Confidence Intervals for One Variance using Relative Error

Transcription

1 Chapter 653 Confidence Interval for One Variance uing Relative Error Introduction Thi routine calculate the neceary ample ize uch that a ample variance etimate will achieve a pecified relative ditance from the true population variance at a tated confidence level when the underlying data ditribution i normal. Caution: Thi procedure control the relative width of the interval a a proportion of the true population variance. For controlling the abolute width of the interval ee the procedure Confidence Interval for One Variance uing Variance and Confidence Interval for One Variance with Tolerance Probability. Technical Detail Following the reult of Deu and Raghavarao (1990) and Greenwood and Sandomire (1950), let be the variance etimate baed on a ample from a normal ditribution with unknown µ and unknown σ. Let r be the proportion of σ uch that i within rσ of σ with deired confidence. If and ( > σ + rσ ) = Pr( > (1 )) p 1 = Pr σ + r ( < σ rσ ) = Pr( < (1 )) p = Pr σ r The confidence level for etimating σ within proportion r i 1 p 1 p. Since and ( n 1) σ ~ χ n 1, it i ueful to rewrite p 1 and p a p ( n 1) = Pr > ( n 1)(1 + ) σ 1 r ( n 1) p = Pr < ( n 1)(1 r) σ Uing the chi-quare ditribution, thee equation can be olved for any of the unknown quantitie (n, r, p 1 + p ) in term of the other

2 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 the variance etimate are obtained from each ample, the proportion of thoe etimate that are within rσ of σ i 1 α. 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. 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. In each cae the limit are baed on the relative error of the true population variance. Confidence 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 the variance etimate are obtained from each ample, the proportion of thoe etimate that are within rσ of σ 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 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. 653-

3 Preciion Relative Error Thi i the ditance from the true variance a a proportion of the true variance. You can enter a ingle value or a lit of value. The value() mut be between 0 and 1. Example 1 Calculating Sample Size Suppoe a tudy i planned in which the reearcher wihe to be 95% confident that etimated variance i within 10% of the true population variance. In addition to 10% relative error, 5%, 15%, 0% and 5% will alo be conidered. The goal i to determine the neceary ample ize. Setup Thi ection preent the value of each of the parameter needed to run thi example. Firt, from the PASS Home window, load the procedure window by expanding Variance, then clicking on One Variance, and then clicking on Confidence Interval for One Variance uing Relative Error. 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 Solve For... Sample Size Interval Type... Two-Sided Confidence Level Relative Error to 0.5 by 0.05 Annotated Output Click the Calculate button to perform the calculation and generate the following output. Numeric Reult Numeric Reult for Two-Sided Relative Error Confidence Interval Target Actual Sample Confidence Confidence Size Relative Level Level (N) Error Reference Deu, M. M. and Raghavarao, D Sample Size Methodology. Academic Pre. New York. Greenwood, J. A. and Sandomire, M. M 'Sample Size Required for Etimating the Standard Deviation a a Per Cent of it True Value', Journal of the American Statitical Aociation, Vol. 45, No. 50, pp

4 Report Definition Confidence Level i the proportion of variance etimate that will be within the relative error of the true variance. Target Confidence Level i the value of the confidence level that i entered into the procedure. Actual Confidence Level i the value of the confidence level that i obtained from the procedure. Sample Size (N) i the ize of the ample drawn from the population. Relative Error i the ditance from the true variance a a proportion of the true variance. Summary Statement With a ample ize of 3074, the probability i (95% confidence) that the etimate of the variance will be within 5% of the true population variance. Thi report how the calculated ample ize for each of the cenario. Plot Section Thi plot how the ample ize veru the relative error

5 Example Validation uing Deu and Raghavarao Deu and Raghavarao (1990) page 6 give an example of a calculation in which the deired confidence level i 95% and the relative error i 0%. Thi calculation i baed on a large ample approximation formula. The neceary ample ize i 194. Setup Thi ection preent the value of each of the parameter needed to run thi example. Firt, from the PASS Home window, load the procedure window by expanding Variance, then clicking on One Variance, and then clicking on Confidence Interval for One Variance uing Relative Error. 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 Solve For... Sample Size Interval Type... Two-Sided Confidence Level Relative Error Output Click the Calculate button to perform the calculation and generate the following output. Numeric Reult Target Actual Sample Confidence Confidence Size Relative Level Level (N) Error PASS calculated the neceary ample ize to be 19, but did not ue the approximation formula. Uing direct calculation with the non-approximate chi-quare formula with a ample ize of 194, the confidence level i 1 ( ) = Uing direct calculation with the non-approximate chi-quare formula with a ample ize of 19, the confidence level i 1 ( ) = , which i cloer to the precribed confidence level. Thu, 19 i the correct value