# SEE
SAMPLING PLANS UNDER CONSIDERATION FOR COMPLIANCE TESTING

by Oskars Petersons and Ken Stricklett Electricity Division National Institute of Standards and Technology

February 10, 1998

Note: This draft material is prepared for a presentation at the Workshop on Distribution Transformer Test Procedures organized by the Department of Energy, February 10, 1998 to stimulate discussion at the workshop and submission of peer review comments. Since the draft is of a preliminary nature and has not received full review by NIST, it may not be cited and referenced for the purposes other than those stated here. 

# SAMPLING PLANS UNDER CONSIDERATION FOR COMPLIANCE TESTING
by Oskars Petersons and Ken 8tricklett Electricity Division National Institute of Standards and Technology February 10, 

## 1998

Note: This draft material is prepared for a presentation at the Workshop on Distribution Transformer Test Procedures organized by the Department of Energy, February 10, 1998to stimulate discussion atthe workshop and submission of peer review comments. Since the draft is of a preliminary nature and has not received full review by NIST, it may not be cited and referenced forthe purposes otherthan those stated here.

## METHOD 1

AS DEFINED IN 10 CFR 430, § 430.32 "A sample shall be selected and tested comprised of units which are production units, or are representative of production units of the basic model being tested and shall meet the following applicable criteria. For each basic model of distribution transformers, as defined in §430.2, a sample of sufficient size shall be tested to insure that any represented value of transformer efficiency shall be no greater that the lower of (A) the mean of the sample or (B) the lower 95% confidence limit of the estimated true mean divided by 0.999." 

### DETERMINE THE REPRESENTED VALUE OF EFFICIENCY ® 

### the smaller of - (A) the mean of the sample (X), or - (B) the lower 95% confidence limit of the mean of the population divided by 0.999 CRITERION FOR COMPLIANCE: 

### METHOD 1 (CONTINUED) STATISTICAL METHOD, SAMPLE SIZE ® not specified ® if t-test selected select sample size, measure losses, compute efficiency compute the mean (X), standard deviation, standard error of the sample, the lower 95% confidence limit (LCL), and (LCL)/0.999 test the represented value for compliance 0 Represented value ? Mandated efficiency 

### .... 

### METHOD 1 AS DEFINED IN 10 CFR 430, 

### § 430.32 ' lOA sample shall be selected and tested comprised of units which are production units, or are representative of production units of the basic model being tested and shall meet the**followin~** applicable criteria. For each basic model of distributIon transformers, as defined in §430.2, a sample of sufficient size shall be tested to insure that any represented value of transformer efficiency shall be no greater that the lower of (A) the mean ofthe sample or (6) the lower 95% confidence limit of the estimated true mean divided by 0.999." 

### DETERMINE THE REPRESENTED VALUE OF EFFICIENCY the smaller of (A) the mean of the sample (X), or (8) the lower 95% confidence limit of the mean of the population divided by 0.999 CRITERION FOR COMPLIANCE:
III 

### Represented value 2! 

### Mandated efficiency 

### METHOD 1 (CONTINUED) STATISTICAL METHOD, SAMPLE SIZE

III 

### not specified • if**t~test** 

### selected select sample size, measure losses compute efficiency , compute the mean (X), standard deviation, standard error of the sample, the lower 95% confidence limit (LCL), and (LCL)/O.999 test the represented value for compliance

## i NORMALLY DISTRIBUTED EFFICIENCIES Increasing Effidency -•

ME Mandated efficiency, ME Probability density for the population, Po Probability density for the mean of 5 units, PS True mean efficiency, a

1 

### METHOD I: Sampling Plan Increasing Efficiency ----

ME LCL

0.999

Sample mean, X Standard error, SE(X) Lower 95% confidence limit, LCL =X - tSE(X) For Compliance: A) X> ME B) X-tSE(X) > ME 0.999 2 NORMALLY DISTRIBUTED EFFICIENCIES looreaslng Elllclency -

ME Mandated efficiency, ME Probability densityforthe population, Po Probability densityforthe mean of 5 units, P s True mean efficiency, J.1. METHOD I: Sampling Plan

Sample mean,

### X Standard error, 

### SE(X) Increasing ettrclency -

ME LCL Q.999

Lower95% confidence limit, 

### LCL 

## =X
- tSE(X) For Compliance:

A) 

### X**~** ME B.) i -tSE(X) 

## > ME 0.999
- 2

### METHOD 2 BASED ON • statistical t-test ® requirement of minimum sample size CRITERIA FOR COMPLIANCE 0 

### X Z ME ®tSE(X ) n = 5or n ^ (tS/T]2 T is the maximum error in the determination of the mean (7) STEPS IN IMPLEMENTATION (SIMPLIFIED) ® measure losses, compute efficiency compute the mean (X ), standard deviation, standard error of the sample, the lower 90% confidence limit (LCL), and ME - tSE(X ) ® determine adaquecy of sample size (in most cases a sample of 5 will suffice)

® 

### test X for compliance METHOD II : Sampling Plan increasing Efficiency
L
ME - tSE(X) ....'

meME Sample mean, .k Standard error, SE(X) Lower 95% confidence limit, LCL =X - tSE(X) For Compliance: A) X .> ME - tSE(X) 

### B) n • Z [tS/T]z
4 t.;..•. 

### METHOD 2 BASED ON
CD 

### statistical t-test
CD 

### requirement of minimum sample size
, 

### CRITER'IA FOR COMPLIANCE

CD 

### X**~** 

### ME -**,tSE{~)** CD 

### n 

### = 5 or 

### n ;?; 

### [tS/T]2 T is the maximum error in the determination ·of the mean (X) STEPS IN

### IMP~EMENTATION' (SIMPLIFIED) • measure losses, compute efficiency • compu:te the mean 

### (X). standard deviation, standard error ofthe sample, the lower

### 90%**confid~nce** 

### limit (LCL), and ME - tSE(X).,.

CD 

### determine adaquecy of sample size (in most cases a sample of

### 5 will suffice)

CD 

### test X for compliance METHOD II: Sampling Plan 

# I ! 

# I f e t 11.' ME·tSE(X)

### ~

Sample mean, 

### X Standard error, 

### SE(X)IlIOI'lIulng Efficiency -

ME

Lower 95% confidence limit,

### LCL 

## =X
- tSE(X) For Compliance:



## . A)**X.~** ME- 

### tSE(X) B) 

### n·**~** 

### [tSffJ2 4

### SUMMARY FEEDBACK DESIRED Method 1 1. General comments on the two plans
-- perceived Advantages: advantages and disadvantages • Simple, flexible, 2. Statistical significance of the 6% loss tolerance Is it • High assurance of meeting EPCA requirements . applicable to 50% and 35% loadings? • Determination assurance increases with sample size 3. Suggestions for the values of constants: the divisor in method 1 and 'T' coefficient in method 2 Disadvantages: 3. Do the actual test data support the modeling results? ® High burden on producers; risk of false determination of noncompliance can be as high as 4. Other sampling plans that should be considered 50% . ® Statistical methods for analyses of operating characteristics are not readily available Method 2 Advantages: Rigorous and widely used statistical method • Can balance producer and consumer risks Disadvantages:

0 Complexity Method 1 Advantages: 

### SUMMARY

CD Simple, flexible. • High assurance of meeting EPCA requirements . CD Determination assurance increases with sample size Disadvantages: • High burden on producers; risk offalse determination of noncompliance can be as high as 50%
e Statistical methodsfor analyses of operating characteristics are not readily available Method 2 Advantages: • Rigorous and widely used statistical method • Can balance producer and consumer risks Disadvantages: • Complexity 

### FEEDBACK DESIRED 1. General comments on the two plans -- perceived advantages and disadvantages 2. Statistical significance ofthe 6% losstolerance. Isit applicable to 50% and 35% loadings? 3. Suggestions forthe values of constants: the divisor in method 1 and "T" coefficient in method 2 3. Do the actual test data supportthe modeling results? 4. Othersampling plans that should be considered.