ISSN -58 (Paer) ISSN 5-5 (Online) Vol., No.9, SINGLE SAMPLING PLAN FOR VARIABLES UNDER MEASUREMENT ERROR FOR NON-NORMAL DISTRIBUTION Dr. ketki kulkarni Jayee University of Engineering and Technology Guna (M.P.) ketki@rediffmail.com Abstract In this aer, the single samling lan for variables under measurement error for non-normal distributions reresented by the first four terms of an Edgeworth series is studied for known case. The effect of using the normal theory samling lan in a non-normal situation is studied by obtaining the distorted errors of the first and second kind. As one will be interested in having a suitable samling lan under measurement error for non-normal variables, the values of n and k are determined. Keywords: Non-Normal distribution, Edgeworth series, Measurement error, Single Samling Plan. Introduction The most imortant area for studying the effect of error due to measurement and misclassification is the samling insection lan. In general, imbedded within the design of accetance samling lans is the assumtion that insection rocedure are free from error and normal distribution. In the area of samling insection lan by variables very little attention has aid so for to investigate the effect of measurement error. Walsh (96) studied the effect of measurement error on the characteristic function of a single samling insection by variables with one-sided secification limit. The assumtion made by them to investigate the effect of measurement error are (i) the error is unbiased and indeendent of the actual value of the characteristics measured (ii) both the true value of the characteristic measured and the measurement error follows normal distributions (iii) the ratio of the oulation variance to the error variance has a ositive uer bound. Singh (966) studied the effect of measurement on the oerating characteristic function of a single samling by variables for normal as well non-normal roduct distributions assuming one-sided secification limit, viz, the uer secification limit. His assumtions were almost same as those of Walsh (96). Bennett et al. (97) investigated the effect of Tye-I and Tye-II insection errors on the economic design of a single samling lan by attributes. Case et al. (975) develoed the formula for evaluating the average outgoing quality function when attributes examination is subject to Tye-I and Tye-II insection errors. The following hyerbolic remarks of Geary (97b) strike out: "Normality is a myth, there never was, and never will be normal distribution". Now the question arises whether such otimistic assumtion of normality is likely to be seriously misleading under non-normal situations. In other words, to what extent the standard statistical rocedures develoed for normal universes are alicable, when samles, in real sense, come from other than normal oulations. A statistical rocedure which is insensitive to deartures from the assumtions, which underlie it is called "Robust" an at term introduced by Box (95) and now widely used. An early survey of the literature on non-normality and robustness was given by Box and Anderson (955). In recent years, distribution-free or non-arametric methods have become quite oular because they are readily comutable and ermit freedom from worry about the classical assumtions of the standard normal theory. It should, however, be ointed out that in cases where classical assumtions hold entirely or even aroximately, the analogous standard normal rocedures are generally more efficient for detecting deartures from the null hyothesis..samling Distribution of Observed Mean Assuming that the true measurement x and the random error of measurement e are additive, we can write the observed measurement as : 5
ISSN -58 (Paer) ISSN 5-5 (Online) Vol., No.9, = x + e, (.) where x and e are indeendent. where, We now assume the density function of x to be secified by the first four terms of the Edgeworth series as follows: xµ () xµ () xµ (6) xµ f ( x) = + +, (.) 6 7 d ( t) = ex ( t ) r π dt r r [ t / ] and = (t). Then the density function of can be written from equation (.) as f () = µ ρ 6 (6) µ, 7 () µ +ρ 6 + ρ (.) which is again an Edgeworth series. () µ The distribution of for the observed samles of size n drawn from the oulation (.) is found by following Gayen (95) as, f () = n µ ρ / n 6 n () µ +ρ / n n () µ / n (6) µ, 7n / n 6 + ρ (.) where r= =. e ρ (ρ ) We now examine the effect of the measurement error on the usual test criterion of single samling lan described below: Accet the lot if x+ k U, And reject otherwise, for a given set of values of the roducer's risk, consumer's risk, Accetable Quality Level (AQL) and Lot Tolerance Proortion Defective (LTPD), the values of n and k are determined by the formulae 6
ISSN -58 (Paer) ISSN 5-5 (Online) Vol., No.9, where (K K ) n (K K ) α + β = (KαK k= (K α + K β, + K K β ) ) K, K K and K are determined by the equation (.5) (.6) kθ ex π t dt=θ, (.7) for different choices of fraction defective θ. If θ is the roortion defective in the lot, we know that U µ = K θ. The OC function L corresonding to a fraction defective is found out as follows. Under the assumtion of normality, a lot having ercent defective items will be acceted, if x + k U =µ+ K (.8) where K is given by equation (.8) for θ =. The exression for robability of accetance under measurement error L = Pr ob. [x+ k µ+ K ] k is derived by recalling the normality of the statistic ( x+ ). The above robability after some simlification works out to be L = Φ[ n ρ (K k)], (.9) where Φ(t) = t π ex x dx.. Known Plan Under Measurement Error for Non-Normal Situation Let the true quality characteristic x follows the first four term of an Edgeworth series under measurement error. The OC function of the lan is given by L' = Pr ob [ x+ k U = µ = µ '], where ' = U K', µ 7
ISSN -58 (Paer) ISSN 5-5 (Online) Vol., No.9, K' being the uer ercent oint of the standardized Edgeworthian oulation. i.e., K' is given by K ' () () (6) ( x) ( x) + ( x) + ( x) dx=. 6 7 (.) µ (µ) = ρ / n where x=, x, (x) e. = Using the distribution of x from equation (.), the OC function of the known lan is obtained as L' π () () 6 (5) = Φ( z) ρ ( z) + ρ ( z) + ρ ( ). 6 n n 7n z (.) where z = n ρ K ' k ). and ( The equation for determining the value of the lan arameters n and k are x L '( ) = α, (.) L ( ) = β. (.) ' Exlicit exression for n and k cannot be obtained under measurement error and non-normal situations. Equation (.) and (.) can however, be solved numerically. If n o and k o are the initial solutions than the imroved solution can be obtained as (n +δ n ) and (k +δ k ) where δ n and δk are the solution of linear equations A ( ) δ n B( ) δk = α c( ) (.5) and A( ) δ n B( ) δk =β c( ), (.6) where A() ρ = n [z Φ(z ) ρ (z (z ) (z )) + () () 6 n n () () 6 (6) (5) z (z ) (z )) + ρ (z (z ) (z )] (.7) 7n ( () () () = n ρ Φ(z ) ρ (z ) + ρ (z ) 6 n n B ρ 8
ISSN -58 (Paer) ISSN 5-5 (Online) Vol., No.9, (6) + ρ (z ) 7n () () C() = Φ(z ) ρ (z ) + ρ (z ) 6 n n (.8) 6 (5) + ρ (z ) (.9) 7n and z = n ρ K ' ). ( k The required value of n and k can be obtained by taking the normal theory values as the initial solution and reeating the rocess of interaction for equation (.5) and (.6) till the desired accuracy is obtained..discussion of Numerical Results and Conclusions For the urose of illustrating the effect of measurement error and non- normality on the error of the first and second kind and the lan arameters n and k, we have determined the values of these quantities for few chosen sets of values of,, α β,, and for different error sizes. The values of n and k are determined from the equations (.5) and (.6). The actual error of the first and second kind are given by α = L'( ) and β' L' ( ). ' = The actual errors of the first kind when normal theory known lan is used under measurement error and non-normal situations are determined and resented in the Table (). As is evident from the Table, for letokurtic, latykurtic oulation under different error sizes the marked difference is seen in α' and β'. The effect of skewness, kurtosis and measurement error are serious. For few non-normal situations and measurement error the value of the lan arameters n and k are given in Table (). The values of n are rounded u and the values of k are given u to decimal laces which is correct u to the third lace of decimal. As can be seen from the Table the negative skewness and negative kurtosis increases n where as the ositive skewness and ositive kurtosis decreases the value of n. The value of k is also considerably affected by measurement error and the non-normality of the oulation. It may inferred that the use of normal theory samling lan is not valid for the non-normal situations and measurement error roblems. Even when there is slight dearture from normality and different error sizes it is advisable to take into account the measurement error and the non-normality of the arent oulation while choosing the samling lan arameters n and k. References ENNETT, G.K., CASE, K.E. and SCHMIDT, J.W. (97). The economic effects of insection error on attribute samling lans noval research logistics quarterly. Vol., No., -. BO, G.E.P. (95). Non-Normality and tests of variances. Biometrika,, 8-5. 9
ISSN -58 (Paer) ISSN 5-5 (Online) Vol., No.9, BO, G.E.P. and ANDERSON, S.L. (955). Permutation theory in the derivation of Robustness criteria and the study of deartures from assumtions, JRSS, B, 7, -. CASE, K.E., BENNETT, G.K. and SCHMIDT, J.W. (975). The effect of insection error on average outgoing quality journal of quality technology, vol. 7, No. -. GAYEN, A.K. (95). The Inverse Hyerbolic sine Transformation on Student's - t for non-normal, samles, Sankhya,, 5-8. GEARY R.C. (97b). Testing for normality, biomatrika,,. 9-. SINGH, H.R. (966). Test and Measurement Error in statistical quality control, unublished, Ph.D. Thesis, IIT, Kharagur. WALSH, J.E. (96). Loss in test efficiency due to disclassification into Tables. Biometrika, 9, 58.
ISSN -58 (Paer) ISSN 5-5 (Online) Vol., No.9, Table Value of α' and β' (Underline) for known lan α =.5, β =., =.5, =. / r = r = r = 6 r =.5.5.5.5..695.9.57.9.56.7.969.58....867..8..8 -.5.659.7.57.7.86.5.7.8.69.7.8.98..9..9.5.695.98.57.9.56.7.85.58..9..85..87.97.8.5.778.58.6.9.585..56..87.7.78.59.668.558.6.58.5.85.9.67.86.66.67.6.9.67.576.5..9.5.59.9
ISSN -58 (Paer) ISSN 5-5 (Online) Vol., No.9, Table Values of n (integer) and k for known lan α =.5, β=., =.5, =. / r = r = r = 6 r =.5.5.5.5. 7 6 8 7 7 8 7 6.9.96...98.979.5.97 -.5 7 6 9 8 8 7 8 7.68.999.786.5.5..9..5 7 6 8 7 7 6 7 6.98.956.55..6.975.6.967.5 5 5 8 5 5 5.97.98.9759.96.97.89.969.87.5 5.8.77.9.865.887.76.865.79