Sec. 4.3 FRACTION OF Z'S WITHIN GIVEN LIMITS 99 



resemble those of Figures 4.23 and 4.31 rather closely if the former 

 were smoothed curves instead of broken-line graphs. Hence, it 

 appears, superficially, that the population of ACE scores follows a 

 normal distribution fairly well if the more general and important 

 features are the only ones considered. To be more definite, consider 

 the following information: 



(a) For the ACE scores, ^ = 96, approximately, and the median 

 is 97. These averages are equal in a normal distribution but the 

 discrepancy is not at all large. 



(6) The following table shows the corresponding proportions 

 within stated, and important, intervals on X: 



Percentage of the Population Included 



Interval on X ACE Normal Difference 



M±0.5(T 37.6 38.3 -0.7 



M±1.0<r 67.1 68.3 -1.2 



/xdb 1.5o- 85.3 86.6 -1.3 



M±2.0cr 95.2 95.4 -0.2 



H±2.5<r 99.2 98.8 +0.4 



M±3.0cr 99.8 99.7 +0.1 



Although the deviations from normal expectancy are somewhat 

 systematic, there being a small deficiency in the middle and a smaller 

 excess in the tails of the distribution, the ACE distribution still 

 seems to be approximated by the normal quite well. 



If, then, it is assumed that the ACE scores do essentially conform 

 to a normal distribution, the substitution A = {X — 96) /26 would 

 convert the scores of Table 2.01 into standard normal measurements. 

 The graph of their frequency distribution essentially would be Fig- 

 ure 4.23, the r.c.f. curve would be given approximately by Figure 

 4.31, and Table III would present the distribution in tabular form. 

 The statistical analysis of these data then might be more easily and 

 efficiently accomplished than would otherwise be the case, and little 

 or no important information would be lost in the process. 



PROBLEMS 



1. If a binomial frequency distribution has p = 1/4 and n = 80, calculate 

 P(r > 25) by means of the normal approximation to this binomial distribution. 



2. Suppose that all the residents of a certain city definitely have made up 

 their minds about a particular civic issue, and that 55 per cent favor one specific 

 decision. What is the probability that on a random sample of 100 interviews 



