96 



BINOMIAL AND NORMAL DISTRIBUTIONS 



Ch. 4 



distribution function. The mathematical procedures needed in this 

 process are beyond the level of this course ; but the reader can under- 

 stand that the r.c.f. curve of Figure 4.31 plays the same general role 

 in the analysis of normal data that the r.c.f- curves did in Chapter 2. 



1.00 



Figure 4.31. Relative cumulative frequency distribution for the standard nor- 

 mal frequency distribution described by formula 4.23. 



The following problems will illustrate the uses to which Figure 

 4.31 can be put. 



Problem 4.31. Determine the limits on X for the third quartile of a standard 

 normal population of measurements. 



The limits required are obviously the median and Q 3 , respectively. 

 If we read horizontally from .50 on the vertical scale over to the 

 normal r.c.f. curve and then downward to the horizontal scale, we 

 find that A = 0, as is to be expected. Doing likewise for .75 on the 

 vertical scale, we find that A = 0.68; therefore, the limits on the third 

 quartile are A = to A = 0.68. Since these limits apply to any 

 standard normal distribution, the limits of the third quartile for any 

 particular normal distribution in terms of a measurement, X, can be 

 obtained from the relation: A = (X — /*)/<r. 



Problem 4.32. What is the probability that a measurement chosen at random 

 from a normal population with /j. = 50 and a = 5 will be found to lie between 

 50 and 52? Between 48 and 50? Between 60 and 65? 



To reduce this specific normal distribution to the standard normal 

 distribution, substitute /a = 50 and a = 5 into A = {X — fi)/a so 



