variance with Tolerance Probability Sample Size N (Sample Size) Enter one or more value for the ample ize. Thi i the number of individual elected at random from the population to be in the tudy. You can enter a ingle value or a range of value. Preciion Confidence Interval Width (Two-Sided) Thi i the ditance from the lower confidence limit to the upper confidence limit. The ditance from the variance to the lower and upper limit i not equal. You can enter a ingle value or a lit of value. The value() mut be greater than zero. Ditance from Var to Limit (One-Sided) Thi i the ditance from the variance to the lower or upper limit of the confidence interval, depending on whether the Interval Type i et to Lower Limit or Upper Limit. You can enter a ingle value or a lit of value. The value() mut be greater than zero. Variance Variance Source Thi procedure permit two ource for etimate of the variance: V i a Population Variance Thi option hould be elected if there i no previou ample that can be ued to obtain an etimate of the variance. In thi cae, the algorithm aume that the future ample obtained will be from a population with variance V. V from a Previou Sample Thi option hould be elected if the etimate of the variance i obtained from a previou random ample from the ame ditribution a the one to be ampled. The ample ize of the previou ample mut alo be entered under 'Sample Size of Previou Sample'. Variance V i a Population Variance V (Variance) Enter an etimate of the variance (mut be poitive). In thi cae, the algorithm aume that future ample obtained will be from a population with variance V. You can enter a range of value uch a 1 3 or 1 to 10 by 1. Pre the Standard Deviation Etimator button to load the Standard Deviation Etimator window. 65-4

5 Confidence Interval for One Variance with Tolerance Probability Variance V from a Previou Sample V (Var Etimated from a Previou Sample) Enter an etimate of the variance from a previou (or pilot) tudy. Thi value mut be poitive. A range of value may be entered. Pre the Standard Deviation Etimator button to load the Standard Deviation Etimator window. Sample Size of Previou Sample Enter the ample ize that wa ued to etimate the variance entered in V (Var Etimated from a Previou Sample). Thi value i entered only when 'Variance Source:' i et to 'V from a Previou Sample'. Example 1 Calculating Sample Size A reearcher would like to etimate the variance of a population with 95% confidence. It i very important that the interval width i le than 10 gram. Data available from a previou tudy are ued to provide an etimate of the variance. The etimate of the variance i 35.4 gram, from a ample of ize 16. The goal i to determine the ample ize neceary to obtain a two-ided confidence interval uch that the width of the interval i le than 10 gram. Tolerance probabilitie of 0.70 to 0.95 will be examined. Setup Thi ection preent the value of each of the parameter needed to run thi example. Firt, from the PASS Home window, load the Confidence Interval for One Variance with Tolerance Probability procedure window by expanding Variance, then clicking on One Variance, and then clicking on Confidence Interval for One Variance with Tolerance Probability. You may then make the appropriate entrie a lited below, or open Example 1 by going to the File menu and chooing Open Example Template. Option Value Deign Tab Solve For... Sample Size Interval Type... Two-Sided Confidence Level Tolerance Probability to 0.95 by 0.05 Confidence Interval Width Variance Source... V from a Previou Sample V Sample Size of Previou Sample

6 Confidence Interval for One Variance with Tolerance Probability Annotated Output Click the Calculate button to perform the calculation and generate the following output. Numeric Reult Numeric Reult for Two-Sided Confidence Interval Sample Confidence Size Target Actual Tolerance Level (N) Width Width Variance Probability Sample ize for etimate of the variance from previou ample = 16. Reference Hahn, G. J. and Meeker, W.Q Statitical Interval. John Wiley & Son. New York. Zar, J. H Biotatitical Analyi. Second Edition. Prentice-Hall. Englewood Cliff, New Jerey. Harri, M., Horvitz, D. J., and Mood, A. M 'On the Determination of Sample Size in Deigning Experiment', Journal of the American Statitical Aociation, Volume 43, No. 43, pp Report Definition Confidence Level i the proportion of confidence interval (contructed with thi ame confidence level, ample ize, etc.) that would contain the population variance. N i the ize of the ample drawn from the population. Width i the ditance from the lower limit to the upper limit. Target Width i the value of the width that i entered into the procedure. Actual Width i the value of the width that i obtained from the procedure. Variance i the etimated variance baed on a previou ample. Tolerance Probability i the probability that a future interval with ample ize N and correponding confidence level will have a width that i le than or equal to the pecified width. Summary Statement The probability i that a ample ize of 641 will produce a two-ided 95% confidence interval with a width that i le than or equal to if the population variance i etimated to be by a previou ample of ize 16. Thi report how the calculated ample ize for each of the cenario. 65-6

7 Confidence Interval for One Variance with Tolerance Probability Plot Section Thi plot how the ample ize veru the tolerance probability. 65-7

8 Confidence Interval for One Variance with Tolerance Probability Example Validation uing Simulation We could not find a publihed reult for confidence interval for a variance with tolerance probability. Thi procedure i validated uing a imulation. A imulation wa run with a confidence level of 95%, ample ize of 57, a pecified confidence interval width of 0.989, and aumed population variance of 1. The number of imulation wa 100,000. The proportion of interval with width le than (tolerance probability) wa Setup Thi ection preent the value of each of the parameter needed to run thi example. Firt, from the PASS Home window, load the Confidence Interval for One Variance with Tolerance Probability procedure window by expanding Variance, then clicking on One Variance, and then clicking on Confidence Interval for One Variance with Tolerance Probability. You may then make the appropriate entrie a lited below, or open Example by going to the File menu and chooing Open Example Template. Option Value Deign Tab Solve For... Tolerance Probability Interval Type... Two-Sided Confidence Level N (Sample Size) Confidence Interval Width Variance Source... V i a Population Variance V... 1 Output Click the Calculate button to perform the calculation and generate the following output. Numeric Reult Numeric Reult for Two-Sided Confidence Interval Sample Confidence Size Target Actual Tolerance Level (N) Width Width Variance Probability PASS calculated the tolerance probability to be 0.900, which i well within the imulation error of the imulation etimate of