容忍區間及新方法探討貨品接受之研究

國 立 交 通 大 學

統 計 學 研 究 所

碩 士 論 文

容忍區間及新方法探討貨品接受之研究

Tolerance Interval And a New Approach

for Lot Production Inspection

研 究 生 ：吳志文 (Chih-Wen Wu)

指導教授 ：陳鄰安 博士 (Dr. Lin-An Chen)

容忍區間及新方法探討貨品接受之研究

Tolerance Interval and a New Approach

for Lot Production Inspection

研 究 生：吳志文

Student： Chih-Wen Wu

指導教授：陳鄰安 博士

Advisor： Dr. Lin-An Chen

國 立 交 通 大 學

統計學研究所

碩 士 論 文

A Thesis

Submitted to Institute of Statistics College of Science

National Chiao Tung University in Partial Fulfillment of the Requirements

for the Degree of Master

in Statistics

June 2007

Hsinchu, Taiwan, Republic of China

中華民國九十六年六月

I

容忍區間及新方法探討貨品接受之研究

研究生：吳志文 指導教授：陳鄰安 博士

國立交通大學統計學研究所

摘 要

長期以來容忍區間被用於檢驗每批貨品是否有固定比率的 合格產品且其信心是否夠高。我們在這篇文章正式把信心寫 成參數的函數。因此，任何預測信心的方法都可用來檢驗它 的有效性。我們研究了容忍區間的檢力，其中我們發現當參 數已知時此一方法太過樂觀當參數未知時其方法又太悲 觀。主要原因是此一方法未使用一批貨品之量的大小。我們 進一步發展了信心的點估計方法其表現似乎很好。 關鍵字：假設檢定; 檢定力; 品質管制; 容忍區間。

II

for lot Production Inspection

Student

：Chih-Wen Wu Advisor：Dr. Lin-An Chen

Institute of Statistic

National Chiao Tung University

ABSTRACT

The tolerance interval has long been a technique for manufacturer to verify if there is confidence large enough that ensures a proportion of production lot conforming to specification limits. This paper formally formulate this "confidence" in terms of lot size and parameters involved in the underlying distribution of the characteristic. With this formulation, any technique for prediction of manufacturer's confidence may be evaluated for its efficiency and it also provides a wide room for this prediction through statistical inferences for the unknown confidence. We then study the power of the tolerance interval in detecting if there is a reasonably large manufacturer's confidence. We found that when the parameters involved in the distribution are known, the predicted manufacturer's confidence is too optimistic in a value much higher than the true value and when the parameters involved in the distribution are unknown, the predicted manufacturer's confidence is too conservative in a value much lower than the true one. The inefficiency partly comes from the fact that tolerance interval does not use the information of lot size in prediction which is an ancillary statistic in the considered statistical model. For statistical inference of this unknown confidence, we introduce a point estimation technique that its results shown a power comparison seems to be very promising for the manufacturer.

III

誌 謝

可以完成這篇論文，首先要感謝我的指導教授--陳鄰安 老師，在這一年裡細心的指導，不論是課業或者是處事態度 上都給於很好的建議，讓我獲益良多。更要感謝交大統計所 的所有老師們，也因為您們才能讓我更了解統計哲學。特別 感謝--洪慧念老師、蔡明田老師以及黃信誠老師三位口試委 員，撥冗指導。 更要感謝在這兩年裡和我一起成長的同學們。永在，提 供解決有關電腦問題方法。建威，提供解決摸擬程式上的害 蟲建議。益銘及俊睿，讓我的眼界更加的遼闊。煜淳及育辰， 陪我一起運動健身。以及曾經給於我協助、一起打拼及一起 同樂的好同學們。 最後要感謝我的雙親、哥哥和姊姊。感謝爸爸媽媽辛苦 的供我讀書及給於我一個盡情發揮的空間，讓我在就學上能 一路順遂。感謝哥哥姊姊給於我支持及鼓勵。 在此，將以本篇論文獻給我親愛的家人、師長及同學們，並 致上我最誠摯的謝意。 吳志文 謹誌于 國立交通大學統計學研究所 九十六年六月

IV 英文摘要……….….………II 誌謝……….……III 目錄……….……IV

Contents

1 Introduction … … ……….…1

2 Formulation of Confidence and an Evaluation of

Tolerance Interval When Parameters Are Known ………. 5

3 Minimum Item Reliability for Achieving Proportion

Acceptable Products with Confidence q … ………..…10

4 Power Study for the Classical Tolerance Intervals ……... 12

5 Confidence Estimation and Its Power for a Process of

γ-Content Acceptable Products … … … ……….……17

6 Power Study and Robustness of Lot Size for

Confidence estimation Technique ……….………20

Tolerance Interval and a New Approach for Lot Production Inspection

Abstract

The tolerance interval has long been a technique for manufacturer to verify if there is condence large enough that ensures a proportion of pro-duction lot conforming to specication limits. This paper formally formu-late this \condence" in terms of lot size and parameters involved in the underlying distribution of the characteristic. With this formulation, any technique for prediction of manufacturer's condence may be evaluated for its e ciency and it also provides a wide room for this prediction through statistical inferences for the unknown condence. We then study the power of the tolerance interval in detecting if there is a reasonably large manu-facturer's condence. We found that when the parameters involved in the distribution are known, the predicted manufacturer's condence is too opti-mistic in a value much higher than the true value and when the parameters involved in the distribution are unknown, the predicted manufacturer's con-dence is too conservative in a value much lower than the true one. The ine ciency partly comes from the fact that tolerance interval does not use the information of lot size in prediction which is an ancillary statistic in the considered statistical model. For statistical inference of this unknown condence, we introduce a point estimation technique that its results shown a power comparison seems to be very promising for the manufacturer.

Key words: Hypothesis testing power quality control tolerance interval.

1. Introduction

In manufacturing industry, specication limits for one characteristic of an

item, sayingLSLandUSL, dene the boundaries of acceptable quality for a

manufacturing item (component). An item is said to be non-defective if the measured characteristic is between the limits, otherwise it is defective. For a manufacturer of a mass-production item, the tolerance interval is designed for a quality assurance problem. For a production lot, the manufacturer

TypesetbyA M

S-T E

knows that unless a proportion, saying , of this lot is acceptable in the sense that the corresponding items are non-defective, he will lose money in this production. In practice, the manufacturer further expects a process that, in the long run, he or she will have lots of production not losing money

at least a percentage, sayingq0, of the time. If this is true, the manufacturer

is guaranteed that, in the long run, he or she will have lots of production

of proportion  or more conforming to specications at least 100q0% of

the time. The tolerance intervals are widely used in quality control and related prediction problems to monitor manufacturing processes to ensure product compliance with specications, etc. An important application of the tolerance interval is to examine if the manufacturer's expectation is accomplished.

We summarize the application of tolerance interval for predicting the manufacturer's condence into three steps:

(a) The rst step is to set up a hypothesis. Suppose that the characteristic

variable has a probability density functionf(x). The common purpose to

construct the tolerance interval is to test the following hypotheses:

H0 :

Z USL

LSL f(x)dx  vs. H1 :

Z USL

LSL f(x)dx >  (1.1)

(see Goodman and Madansky (1962) for reference). In fact, a level  test

for this hypothesis tries to answer if there is at least a proportion  of the

population conforming to specication limits with a condence level 1;.

(b) The second step is to construct the tolerance interval as a test statistic.

Suppose that we have a random sample X = (X1:::Xn)

0 from the

distri-bution of the characteristic variable. The pioneer article by Wilks (1941)

introduced a-content tolerance interval with condence 1;as an interval

(T1T2) = (t1(X)t2(X)) that satises PfP(X 0 2(T 1T2) jX)g1; for  2 (1.2)

where  is the parameter space and X0 represents the future observation

with the same distribution. A vast literature on developing tolerance inter-val has been proposed (see for example Wilks (1941), Wald (1943), Paulson

(1943), Guttman (1970) and, for a recent review, Patel (1986)). Comparing tolerance intervals based on criterion of expected length is the most popu-larly used selection technique. For normal tolerance interval, Eisenhart etc. (1947) constructed the shortest one. With the appealing property of short-est length, it is now popularly implemented in manufacturing industry and introduced in engineering texts. This criterion has also been a guide line for developing regression tolerance interval (see Goodman and Madansky (1962), Liman and Thomas (1988) and Mee et. al. (1991)).

(c) The third step is to set up a rule for hypotheses in (1.1) based an observed

tolerance interval. Let (t1t2) be the observation of a -content tolerance

interval at condence 1;=q

0. The test for hypotheses H0 vsH1 of (1.1)

is:

We reject

H0

if

(t1t2)

(LSLUSL): (1.3)

When H0 is rejected, statistically we are 100(1

;)% sure that at least

100% of the population is conforming to the specication limits. So, in a

long run, we will have lots having at least proportion  of the distribution

conforming to the specication limits at least a percentage 1 ; of the

time. With the interest of resolving the manufacturer's question, it is also generally making an extending conclusion as the following:

When

H0

is rejected, the lot of product is acceptable

because we have condence of a reasonably large value

that at least

100%

of the population is

conforming to specication limits

(1.4)

where Papp (1992) further interpreted that the reasonably large value is

1;. For other references, see Bowker and Goode (1952), Owen (1964) and

Schilling (1982). Our interest concerns the question: When a sample data

support to rejectH0, is making the extending inference in (1.4) appropriate?

The manufacturer wants to see if there is proportionof acceptable

prod-ucts with a reasonable large condence q0. Probabilitically it is known that

guaranteeing proportion of the population conforming to the specication

limits is not guaranteeing proportion  of the products in a lot or even the

of reasonably large condence 1; for tolerance interval indicates a

rea-sonably large condence q0 for the manufacturer? We concern this question

since manufacturers want to assure a good chance to have lots accepted when lots are produced at permissible levels of quality. On the other hand, it is also important for the consumer to see if there is irregular degrada-tion of levels of process quality in submitted lots. Henceful, any technique used for this prediction should be able to protect the benets of both the consumer and manufacturer.

With setting that the characteristic variable of a product obeys some

xed probability distribution, the condence that there is proportion  or

more non-defective items, i.e., conforming to specication limits, is com-pletely determined by the underlying distribution and the lot size, which is an unknown constant involving unknown parameters. The construction of tolerance interval for solving the manufacturer's problem aims to predict this unknown condence. However, the tolerance interval in (1.2) is con-structed primarily for the condence that the interval contains a proportion

 of the distribution. There may exists a big discrepancy between the

re-sulted proportion of the sample space and the proportion of product lot.

The size of discrepancy may be determined from the underlying distribution and the lot size, however, it is desired to discover.

The aim of this paper is to study this discrepancy and investigate if there is an alternative technique that is promising for predicting the manufac-turer's condence. We rst explicitly formulate (in Section 2) the manu-facturer's unknown condence in terms of distribution parameters and the lot size. With this formulation, it provides a room for statistical inferences for this true condence. Next, we study (in Sections 2 and 4) the power of the test using the tolerance interval to detect if there is a reasonably

large condence q0. We will see that the tolerance interval leads to

pre-dicted condence too optimistic in way that it is higher much more than the true one when the distribution parameters are known. On the other hand, when the unknown parameters are unknown the predicted condence is too conservative in the way that its predicted condence is lower much

smaller than the true one. This veries that the use of tolerance interval to predict the manufacturer's condence is not appropriate. Third, we will in-troduce a point estimation technique (in Section 5) for prediction of the true condence which provides a new approach for answering the manufacturer's question. We then further investigate the power of detecting the manufac-turer's condence through this new technique which leads to very promising results. Fourth, one observation for the ine ciency of the tolerance inter-val in detecting the manufacturer's condence is that this interinter-val does not use the information of lot size which represents an ancillary statistic in the statistical model. This provides another evidence that ancillary statistic is important in several statistical inference problems.

2. Formulation of Condence and an Evaluation of Tolerance

In-terval When Parameters Are Known

LetX0 be a random variable, representing the characteristic of a product

from a manufacturing process, having a distribution. There are production lots produced from this process. For simplicity, the lots are all assumed

to have the same size k. We want to see if there is condence q0 that

there is proportion  of products in lots conforming to specication limits

fLSLUSLg. To evaluate this, we have a random sample X

1:::Xn from

the same process to evaluate the manufacturer's condence. We need rst introduce appropriate criterions for evaluation of a technique for this pur-pose. We consider its ability in correctly identifying the true condence for reducing the following two errors:

(a) The rst error is that the inferenced condence is much higher than the true condence. In this situation, the resulted condence is too optimistic for the manufacturer.

(b) The second error is that the inferenced condence is much lower than the true condence. In this situation, the resulted condence is too conservative for the manufacturer.

Before consideration of any evaluation, let's formulate the framework of the statistical model we want to consider. We consider a manufacturing

process that the products forms a sequence of independent and identically distributed random variables from a distribution with distribution function

F. We are interesting to see if a production lot of sizek from this process

is with proportion  conforming to the specication limits fLSLUSLg.

With this purpose, we have a random sample X1:::Xn drawn from this

process for prediction. This random sample of size n is the only observable

variables.

We rst introduce a formulation of the true condence. Let X be a

random variable having distribution function F with probability density

function f(x). Then the item (product) reliability representing the

prob-ability that an item is non-defective is

pitem() =

Z USL

LSL f(x)dx=F(USL);F(LSL) (2.1)

whereFis the distribution function. Suppose that the lot size isk(usually a

large number). For this production lot, the number of acceptable products

is with binomial distribution b(kpitem()). Then the true condence for

having proportion  of production lot conforming to specication limits is

q= Xk i=k]  k i  pitem()i(1;pitem())k ;i: (2.2)

Treating q as a function of pitem(), 1;q has the same properties as the

operating characteristic (OC) curve. However, 1;qand OC curve are

dier-ent since OC curve doesn't involve lot sizek. Since condence function q is

increasing in item reliabilitypitem(), the interest of the manufacturer then

is to see if the true item reliability makesq larger or equal to a prespecied

value.

We assume that the manufacturer expects to have condence q0. Then,

when a lot of production is with q q

0 this lot attains the manufacturer's

expectation. In this situation, in the long run, it will accept lots having at

least proportion  acceptable products at least a proportion q0 of the time.

The true condence q often relies on the unknown parameters involving

predicted. The tolerance interval was designed to do this through the stated rule of the hypothesis testing. We will study the appropriateness of using the tolerance interval to detect the condence with a simple situation.

Suppose that now X has a normal distribution N( 2) where  and

are both known. From Wilks (1941), the natural tolerance interval as

(;z1+

2 

+z1+

2

) (2.3)

is a -content tolerance interval with condence 1. It is a 100% condence

tolerance interval. Suppose that we further assume that (LSLUSL) = (;z1+

2 +z 1+

2 ): (2.4)

It is generally accepted the statement that when an item reliability 100% is

required, the production lot satisfying (2.4) should be accepted (see Schilling (1982)) and we may conclude that with 100% condence that there is

pro-portion of acceptable products in lots. Is this conclusion really true? With

lot size k, the true condence of this production lot with proportion  or

more acceptable products is

q = Xk i=k]  k i  i(1;)k ;i:

We assume that the manufacturer requires item proportion  of acceptable

products which, in this case of (2.4), is identical to the item reliability. We

list the corresponding values of true condence for  = 0:90:95 and several

values of k.

Table 1.

True condence q for -content acceptable products

 k = 100 1000 10000 100000 0:5 0:5397 0:5126 0:5039 0:5012 0:6 0:5432 0:5137 0:5043 0:5013 0:7 0:5491 0:5155 0:5049 0:5015 0:8 0:5594 0:5189 0:5059 0:5018 0:85 0:5683 0:5217 0:5068 0:5021 0:9 0:5831 0:5265 0:5084 0:5026 0:95 0:6159 0:5375 0:5118 0:5037

We have several conclusions drawn from the results in the above table:

(a) Although in all cases of  and lot size the condences are greater than

0:5, but are all less than 0:65. In this situation that the tolerance interval

co-incides with the specication limits, the manufacturer, in the long run, may

accept lots having at least proportion  acceptable products guaranteeing

only proportion q less than 70% of the time. This result is not a surprise.

Knowing the coverage interval only helps in knowing the item reliability that, in fact, doesn't improves in enlarging the true condence.

(b) The true condence q relies on lot size k, specication limits and the

true item reliabilitypitem() no matter if the parameter  is known or not.

(c) For a given item reliabilitypitem() =, the true condence q is

increas-ing in lot size. On the other hand, for a given lot size, the true condence

decreases when  increases. Furthermore, when the lot size increases to

innity +1, the true condence q approaches to 0:5.

(d) It seems that it is too optimistic by using the tolerance interval to interpret the condence for that the manufacturer may accept lots having

at least proportion  acceptable products.

Example 1.

Schilling (1982) considered a case that the characteristic

obeys the normal distribution N(101) which indicated that (8:0411:96) =

(10;1:9610 + 1:96) = (8:0411:96) is a  = 0:95-content tolerance

inter-val with condence 100%. The author then assume that the specication

limits are LSL= 8:0 and USL= 12:0. Since the tolerance interval is fully

contained in the limits, he or she then claim that the product lot has to be accepted. However, the probabilities that there are more than proportion

0:95 of products conforming to the limits is

q= Xk i=k0:95  k i  i(1;)k ;i:

which given q = 0:77890:98441 when k = 100010000100000

respec-tively. Let's consider the case that the specication limits be LSL= 8:02

and USL = 11:98. We may see that the condences for the same k0s are

Let the proportion of acceptable products requiring by the manufacturer

be xed with . It is interesting to see, when the specication limits are

wider or shorter than the tolerance interval of (2.3), the performances of the true condence. From (2.2), any production lot has condence a positive

value when its item reliability pitem() is in interval (01). That is, no

matter how small the item reliability is, this process has chance to produce

proportion  acceptable products in a lot and, no matter how large (<

1) the item reliability is, this process has chance to produce very small proportion acceptable products in a lot. Then, it is interesting to investigate the sensitivity of the condence as a function of the item reliability. We

set some values of pitem() and proportion  and list the corresponding

condences q in Table 2.

Table 2.

Condence q for -content acceptable products

pitem()  = 0:8 0:85 0:9 0:95 0:99 0:7 4:986e;13 0 0 0 0 0:75 1:089e;4 8:770e;15 0 0 0 0:8 0:5189 2:644e;5 0 0 0 0:85 0:9999 0:5217 2:038e;06 0 0 0:9 1 0:9999 0:5265 5:995e;09 0 0:95 1 1 1 0:5375 2:797e;12 0:99 1 1 1 1 0:5830

We have several conclusions drawn from Table 2:

1. When pitem() is moderately below , the condence q is approximately

equal zero and whenpitem() is moderately larger than, it is approximately

equal one.

2. When item reliabilitypitem() is or lesser, the corresponding condence

q is about equal or smaller than 0:6.

3. For given a value of , the curve representing the condence as a

func-tion of pitem() is started from zero and rapidly climb up when pitem() is

increasing.

4. The interest is that when the true condence q may be moderately large

it is the case that item reliabilitypitem() is larger than. This tells us how

to improve the process.

3. Minimum Item Reliability for Achieving Proportion



Accept-able Products with Condence

q0

The use of tolerance interval to predict the manufacturer's condence is shown to be too optimistic when the parameters involved in the underlying distribution are known. For case that parameters are unknown, can the classical tolerance intervals be also too optimistic or on the opposite way? To investigate this question, we need to prepare some further results on the

relation between item reliability and manufacturer's condence q.

Recall that the manufacturer expects to have proportion  or more

ac-ceptable products with condence q0 or more. Let's dene the minimum

item reliability that guarantees proportion  acceptable products with

con-dence q0. That is, pq

0 satises k X i=k]  k i  (pq0) i₍₁;pq 0) k;i =q 0: (3.1)

Consider that the product's characteristic variable has a distribution

func-tion F. Then, the manufacturer may expect to have proportion 

accept-able products with condence q0 only if the item reliability satises

pitem() =F(USL);F(LSL)pq

0: (3.2)

For a given pairs (q0), we list the item reliabilities to achieve exactly

proportion of acceptable products with condenceq0in the following table.

Table 3.

Minimum item reliability (k=1,000)

q  = 0:8 0:85 0:9 0:95 0:99 0:8 0:8099 0:8587 0:9071 0:9549 0:9918 0:85 0:8123 0:8608 0:9089 0:9562 0:9923 0:9 0:8152 0:8634 0:9111 0:9577 0:9929 0:95 0:8196 0:8673 0:9142 0:9599 0:9938 0:99 0:8277 0:8744 0:9200 0:9638 0:9952

Suppose that the characteristic variable of interest obeys a normal

dis-tribution N( 2). Then item reliability, the probability that an item

con-forming to specications, is pitem( ) = Z USL LSL p1 2 e; (x;) 2 2 2 dx

and then the true condence to have proportion  acceptable products is

q = Xk i=k]  k i  pitem( )i(1;pitem( ))k ;i = Xk i=k]  k i  ((USL; );( LSL; ))i(1;(( USL; );( LSL; )))k;i:

We denote the minimum item reliability as pq0. A production lot to

have proportion  acceptable products with condence q0 requires that

pitem( ) = (USL;

);(

LSL;

)pq

0 (3.3)

where pq0 may be found from Table 3.

One question very interesting is how short the specication limits that guarantees the minimum item reliability. Let the specication limits be the

special form fLSLUSLg=f;l +l g and we denote lq

0 as the l so

that the item reliability produce item reliability exactly as

pq0 =P( ;lq

0 X +lq0 ):

We listlq0 in this design in the following table.

Table 4.

Specication limits (LSLUSL) = (;lq

0 +lq0 ) to achieve

item reliability exactly equal to pq0 (k=1,000)

q  = 0:8 0:85 0:9 0:95 0:99 0:8 1:3103 1:4710 1:6807 2:0043 2:6451 0:85 1:3172 1:4789 1:6898 2:0161 2:6672 0:9 1:3263 1:4889 1:7013 2:0309 2:6953 0:95 1:3397 1:5037 1:7184 2:0531 2:7380 0:99 1:3649 1:5317 1:7509 2:0954 2:8212

This table will be used in next section to study the power of the classical tolerance interval in detection of manufacturer's condence.

4. Power Study for the Classical Tolerance Intervals

Suppose that we have X1:::Xn representing a random sample for the

characteristic variable of interest and the specication limits for the

charac-teristic arefLSLUSLg. Let (T

1T2) be a tolerance interval of Wilks (1941)

constructed from the random sample. To test the hypotheses of (1.2), we consider the popularly rule ( see Bowker and Goode (1952)) setting: re-jecting H0 if (T1T2)

 (LSLUSL). Following the classical evaluation of

hypothesis testing, we may analogously dene a power function, in terms of specication limits, as

(LSLUSL) =P((T1T2)

(LSLUSL)) (4.1)

which, from Papp(1992), provides the probability of concluding that there

is proportion  or more acceptable products in a lot with condence 1;

at the specied specications. Basically (4.1) is to evaluate the rule that

we conclude that there is proportion  or more acceptable products with

condence q0 when (T1T2)

 (LSLUSL) occurs. Hence, we expects a

tolerance interval to have lower power (LSLUSL) when pitem() < pq0

is true and large power when pitem()pq

0 with q 0 = 1

;. We want to

simulate the powers for the Eisenhart et al.'s tolerance interval for several combinations of specication limits. We also consider a Monte Carlo study

with replication numberm. By letting (LSLUSL) = (;bb), the simulated

power of a tolerance interval (T1T2) is dened as

 = 1_mXm j=1 I((tj1t j 2) (;bb)) (4.2) where (tj1t j 2) is the observation of (T

1T2) from the jth sample. The power

of (4.2) simulates the chance of (4.1) that the tolerance interval (T1T2)

may conclude that the production lot includes a proportion  of acceptable

To study (4.2), suppose that we have a random samapleX1:::Xndrawn

from a distribution normal distribution N( 2) where both  and are

unknown. The general form of a prediction interval for a future normal random variable is of the form

( X;m

SX +mS) (4.3)

where the 100(1;)% condence interval (prediction interval) is the form

with m = t 1;  2(n ;1) q 1 + 1 n and where t1;  2(n ;1) represents the 1; 

2th quantile of the central t-distribution with degrees of freedom. For the

Wilks' tolerance interval, Eisenhart et al (1947) developed the shortest one which is now the most popular version of tolerance interval to deal with the manufacturer's problem when the characteristic variable does obey a normal

distribution. We select values m from the table in Eisenhart et al. (1947).

With replication m = 100000, we generate random sample of size n from

distributionN(01). Let xj ands2

j be the sample mean and sample variance

for thejth sample. We compute this tolerance interval and study its powers

of (4.2) with several sample sizes n= 203050 and 100 and various values

b. The simulated results are listed in Table 5.

Table 5.

Powers of the minimum-width tolerance intervals ((1;) =

Limits n= 20 n= 30 n= 50 n= 100 (k) (2:152) (2:025) (1:916) (1:822) b= 1:4 q= 1:371e;08 0:0042 0:0024 0:0006 0:0000 b= 1:5 q = 0:00071 0:0107 0:0080 0:0041 0:0016 b= 1:6 q = 0:17899 0:0237 0:0217 0:0187 0:0151 b= 1:645 q = 0:52786 0:0325 0:0330 0:0328 0:0341 b= 1:7013 q = 0:9 0:0480 0:0542 0:0646 0:0863 b= 1:8 q= 0:9995 0:0834 0:1064 0:1484 0:2536 b= 2:0 q = 1:0 0:2129 0:3032 0:4714 0:7657 b= 2:2 q = 1:0 0:4043 0:5757 0:8067 0:9790 b= 2:5 q = 1:0 0:7113 0:8842 0:9860 0:9999 b= 3:0 q = 1:0 0:9649 0:9970 1:0000 1:0000 b= 3:5 q = 1:0 0:9986 0:9999 1:0000 1:0000

According to Papp(1992), when a-content tolerance interval with

con-dence 1; is contained in the specication limits, we may claim that there

is percentage  of acceptable products with condence 1;. We then have

several conclusions drawn from the simulation results in above table:

is increasing the power for the tolerance interval to detect the manufacturing

process to have condence 1; is increasing.

(b) Whenb < 1:7013 the true condence for a-content acceptable products

is less than 1;= 0:9. In this situation, the powers of this tolerance interval

in all cases of sample size are all less than 0:05. That is, it is fewer than

0:05 to claim to have condence 0:9.

(c) Whenb1:7013 the true condence for a-content acceptable products

is greater than or equal to 1; = 0:9. In this situation, although the

powers are increasing when b increases. However, they are still with very

low percentages for cases such as 1:7013 b 2:0 to claim that it is a

process of = 0:9-content acceptable products with condence 1;= 0:9.

This is a plan too conservative to both the consumer and manufacturer. (d) If, in the long run, there is a large proportion that the tolerance

inter-vals rejected the null hypothesis H0, the process must be in well control so

that the specication limit interval (LSLUSL) is wide enough to have the

tolerance intervals contained in the interval of specication limits.

(e) When there is larger chance that the tolerance interval will claim to

have percentage  of acceptable products with condence 1;. The actual

condence often is larger far from 1 ; . For example, b = 2:5 is the

situation that there is good chance to claim the manufacturer's expectation

with condence 0:9, however, the condence is approximately near 1. This

indicates that detection of manufacturer's question by the tolerance interval is too conservative.

Alternatively Huang, Chen and Welsh (2005) showed that ( X;t 1;  2(n ;1 p nz1+ 2 )pS nX +t1;  2(n ;1 p nz1+ 2 )pS n) (4.3)

is also a  content tolerance interval with condence 1; where t(`m)

is the -th quantile of noncentral t-distribution with degrees of freedom `

and noncentrality parameter m. The interest is that this tolerance

inter-val is explicitly formulated. By letting kt = t1;

 2 (n;1 p nz1+ 2 ) p n , we list the

simulation results of the powers of (4.1) for this tolerance interval in Table 5.

Table 6.

Power of coverage-interval tolerance intervals ((1;) = (0:90:9)) Limits n= 20 n= 30 n= 50 n= 100 (kt) (2:3960) (2:2198) (2:0649) (1:9265) b= 1:4 0:0009 0:0004 0:0001 0:0000 b= 1:5 0:0015 0:0009 0:0002 0:0000 b= 1:6 0:0065 0:0056 0:0041 0:0024 b= 1:645 0:0098 0:0088 0:0081 0:0063 b= 1:7013 0:0154 0:0160 0:0162 0:0206 b= 1:8 0:0291 0:0337 0:0477 0:0868 b= 2:0 0:0887 0:1288 0:2325 0:5043 b= 2:2 0:2023 0:3255 0:5669 0:9071 b= 2:5 0:4701 0:6979 0:9273 0:9991 b= 3:0 0:8675 0:9785 0:9997 1:0000 b= 3:5 0:9878 0:9996 1:0000 1:0000

Let's investigate the probability thatH0 will be accepted when there does

exist a  coverage interval contained in specication limits.

1 m m X j=1 I(PX0 f(t 1t2) g and (t 1t2) (;bb)):

Table 7.

Power with content acceptable products for the minimum-width

tolerance intervals ((1;) = (0:90:9)) Limits n= 20 n= 30 n= 50 n= 100 (k) 2:152 2:025 1:916 1:822 b= 1:645 0:0000 0:0000 0:0000 0:0000 b= 1:8 0:0225 0:0373 0:0704 0:1589 b= 2:0 0:1270 0:2120 0:3750 0:6661 b= 2:2 0:3052 0:4748 0:7033 0:8797 b= 2:5 0:6084 0:7845 0:8842 0:9001 b= 3:0 0:8608 0:8965 0:8983 0:9002 b= 3:5 0:8952 0:8989 0:8983 0:9002 b= 4:0 0:8965 0:8989 0:8983 0:9002 b= 4:5 0:8966 0:8989 0:8983 0:9002

Table 8.

Power with content acceptable products for the coverage-interval

Limits n= 20 n= 30 n= 50 n= 100 (kt) 2:3960 2:2198 2:0649 1:9265 b= 1:645 0:0000 0:0000 0:0000 0:0000 b= 1:8 0:0092 0:0149 0:0268 0:0659 b= 2:0 0:0598 0:1036 0:2052 0:4808 b= 2:5 0:4317 0:6626 0:8986 0:9766 b= 3:0 0:8279 0:9473 0:9725 0:9773 b= 3:5 0:9504 0:9683 0:9727 0:9773 b= 4:0 0:9630 0:9686 0:9727 0:9773 b= 4:5 0:9634 0:9686 0:9727 0:9773

5. Condence Estimation and Its Power for a Process of



-Content

Acceptable Products

Let the interest of characteristic be a random variable having a

distribu-tion funcdistribu-tion F and the specication limits arefLSLUSLg. We also have

a random sample X1:::Xn drawn from this underlying distribution.

Let ^ be an estimator of the unknown parameter . Replacing F by F^

and from (3.1), we have an estimator of the probability of a product to be acceptable as

^

pitem =pitem(^) =F^(USL) ;F

^

(LSL):

For each (q), there is item reliabilitypq such that whenpitem =pq

iden-tity of (3.1) holds. First we want to simulate the e ciencies of estimating

the item reliabilitypitem =pq in Table 3.

Consider that we are dealing with the normal distribution N( 2). For

each (q) there is`qsuch thatpq =P(;`q X +`q ) (see Table

4). When the item reliability is pq we may guarantee that the condence

of a lot with proportion  products conforming in specication limits is q.

Our aim is to see if the estimate ^pitem is e cient for estimating the true

reliabilitypitem =pq. Suppose that now this random sample is drawn from

a normal distribution N( 2) and choose the classical estimators ^ = X

and ^ 2 =S2 = 1 n;1 Pn i=1(Xi ;X) 2. We then have ^ pitem = (USL;X S );( LSL;X S ): (5.1)

With the above process of parameter estimation, we perform this

simula-tion with replicasimula-tion m = 100000 and sample size n = 30. Let ^p_j be

the condence estimate corresponding to the j-th, j = 1:::m, observation

x1j:::xnjdrawing from the standard normal distributionN(01) where we

have then (LSLUSL) = (;`q`q). Let ^pjitem be the estimate of the item

reliability at the jth replication. We then dene the empirical average of

ietem reliability estimation as 

pitem = 1_mXm

j=1

^

pjitem:

In the following table, we display the simulated results of the averaging item

reliability and true reliabilitypq in the vector form

  pitem (pq)  .

Table 9.

Empirical average for the estimation of item reliabilitypq when

the underlying distribution is normal (k=1,000, m=100,000, n=30)

q  = 0:8  = 0:85  = 0:9  = 0:95  = 0:99 0:8 ₍₀0:_:8070_{8099) (0}0:_:8547_{8587) (0}0_::9020_{9071) (0}0:_:9497_{9549) (0}0:_:9887₉₉₁₈₎ 0:85 ₍₀0:_:8093_{8123) (0}0:_:8568_{8608) (0}0_::9039_{9089) (0}0:_:9510_{9562) (0}0:_:9893₉₉₂₃₎ 0:9 ₍₀0:_:8121_{8152) (0}0:_:8594_{8634) (0}0_::9061_{9111) (0}0:_:9526_{9577) (0}0:_:9900₉₉₂₉₎ 0:95 ₍₀0:_:8167_{8196) (0}0:_:8630_{8673) (0}0_::9094_{9142) (0}0:_:9548_{9599) (0}0:_:9911₉₉₃₈₎ 0:99 ₍₀0:_:8245_{8277) (0}0:_:8700_{8744) (0}0_::9149_{9200) (0}0:_:9588_{9638) (0}0:_:9928₉₉₅₂₎ In the next, we consider the mean square error

MSE = 1_mXm

j=1

(^pjitem;pq) 2:

The simulated results are displayed in the following table.

Table 10.

Mean square error for the Estimation of item reliability pitem

q  = 0:8  = 0:85  = 0:9  = 0:95  = 0:99 0:8 0:0031 0:0025 0:0017 0:0008 1:08e;4 0:85 0:0031 0:0025 0:0017 0:0007 1:01e;4 0:9 0:0031 0:0025 0:0016 0:0007 9:09e;5 0:95 0:0030 0:0024 0:0016 0:0007 7:85e;5 0:99 0:0029 0:0023 0:0015 0:0006 5:68e;5

From (3.2), we further have condence estimator for a process of 

-content acceptable products as ^ q = Xk i=k]  k i  ^ piitem(1;p^item)k ;i: (5.2)

The estimation of the unknown condence uses the estimation of unknown

parameters  and lot size where the latter one is an ancillary statistic.

Suppose that now this random sample is drawn from a normal

distribu-tion N( 2) and choose the classical estimators ^ = X and ^ 2 = S2 =

1 n;1 Pn i=1(Xi ;X) 2. We then have ^ pitem = (USL;X S );( LSL;X S ):

With this, we further have ^ q= Xk i=k]  k i  ((USL;X S );( LSL;X S ))i(1;(( USL;X S ) ;( LSL;X S )))k;i: (5.3)

Let's study the e ciency of condence estimation for having a

propor-tion  or more acceptable products. Let's rst x  = 0:9. We

gener-ate observation of a random sample from the standard normal

distribu-tion and then compute its sample mean x and sample standard deviation

s= ( 1

n;1 Pn

i=1(xi ;x)

2)1=2. With specication limits (LSLUSL) = (

;``)

andk = 1000, the condence estimate for this sample is ^p of (5.3) with X

andS be replaced by x ands, respectively. We choose (pq`) from Table 4

With the above process of parameter estimation, we perform this

simu-lation with replication m = 100000 and let ^p_j be the condence estimate

corresponding to the j-th observation x1j:::xnj for j = 1:::m. Now we

are interesting in the following average condence, 

p = 1_mXm

j=1

^

pj:

The following table displays the simulated results of average condence.

Table 11.

Simulated averaging condences (k=1,000, m=100,000)

Spec L n= 20 30 50 100 l = 1:4 0:1664 0:1199 0:0682 0:0217 l = 1:5 0:2790 0:2459 0:1915 0:1209 l = 1:6 0:4219 0:4085 0:3916 0:3621 l = 1:7013 0:5697 0:5902 0:6245 0:6732 l = 1:8 0:7043 0:7504 0:8075 0:8881 l = 2:0 0:8932 0:9376 0:9758 0:9964 l = 2:2 0:9725 0:9910 0:9986 0:9999 l = 3:0 0:9999 1 1 1

6. Power Study and Robustness of Lot Size for Condence

esti-mation Technique

We have seen that the tolerance interval is not really appropriate in

de-tecting the manufacturer's condence for having a proportionof acceptable

production lots for that it is too optimistic when the parameters involved in the distribution are known and it is too conservative when the param-eters are unknown. It is then interesting to evaluate the manufacturer's condence through the estimate of the unknown condence. Consider a simulation that we randomly select observations from the standard normal

distribution. We consider replication m= 100000,  = 0:9 and condence

q0 = 0:9. By letting ^q

j _{as the jth estimate of the condence, j = 1:::m.}

We dene the power of this point estimator as

 = 1_mXm

j=1

I(^qj q 0):

The following table list the simulation results of this study.

Table 12.

Power for the Estimation of the condence for having a

pro-portion  or more acceptable product when the underlying distribution is

normal (k=1,000, m=100,000) Spec L n= 20 30 50 100 l = 1:4 0:1180 0:0744 0:0322 0:0049 1:5 0:2161 0:1682 0:1087 0:0429 1:6 0:3344 0:3061 0:2653 0:1927 1:7 0:4802 0:4838 0:4864 0:4850 1:7013 0:4890 0:4928 0:4959 0:5062 1:8 0:6208 0:6555 0:7060 0:7849 1:9 0:7460 0:7999 0:8643 0:9445 2:0 0:8455 0:8978 0:9518 0:9916 2:2 0:9548 0:9821 0:9967 0:9999 2:5 0:9966 0:9995 1:0 1:0 3:0 1:0 1:0 1:0 1:0

We have several conclusions that may be drawn from the results in the above table:

(a) For every given sample size n, the e ciency is strictly increasing in the

specication limitl. This fullls our expectation.

(b) In the setting = 0:9 andq0 = 0:9 by the manufacturer,l = 1:7013

guar-antees to have proportion  or more acceptable products with condence

exactly q = 0:9. In this situation, the e ciencies are about 0:48. There is

probability 0:48 that we can detect that the lot of interest is acceptable. In

Table 5, we see that the tolerance interval techniques can observe this fact

with chance less than 0:1.

(c) When the specication limit l is a bit wider we have large chance to

detect that the lot is acceptable. Comparing this table with Table 5, we see that this technique of condence estimation is better than the tolerance interval technique casewise.

For this condence estimation technique, it may be argued that the lot

sizek may not be correctly predicted or counted and we may have, especially

when the size is huge, only an approximate number. Then, it is interesting to see how robust the estimation technique in its estimated condence when

the lot size has not been correctly used. We design a simulation computing the average condences with various large sizes to see its sensitiveness in terms of lot size.

Table 12.

Simulated averaging condences for some large lot sizes

Spec L n= 20 30 50 100 k = 100000 l = 1:5 0:2645 0:2272 0:1715 0:0918 l = 1:6474 0:4756 0:4798 0:4775 0:4808 l = 1:6482 0:4783 0:4788 0:4817 0:4823 l = 1:6507 0:4787 0:4808 0:4842 0:4894 l = 1:8 0:6935 0:7389 0:8030 0:8892 l = 2:0 0:8905 0:9365 0:9775 0:9979 l = 2:5 0:9981 0:9998 1 1 k = 300000 l = 1:5 0:2669 0:2281 0:1739 0:0947 l = 1:6474 0:4787 0:4858 0:4874 0:4858 l = 1:6482 0:4807 0:4843 0:4863 0:4906 l = 1:6507 0:4850 0:4901 0:4943 0:4996 l = 1:8 0:6989 0:7440 0:8090 0:8948 l = 2:0 0:8910 0:9375 0:9780 0:9982 l = 2:5 0:9985 0:9999 1 1 k = 500000 l = 1:5 0:2723 0:2307 0:1745 0:0965 l = 1:6474 0:4802 0:4825 0:4872 0:4892 l = 1:6482 0:4827 0:4848 0:4898 0:4939 l = 1:6507 0:4865 0:4897 0:4963 0:5028 l = 1:8 0:6978 0:7420 0:8080 0:8963 l = 2:0 0:8921 0:9387 0:9783 0:9981 l = 2:5 0:9984 0:9998 1 1

7. Concluding Remarks

We have several remarks illustrating our further concern and clarication of our study:

(a) We show that the use of tolerance interval to evaluate for the

manu-facturer the condence for a proportion  of production lot conforming to

specications is not appropriate. This does not imply any in-appropriateness for other purposes.

(b) Concerning condenceqas a parameter in terms of unknown parameters

, there are condence interval and hypothesis testing that could be

devel-oped to infer the unknownq. Concerning the problem, if a process that may

generally produce products with condenceq0 a proportion of production

lot conforming to specications, we may want to test the hypothesis that if one other process may also produce the same quality.

(c) The ine ciency of the classical tolerance interval for detecting the man-ufacturer's condence partly comes from the fact that it uses only the infor-mation contained in the random sample. The lot size which is an ancillary statistic in this case is an important information that hasn't been considered in construction of tolerance interval. This is an example that an ancillary statistic provides important information for statistical inference.

(d) There are tolerance interval-like technique for deciding if we will accept a production lot (see Kirkpatrick (1970), Owen and Hua (1977), Weingarten (1982) and Mee (1984)). This technique does not employ the information of lot size and has not been popular in practical use. Hence it is not in our study.

References

Bowker, A. H. and Goode, H. P. (1952). Sampling Inspection by Variables,

McGraw-Hill Book Company, Inc.: New York.

Bucchianico, A. D., Einmahl, J. H. J. and Mushkudiani, N. A. (2001).

Smallest nonparametric tolerance regions. The Annals of Statistics,

29, 1320-1343.

Carroll, R. J. and Ruppert, D. (1991). Prediction and tolerance intervals

with transformation and/or weighting. Technometrics, 33, 197-210.

Eisenhart, C., Hastay, M. W. and Wallis, W. A. (1947). Techniques of

Statistical Analysis. McGraw-Hill Book Company: New York.

Goodman, L. A. and Madansky, A. (1962). Parameter-free and

nonpara-metric tolerance limits: the exponential case. Technometrics. 4, 75-95.

Guttman, I. (1970). Statistical tolerance regions: Classical and Bayesian.

Huang, J.-Y., Chen, L.-A. and Welsh, A. (2005). The mode interval and its application to control charts and tolerance intervals. Submitted. Jilek, M. and Likar, O. (1960a). Tolerance limits of the normal

distribu-tions with known variance and unknown mean. Australian Journal of

Statistics, 2, 78-83.

Jilek, M. and Likar, O. (1960b). Tolerance regions of the normal distribution

with known , unknown 2. Biom. Zeit., 2, 204-209.

Kirkpatrick, R. L. (1970). Condence limits on a percent defective

charac-terized by two specication limits. Journal of Quality Technology. 2,

150-155.

Liman, M. M. T. and Thomas, D. R. (1988). Simultaneous tolerance

inter-vals for the linear regression model. Journal of the American Statistical

Association. 83, 801-804.

Marshall, A. W. (1949). A note on the power function of the Wald-Wolfowitz tolerance limits for a normal distribution. The RAND Corporation, Research Memorandum RM-271.

Mee, R. W. (1984). Tolerance limits and bounds for proportions based on

data subject to measurement error. Journal of Quality Technology. 16,

74-80.

Mee, R. W., Eberhardt, K. R. and Reeve, C. P. (1991). Calibration and

simultaneous tolerance intervals for regression. Technometrics. 33,

211-219.

Odeh, R. E. and Owen, D. B. (1980). Tables for Normal Tolerance Limits,

Sampling Plans, and Screening. Marcel Dekker, Inc.: New York. Owen, D.B. (1964). Control of percentage in both tails of the normal distrib

ution. Technometrics, 6, 377-387. Errata (1960) 8, 570.

Owen, D.B. and Hua, T. A. (1977). Tables of condence limits on the

tail area of the normal distribution. Communications in Statistics

Papp, Z. (1992). Two-sided tolerance limits and condence bounds for

nor-mal populations. Communications in Statistics - Theory and Methods,

21, 1309-1318.

Patel, J.K. (1986). Tolerance limits - A review. Communications in Stat

istics - Theory and Methods, 15, 2719-2762.

Paulson, E. (1943). A note on tolerance limits. Annals of Mathematical

Statistics, 14, 90-93.

Schilling, E. G. (1982). Acceptance Sampleing in Quality Control. Marcel

Dekker, Inc.: New York.

Wald, A. (1943). An extension of Wilks' method for setting tolerance limits.

Annals of Mathematical Statistics, 14, 45-55.

Wald, A. and Wolfowitz, J. (1946). Tolerance limits for a normal

distribu-tion. Annals of Mathematical Statistics. 17, 208-215.

Wallis, W. A. (1951). Tolerance intervals for linear regression.

Proceed-ings of the Second Berkeley Symposium on Mathematical Statistics and Probability, ed. by Neyman, University of California Press. 9, 43-52. Weingarten, H. (1982). Condence intervals for the percent nonconforming

based on variables data. Journal of Quality Technology. 14, 207-210.

Wilks, S. S. (1941). Determination of sample sizes for setting tolerance