Search

In sampling inspection by variables, an item is considered defective if its quality characteristic Y falls below some specification limit L0. We consider switching to a new supplier if we can be sure that the proportion of defective items for the new supplier is smaller than the proportion defective for the present supplier. Assume that Y has a normal distribution. A test for comparing these proportions is developed. A simulation study of the performance of the test is presented.

A new process capability index is proposed that takes into account the location of the process mean between the two specification limits, the proximity to the target value, and the process variation when assessing process performance. The proposed index is compared to other indices on several properties. The proposed index is estimated based on a random sample of observations from the production process when the process is assumed to be normally distributed. The 95% lower confidence limits for the proposed index are derived for given sample sizes and its estimates.

The common measures of process capability, usually indicated by Cp, CPU, CPL, and Cpk are considered. Tables of lower confidence limits are given on these measures where the sample mean, and the sample range (properly adjusted by d2) are substituted for the population mean and population standard deviation in the definition formulas.

A procedure is described for screening using a variable X which will guarantee that 100δ% of Y values for screened items will be between L and U (two specification limits). The random variables (X, Y) are assumed to have a joint bivariate normal distribution. Tables for implementing the procedure are also given. The procedure is then extended to the case where all the parameters are unknown.

In the one-sample Student t-test, the occurrence of a type-I error is dependent on the estimates of the mean and standard deviation for a fixed sample size, n. The statistic can achieve significance either by the sample mean being too different from the hypothesized mean or by the sample standard deviation being too small. The critical region is partitioned to determine the characteristics of samples in the critical region, assuming the null hypothesis is true. As might be conjectured from the use of the t-statistic, mis-estimation of the mean is shown to be the predominant characteristic of samples in the critical region for sample sizes larger than 20 and significance level greater than 0.01. Underestimation of the variance, unless accompanied by a misestimation of the mean, is a far less frequent characteristic of critical region samples.

Some properties are considered of two-sided 3-content tolerance intervals for a normal distribution. We are given the sample mean, x, and the sample standard deviation, s, computedfrom a random sample of size n from a N(l,az) distribution. The tolerance interval x ± ks will cover a proportion, 3, of the sampled distribution with (100)confidence. We consider the effect of sample size on the proportion of the population covered, using a criteria for measuring the precision of the coverage. We also give tables to enable the user to control this precision. 969

One-sided B-content tolerance intervals for the two-parameter double exponential distribution are considered. Monte Carlo tolerance factors along with estimates of their asymptotic standard deviations are presented. Smoothing the Monte Carlo results using least squares regression is investigated. Comparisons among the approximate, Monte Carlo and smoothed tolerance factors are also investigated.

A performance variable with a one-sided specification cannot be measured directly, but one or two related variables (called screening variables) can be measured. The screening variables are used to select product in order to raise the proportion of unit..

The bivariate normal distribution function may be expressed as the product of a marginal normal distribution times a conditional distribution. By approximating this conditional distribution, we obtain a simple method for approximating bivariate normal p..

Representations of the doubly noncentral F-distribution are presented i n terms of the cdf of the noncentral chi-square. These are specialized t o the oncentral F and f o r an even number of degrees of freedom f o r the denominator the noncentral F is zxpressed as a weighted sum of noncentral t - probabilities When either or both of the degrees of freedom are even, this result is then specialized to give the F distribution in terms of Student's t - distribution

Duality between points and lines is examined, and the error distributions corresponding to various forms of duality are simulated. Several different least squares procedures for fitting of a line to several points are dualized to the fitting of a point to several lines and then dualized again to give the coordinate of a point of intersection of several lines.

The effect of measurement error on the operating characteristics of one-sided (normal distribution) variables sampling plans is discussed. A table showing the effect on the operating characteristic curves for the proportion nonconforming is given in ter..

A procedure for Screening is given for the case where an underlying joint bivariate normal distribution is assumed for the screening variate and the performance variate. This is extended to the situation with unknown parameters. The procedure for attaining a preset goal for the performance variable is given. An analysis is also made to determine the effect of a preset selection procedure on the performance when the parameters of the bivariate normal distribution are unknown.

The joint density function of the sample mean and sample variance is recursively derived for samples from a population with density function f where f (x) > 0 cilinost syervi,vheie, everywhere continuous and has certain integral properties. For populations where f does not have these integral properties, this joint density is an approximation. This joint density-function is used to derive the density function of the t-statistic for samples from f. The family of generalized normal density functions is used for an example. The approximation for the t-density is given for that family. For some specific members of the family, the true a probabilities for the approximations are tabled and compared to the results of a simulation study.

A solution is given to the problem of using a variable correlated with a performance variable to screen a product when the correlation is unknown or when the proportion of presently acceptable product is unknown. The procedure involves setting lower confidence limits on these unknown parameters and then relating the confidence coefficient to the probability of the procedure attaining a required proportion of acceptable product. The performance variable and screening variable are assumed to be jointly normally distributed.

Integrals of functions of the univariate, bivariate, trivariate and multivariate normal densities are given. Both indefinite and definite integrals are included.

One-sided two-stage prediction intervals for a normal population are extended to a third sampling stage. Procedures and tables are given for two situations. In the first situation, methods for obtaining such intervals are presented, and tables for calculating such prediction intervals are provided. In the second situation, a two-stage prediction interval has been applied, and a third stage is now required. Sample sizes are given for the third stage.

The power of Student's t-test is partitioned to determine* the proportion of time the sample standard deviation is too small or the mean is misestimated given that rejection of the hypothesis under test has occurred

The conditional distribution given complete sufficient statistics is used along with the Rao-Blackwell theorem to obtain uniformly minimum variance unbiased (UMVU) estimators after a transformation to normality has been applied to data. The estimators considered are for the mean, the variance and the cumulative distribution of the original non-normal data. Previous procedures to obtain UMVU estimators have used Laplace transforms, Taylor expansions and the jackknife. An integration method developed in this paper requires only integrability of the normalizing transformation function. This method is easy to employ and it is always possible to obtain a numerical result.

Tables are given of confidence limits on tail areas, γ, of the normal distribution, where γ = P{Y ≥ L}, and where L is a given number, and Y is normally distributed with unknown mean, μ, and unknown variance, σ2.

The Burr distribution, having two shape parameters in addition to the usual parameters for location and dispersion, is very flexible and has several interesting and often used special forms. A formal statement of the approximate minimum chisquare is made and then developed for the Burr.

When the two-sample t-test has equal sample slies, it is widely considered to be a robust procedure (with respect to the significaoce level) under violatioa of the assuaptioo of equal variances. This paper is coa-earned with a quantification of the amount of robustness which this procedure has under such violations, The approach is through the concept of "religion of robustness" and the resluts show an extremely strong degree of robustness for the equal an extremely strong degree of robustness for the equal sample size t-test, probably more so than most statistyicians realise. This extremely high level of robustness, however, reduces quickly as the sample sizes begin to vary from equality. The regions of robustnes obtained show that while most users would likely be satisfied with the degree of robustness inherent when the two sample sizes each vary by 10% from equality, most would wish to be much more cautions when the variation is 20%. The study covers sample sizes n1 -= n 2 = 5(5)30(10)50 plus 10% and 2...

(This paper was presented at the Journal of Quality Technology Session at the 29th Annual Fall Technical Conference of the Chemical and Process Industries Division of the American Society for Quality Control and Section on Physical and Engineering Scien..

Representations of noncentral chi-square cumulative distribution function and probability density function are reviewed and new repre¬sentations are given. One representation of the cdf in terms of an integral is easily computed on any machine which has an accurate algorithm for computing the normal cdf.

The problem solved in this paper is that of determining the minimum sample size for setting the ‘normal’ range for bodily fluids. The proportions of too low and too high values which are considered ‘abnormal’ are chosen based upon medical considerations. The criterion used to determine the minimum sample size is that the proportions of the too low and too high values will be not exceeded by more than a prescribed amount with a given probability. The resulting limits are β-expectation tolerance limits with the added condition just noted, and are labeled β-expectation inner tolerance limits.

The problem is considered of screening on a random variable correlated with the performance variable to increase acceptable product. Consider a stockpile with a given proportion meeting a set quality standard and the necessity to upgrade the stockpile by testing on a variable correlated with the variable of interest. For example, stcppose lifetime is the variable of interest. Clearly it would be useless to try to screen on the lifetime variable itself. However, if a variable correlated with lifetimes is available and a bivariate normal model is assumed then much can be done. If all of the paramctcrs are known, the solution is given by Owen, McIntire and Seymour [12], but if some of the parameters are unknown then some procedures for handling the problem are given here. Ptlost of the previous work in the area of selection by a correlated variate is concerned with personnel selection and animal improvement. As such, it is widely scattered. Fol example, Owen [9], p. XLIV, contains a bibliography which contai...

We investigate in this article various techniques for determining a tolerance limit such that the probability is γ that at least a proportion P of a population, produced in batches, exceeds that limit. If the variance ratio is known, one may obtain exact tolerance limits. For settings where the variance ratio is not necessarily known, we describe a procedure, based on the Satterthwaite approximation, for obtaining conservative tolerance limits. The factors based on the Satterthwaite approximation are smaller than those obtained using the procedure of Lemon (1977).

This paper is concerned with estimating p = P(X1 < Y …, Xn < Y) or q =P (X < Y1, …, X < Yn) where the X's and Y's are all independent random variables. Applications to estimation of the reliability p from stress-strength relationships are considered where a component is subject to several stresses X1, X2, …, XN whereas its strength, Y, is a single random variable. Similarly, the reliability q is of interest where a component is made of several parts all with their individual strengths Y1, Y2 …, YN and a single stress X is applied to the component. When the X's and Y's are independent and normal, maximum likelihood estimates of p and q have been obtained. For the case N = 2 and in some special cases, minimum variance unbiased estimates have been given. When the Y's are all exponential and the X is normal with known variance, but unknown mean (or uniform between 0 and θ, θ being unknown) the minimum variance unbiased estimate of q is established in this paper.

The conditional probability that Y ≥ L, given that (Y−μ)/sy ≥ −k, is evaluated where Y is N(μ, σ 2), L is a given constant (a specification limit), and is the usual unbiased estimator of σ2 based on a preliminary sample of size n. Tables of k are given for use in raising the proportion of Y’s above L from γ (before screening) to δ (after screening) for δ = 0.90, 0.95, 0.99, and 0.999. These tables correspond to the limiting case as the correlation approaches one for the screening procedure given by Owen and Su (1977). Tables for the case where the correlation is less than one were given by Owen and Haas (1977).

Tables are provided for constructing one-sided simultaneous confidence limits for the two tail areas of a normal distribution with unknown mean and unknown variance. The two tail areas are the proportion of product above an upper specification limit and..

A Cornish-Fisher expansion is used to approximate the per-centiles of a variable of the bivariate normal distribution when the other variable is truncated. The expression is in terms of the bivariate cumulants of a singly truncated bivariate normal distribution. The percentiles are useful in the problem of personnel selection where we use a screening variable to screen applicants for employment and a correlated performance variable to screen employees for rehiring. This paper provides a bivariate cumulants table for determining the cutoff score of the performance variable. The following two problems are also con¬sidered: (1) determine the proportion of applicants who would have been successful had no screening been applied, and (2) determine the proportion of individuals being rejected byscreening who would have been successful had they been hired, The variable that is used to measure job performance and the variable that measures the outcome of an aptitude test are assumed to be jointly normally distribu...

Tolerance limits which control both tails of the normal distribution so that there is no more than a proportion β1 in one tail and no more than β2 in the other tail with probability γ may be computed for any size sample. They are computed from X - k1S and X - k2S, where X and S are the usual sample mean and standard deviation and k1 and k2 are constants previously tabulated in Odeh and Owen [3]. The question addressed is, “Just how accurate are the coverages of these intervals (– Infin;, X – k1S) and (X + k2S, ∞) for various size samples?” The question is answered in terms of how widely the coverage of each tail interval differs from the corresponding required content with a given confidence γ′.

The exact distributions of the estimated process capability indices are presented and their means, variances, and mean-squared errors are given. The basic assumption is that the process measurements are taken from a normal distribution. Theresults in this article are useful in evaluating process capability.

Simultaneous prediction intervals for observations inl future samples from a two-parameter exponential distribution are considered. The prediction limits depend upon a previously available complete or type II censored sample from the same distribution. An equation is derived from which the prediction factor is determined and exact prediction limits are illustrated with a numerical example.

B-expectation tolerance intervals for the double exponential distribution are considered. Monte Carlo B-expectation tolerance factors along with estimates of their asymptotic standard deviations are presented. Smoothing the Monte Carlo results using least squares regression is investigated.

The starship procedure for transformations to normality described by Owen (1988) is implemented using the Johnson (1949) System of transformations, the Slifker-Shapiro (1980) technique for choosing a transformation, and the Shapiro-Wilk (1965) test for normality. This procedure was applied to obtain maximum likelihood point estimates of a mean, to obtain confidence intervals on a mean, and to estimate percentiles of a distribution based on a sample. Simulations of three distributions show that the starship has many desirable properties, and can be compared very favorably with the bootstrap procedures of Efron (1987).

At least I units are required whose values of a performance variable Y fall below the upper specification limit U. They are obtained by screening units on a correlated variable X until m units are obtained whose values of X meet the threshold W. The numbers I and m are specified in advance of screening, and W needs to be determined. Two cases are discussed: one with all parameters known and one with all parameters unknown. The analysis in this paper amounts to the application of a binomial model in addition to the bivariate normal screening model previously discussed in Owen, Mclntire, and Seymour (1975) and in Owen and Su (1977). When the bivariate distribution parameters are unknown, finding W is similar to the prediction interval problem discussed in the literature. Some illustrative tables and examples are included.

Owen (1964) has developed tolerance limits based on the sample mean, x, and the sample standard deviation, s, which control percentages in each tail of the normal distribution. In this paper tolerance limits based on x and the range R or the mean range Rg,r are developed. Tables of factors for the tolerance limits are given.

Sampling Plans are considered which call for the acceptance of a lot of material if both – ks ≥ L and + ks ≤ U, where L and U are the lower and upper specification limits and where and s are the sample mean and standard deviation and k is an appropriate constant. Some errors that need to be corrected in previous work are pointed out in interpretation of the mathematical formulas before plans of the type described can be properly applied. Tables are given for the constant k under two different circumstances. Table II applies when the sum of the proportions defective in the two tails of a normal distribution is to be controlled. Table III applies when each tail is to be controlled below preassigned values, i.e., so that one can say with probability y that the proportion in the lower tail (below L) of accepted lots does not exceed a preassigned value and simultaneously the proportion in the upper tail (above U) does not exceed its preassigned value. The same constants, in Table III, apply when all the defect...

Data on 23 lots of various aircraft programs were gathered. Total engineering man-hours, and information on performance, weight, area, avionics systems, data, and schedule were subjected to least squares analysis. An equation is presented which indicates a relationship between total engineering manhours and a set of seven predictor variables. While the equation derived could only be used with confidence by the manufacturer whose data was analyzed, this article should be looked upon as demonstrating a method of data analysis which others may also find useful, not only for predicting engineering manhours in major aircraft programs, but also in other situations where there is an abundance of possible predictor variables, and the problem is to sort out a meaningful subset of these variables. In order to demonstrate the viability of the formula obtained, comparisons were made with various bid programs.

Birnbaum and McCarty give a distribution-free upper confidence bound on Pr{Y < X} when X and Y are independent and have continuous cumulative distribution functions. In this paper the restriction to a continuous distribution function is removed. The same problem is then considered where X and Y have a joint bivariate normal distribution function. The distribution of the estimator of Pr{Y < X} is non-central t. Upper confidence limits on Pr{Y < X} are examined for sample sizes n = 10(10)100 in two cases—where X and Y are independent and where observations are taken in pairs and X and Y may be dependent.

A special type of tolerance limits for a normal distribution is presented. The limits have the property that with probability y no more than a specified proporation p1 of a normal population is below the lower limit and at the same time no more than a s..

Both tolerance limits and sampling plans are discussed where percentages p/2 in each of the tails of the normal distribution are to be controlled. Tables of constants are given which have been accurately computed using the new techniques developed in the paper. These results are set up from the consumer's viewpoint, but can easily be extended to the producer's viewpoint. Programs on the CDC-1604 computer are available to extend the tables for any special applications not already covered.