Sec. 4.4 APPROXIMATE BINOMIAL PROBABILITIES 101 



12. Suppose that 52 per cent of the voters in a certain city are in favor of 

 a particular one of the possible sites for a new high school. If 100 voters are 

 to be selected at random, what is the probability that less than 50 per cent 

 will vote in favor of this site? If the poll is so taken that 60 per cent of those 

 who favor that site will not participate in the poll, what now is the probability 

 that less than 25 per cent of a sample of 100 will vote for the site in question 

 which 52 per cent of the voters actually favor? Aiis. .31, .12. 



4.4 USE OF THE NORMAL DISTRIBUTION TO 



APPROXIMATE PROBABILITIES FOR A 



BINOMIAL FREQUENCY 



DISTRIBUTION 



Another important use to which the normal r.c.f. distribution can 

 be put has been suggested previously, namely, the approximation of 

 the summation of C n , r p r q n ~ r from r — a to r = b, when n is at all 

 large and p is close to 1/2. It has been indicated that this sum is 

 approximately equal to that area under the normal curve between 

 the points X x = a — 1/2 and X 2 = b + 1/2. Moreover, it has been 

 shown that the area under the normal curve between any two points 

 along the X-axis can be obtained quite easily from an r.c.f. curve 

 or from Table III. 



To illustrate this process and to indicate its accuracy, suppose 

 n = 20 and p = q = 1/2, and that it is required to determine 

 P(r^ 12). For this binomial distribution, \i = np = 10 and o- = 

 yjnpq — y5; hence the normal distribution with these parameters 

 will be employed in the approximation. Also, Xi = 11.5, and X 2 = 

 20.5. In terms of standard normal units, 



Xj = (11.5 - 10)/2.24 = +0.67, and 

 X 2 = (20.5 - 10)/2.24 = +4.69. 



By means of Table III and some interpolation it is found that ap- 

 proximately 25 per cent of a standard normal population has num- 

 bers between these A-limits; hence P(r^ 12) = .25, approximately. 

 Using the last column of Table VII from r = 12 on down, and 

 using a divisor of 2 10 = 1,048,576, the exact probability — to 4 deci- 

 mals — that r will have some size from 12 to 20, inclusive, is found 

 to be .2517. Certainly the normal approximation of .25 is excellent 

 for most purposes, and the labor saved is considerable. 



