The sampling plan for the mean for correlated data is studied. The Operating Characteristic (OC) of the variable sampling plan for mean for correlated data are calculated and compared with the OC of known σ case

Khanday, M. Ahmad

Sankle, R.

Singh, J. R

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Journal of Modern Applied StatisticalMethodsVolume 12 | Issue 2 Article 1011-1-2013Variables Sampling Plan For Correlated DataJ. R. SinghVikram University, Ujjain, India, jrsinghstat@gmail.comR. SankleVikram University, Ujjain, India, rajshreesankle@gmail.comM. Ahmad KhandayVikram University, Ujjain, India, manzoorstat@gmail.comFollow this and additional works at: http://digitalcommons.wayne.edu/jmasmPart of the Applied Statistics Commons, Social and Behavioral Sciences Commons, and theStatistical Theory CommonsThis Regular Article is brought to you for free and open access by the Open Access Journals at DigitalCommons@WayneState. It has been accepted forinclusion in Journal of Modern Applied Statistical Methods by an authorized administrator of DigitalCommons@WayneState.Recommended CitationSingh, J. R.; Sankle, R.; and Khanday, M. Ahmad (2013) "Variables Sampling Plan For Correlated Data," Journal of Modern AppliedStatistical Methods: Vol. 12: Iss. 2, Article 10.Available at: http://digitalcommons.wayne.edu/jmasm/vol12/iss2/10Journal of Modern Applied Statistical Methods November 2013, Vol. 12, No. 2, 184-190. Copyright © 2013 JMASM, Inc. ISSN 1538 − 9472    The authors are in the School of Studies in Statistics. Email them, respectively, at jrsinghstat@gmail.com, rajshreesankle@gmail.com, and manzoorstat@gmail.com (corresponding author).  184 Variables Sampling Plan For Correlated Data J. R. Singh Vikram University Ujjain, India R. Sankle Vikram University Ujjain, India M. Ahmad Khanday Vikram University Ujjain, India   The sampling plan for the mean for correlated data is studied. The Operating Characteristic (OC) of the variable sampling plan for mean for correlated data are calculated and compared with the OC of known σ case.  Keywords: Variable sampling plan, correlation coefficient, operating characteristic function  Introduction Quality control methods are commonly used to determine acceptability of a product with regard to its usefulness at the time it is put into service. It is essential from the consumer point of view, however, that a product return its usefulness for a certain length of time. Acceptance sampling is the testing or the inspection of selected items from a given lot followed by acceptance or rejection of that lot on the basis of the results of the test and its indicator of the lot’s quality. It is assumed that a lot’s quality is determined by the proportion of defective items in the lot. Further, attention will be restricted to those types of defects that are determined by one sided specification limits. For the purpose of exactness, upper specification limits will usually be discussed, i.e. an item will said to be defective if its measured characteristics are greater than some specified value U. Variable plans, however, require that the characteristic of interest be continuous variable. The characteristic is measured and its actual value is recorded. In variable sampling plans an underlying process distribution form is assumed. Then the proportion defective in the lot can be estimated by estimating the parameters of that distribution. The variable model thus requires more restrictive assumptions on the manufacturing process. If these assumptions can be justified there would be a substantial saving in sample size corresponding to a given sampling list. SINGH, SANKLE & KHANDAY  185 When the standard deviation of the lot quality is known, the criteria for acceptance and the associated mathematical computations get simplified. But, we should examine in each case whether treating the lot standard deviation (σ) as known and giving it a particular value are justified. When products are manufactured by automatic machinery whose inherent variation is known and tested, an example is provided where the lot standard deviation is known. When it is assumed that the lot standard deviation is known, and given a particular value σ, it must be remembered that σ is a constant in calcu1ations and discussions. Also, the previous assumption that the lots are formed in such a way as to ensure homogeneity within lots holds good here also; and we assume that the directly measurable quality X follows the normal law of pattern of variation in the lot; these assumptions must be examined and reviewed from time to time when variables plans with known σ are in use. Hapuaxachi and Macephexsan (1992) studied the effect of serial correlation on acceptance sampling plans by variables by comparing Operating Characteristic (OC) curves, sample size and producers risk, α with that of the independent case when the process standard deviation (σ) is known. When σ is unknown and for large n, sampling plans can be constructed using central limit theorem. Several works have studied the effect of correlated data (see Kaiyang & Hancock, 1990; Seal, 1959; and Qiu et al., 2010). This study examines the sampling plan for mean for correlated data. The OC function of the variable sampling plan for mean for correlated data are calculated and compared with the OC function of known σ case. Model Description And OC Function For Correlated Data For a single sampling plan, with known-sigma, the procedure of selection of sample is as in the other single sampling plans. The n units in the sample are measured, and the values x1, x2, x3,..., xn are obtained. The mean x  is calculated. Since the standard deviation sigma (σ) of the lot is known, σ is used. Suppose that observations x1, x2, x3, ..., xn have a multivariate normal distribution with E(xi) = μ and Var(xi) = σ2 and ρ as the common correlation coefficient between any xi and xj, i ≠ j. Then   ( )E x µ=   (1)  and  VARIABLES SAMPLING PLAN FOR CORRELATED DATA 186  ( ) ( )2221 1Var x nnTnσ ρσ= + −  =  (2)  where   ( )2 1 1 .T n ρ= + −     (3)  In connection with a single sampling variable plan, when data are correlated, the following symbols will be used,  L = Lower specification limit, U = Upper specification limit, k = Acceptance parameter, x  = Sample mean of correlated data, ρ = Correlation Coefficient.   2212( )x zeF zx dπ−−∞= ∫   (4)  where z ~ N(0,1). The OC function of single sampling plan can now be calculated. The acceptance criterion for correlated data mean plan is, for upper specification limit U, accept the lot if   k Tx Unσ+ ≤   (5)  reject the lot otherwise. The values of n and k are determined for a given set of values of the producer risk, α and consumer risk, β, AQL and LTPD, by formulae   ( )( )1 22p pK KnK Kα β + =−    (6) SINGH, SANKLE & KHANDAY  187   2 1p pK K K KkK Kα βα β +=  +    (7)  If p is the proportion defective in the lot   ,pU Kµσ−=   (8)  where 2212pzKp e dzπ∞ −= ∫ .  The expression for probability of acceptance (OC) function of the plan in normal case is   ( ) ( ) pTL p Prob x MSE x U Knµ σ = + ≤ = +    (9)  where  T2 = [1 + (n – 1)ρ].  Following Schilling (1982) the OC function for correlated data works out to be as   ( ) ( )pnL p K kTΦ = −    (10)  where  ( ) ( )tt z dzφ−∞Φ = ∫    The usual single sampling plan for known σ is  VARIABLES SAMPLING PLAN FOR CORRELATED DATA 188  ( ) ( )pL p n K k = Φ −    (11)  Numerical Illustration and Result For illustration, consider an example of producers and consumers oriented single sampling plan p1 = 0.01, α = 0.05, p2 = 0.08, and β = 0.10. The values of n and k have been determined from equation (6) and (7) and are 10 and 18.09, respectively. The values of OC function for the above plan have been calculated for correlated data as well as for known standard deviation by using equation (10) and (11). These values of OC function for different values of correlation and usual known standard deviation case are presented in Table 1 and are plotted in Figure 1.   Table 1. Values of OC for different values of ρ  P ρ = 0 ρ = 0.2 ρ = 0.5 ρ = 0.6 ρ = 0.8 ρ = 1 0.005 0.9926 0.8944 0.8056 0.7867 0.7574 0.7354 0.010 0.9500 0.8006 0.7196 0.7042 0.6812 0.6645 0.020 0.7820 0.6553 0.6085 0.6003 0.5882 0.5797 0.030 0.5908 0.5469 0.5323 0.5298 0.5262 0.5236 0.040 0.4271 0.4625 0.4741 0.4761 0.4790 0.4811 0.050 0.3016 0.3949 0.4271 0.4327 0.4408 0.4466 0.060 0.2101 0.3396 0.3878 0.3963 0.4087 0.4176 0.080 0.1000 0.2555 0.3252 0.3380 0.3568 0.3704 0.100 0.0471 0.1953 0.2770 0.2925 0.3159 0.3328 0.120 0.0221 0.1510 0.2384 0.2558 0.2823 0.3017 0.140 0.0104 0.1177 0.2067 0.2253 0.2539 0.2752 0.160 0.0049 0.0923 0.1803 0.1995 0.2296 0.2522 0.180 0.0023 0.0728 0.1579 0.1774 0.2083 0.2319 0.200 0.0011 0.0576 0.1388 0.1582 0.1896 0.2139   SINGH, SANKLE & KHANDAY  189                     Figure 1. OC Curve for different values of Correlated Data ( ρ )   From Figure 1 it is evident that the effect of correlation data on OC increases as ρ increases. As ρ increases, a significant effect is seen in producers risk as well as consumers risk, which is not acceptable. Hence one should maintain the correlation between the observations as low as possible, so as to protect producer as well as consumer. References Hapuaxachchi, K. P., & Macpherson, B. D. (1992). Autoregressive process applied to acceptance sampling by variables. Communication in Statistics – Simulation and Computation, 21(3), 833-848. Qiu, P., Zou. C., & Wang, Z. (2010). Nonparametric profile monitoring by mixed effects modeling. Technometrics, 52(3), 265-277.  VARIABLES SAMPLING PLAN FOR CORRELATED DATA 190 Schilling, E. G. (1982). Acceptance Sampling in Quality Control. New York: Marcel Dekker, Inc. Seal, K. C. (1959). A single sampling plan for correlated variable with a single sided specification limit. Journal of American Statistical Association., 54(285), 248-259. Yang, K., & Hancock, W. M. (1990). Statistical quality control for correlated samples. International Journal of Production Research, 28(3), 595-608. 

Variables Sampling Plan For Correlated Data

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