146 SAMPLING FROM BINOMIAL POPULATIONS Ch. 5 



by insects. Assume the following data were obtained from random samples 

 of 250 of each species: 



Seriously Not 



Species Damaged Damaged Sum 



Yellow 58 192 250 



Short-leaved 80 170 250 



Spruce 78 172 250 



Do the insects studied attack one of these species more than another? Or is 

 the assumption that the percentage of seriously damaged trees is the same for 

 all these acceptable? Am. Chi-square = 11.47, 2D/F, P = .03. 



Reject the assumption. 

 5. For each species of pine studied in problem 4, test the hypothesis that 

 one-third of the trees in each species population are seriously damaged. Then 

 combine these tests by adding the chi-squares, and draw appropriate conclusions. 



5.6 CONTROL CHARTS 



Sampling techniques appropriate to binomial populations have 

 some important applications in industry in addition to those con- 

 sidered previously in this chapter. During a manufacturing process 

 designed to produce marketable goods it is important to check fre- 

 quently upon the quality of these products. Quality control charts 

 provide a simple but effective means for watching both the general 

 level of quality and the consistency with which this level is being 

 maintained. No attempt will be made herein to discuss all the 

 various methods in use because books devoted solely to industrial 

 statistics or to quality control are available on this subject. How- 

 ever, it can be seen rather easily that some of the topics presented 

 earlier in this book are fundamental to this subject. The subsequent 

 remarks in this section are intended to point out some of these funda- 

 mentals. 



Consider first a manufactured item which could be classified as 

 either defective or non-defective with respect to predetermined stand- 

 ards of production. Clearly, a binomial frequency distribution must 

 be involved with some unknown proportion, p, of defective products 

 being manufactured. The number of items inspected and classified 

 as defective or non-defective is the n in the previous discussions of 

 sampling from binomial populations. As indicated earlier the stand- 

 ard deviation of a proportion derived from a sample of n observations 

 is VpU — p) /n. If the manufacturing process is running smoothly 

 with p = .05, say, and then something occurs to increase the fraction 

 defective to .15, this occurrence will reveal itself in two ways: (a) the 



