148 SAMPLING FROM BINOMIAL POPULATIONS Ch. 5 



Table 5.61 and Figure 5.61 were obtained in the following manner: 

 (a) p was fixed at .05 by combining 950 green and 50 red beads in 

 a receptacle, and considering the red beads as defective manufactured 

 items; (6) 25 samples with n = 50 were drawn and the fraction de- 

 fective was plotted over the order number of the sample; (c) start- 

 ing with the twenty-sixth sample and continuing through the seventy- 

 fifth, two green beads in the receptable were replaced by two red 

 beads after each successive sample was drawn and recorded; and 

 (d) starting with the seventy-sixth sample, no additional changes 

 were made. In brief, p = .05 for the first 25 samples; p increased 



.30 



UCL .30 



I i i i I i i i i i i i rri i i i i r 



?>.25 - 



.20 | — o o o 



UCL = .143 * °° ° 



.15 



.10 



.05 







X X X x X x ooooo <> 



x XX xxxxxoOO 



•• • XXXXx XXX o O- 



' • X X X X XXX o 



_••••••• • • X XXX X x 



• • ••• • *x * X X XX XX O 



-L.J I I x l ,. I x I I I I 



10 20 30 40 50 60 70 80 90 100 



Sample number 



Figure 5.61. Control chart for Table 5.61. 



.002 per sample (to simulate slippage or excessive wear) for the next 

 50 samples so that it finally was .15 for the last 25 samples. 



The horizontal lines in Figure 5.61 marked UCL (upper control 

 limit) were obtained from p ± 3\/p(l — p)/50 with p = .05 until 

 the seventy-sixth sample, and p = .15 thereafter. Unless something 

 has occurred unknowingly to change the size of p the fraction defec- 

 tive rarely will go above the UCL; hence, when the observed fraction 

 defective frequently exceeds this limit, it is suspected that the manu- 

 facturing process has broken down to some degree. It can be seen in 

 Figure 5.61 that when the "machine" had "slipped" and p began to 

 increase, the fraction defective soon started an upward trend. 

 Shortly, it exceeded the UCL which had been set on the supposition 

 that p = .05. Then when p ceased to increase and a new UCL was 

 figured with p = .15, the fraction defective again stayed below the 

 UCL. Generally, there also is a lower control limit (LCL) , but in 

 this situation it would have been negative and was taken as zero, as 

 is customary. 



In practice when the percentage of defectives is unknown the frac- 

 tion defective observed on at least 25 samples is used in place of p 



