A METHOD OF SAMPLING INSPECTION 629 



r m=c 



1= n-h (N -n)\ 1 - J: C:(1 - pY~^p^' 

 L »»=o 



For the condition, ^ < .10, which is usual in practice, it has been 

 found satisfactory to replace the Point Binomial by the Poisson 

 Exponential.^ By multiplying both sides of the equation by pt we 

 obtain z in the form, 



which is the function /i desired. 



To obtain /o, we state the probability of finding c or less defects in 

 a sample n taken from a lot N containing M = ptN defective pieces. 

 This is given by the equation, 



p — v r^~M rM 



■'- /-"Af ^ n—m ^m • 



But this equation is too difficult to handle in general computations 

 on a large scale. When pt < .10 and n is sufficiently large, a satis- 

 factory approximation to the above equation may be developed from 

 the first c + 1 terms of the Point Binomial, that is 



An even better approximation is obtained by interchanging ^ n 

 and M in the latter equation giving the expression, 



m=c / „ \ M-m / „ 



CI 7t 



Since — = -^ , we obtain the final form, 



m=c I \M-m / a \m 



which is the function Jo desired. 



Now that we have/i and/2 as expressed in equations (1') and (2') 

 we must explain how Fig. 2 was obtained. When P = .10, for any 

 pair {M, k), a particular pair (c, a) was found which made z a minimum. 

 The acceptance number c may assume only discrete values since any 



* G. A. Campbell, "Probability Curves Showing Poisson's Exponential Sum- 

 mation," Bell System Technical Journal, Vol. II, pp. 95-113, January, 1923. 



* Paul P. Coggins, "Some General Results of Elementary Sampling Theory for 

 Engineering Use," Bell System Technical Journal, Vol. VII, p. 44, Equation (11), 

 January, 1928. 



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