102 BINOMIAL AND NORMAL DISTRIBUTIONS Ch. 4 



PROBLEMS 



1. Given a large number of college grades which follow a normal distribution 

 with n = 65 and <r = 10, what proportion of the grades would you expect to lie 

 in the interval from 50 to 70, inclusive? 



2. Referring to Figure 4.31, how probable is it that 3 random selections from 

 this population will each have \'s ^ 2? Am. P = .000027. 



3. What proportion of the measurements in a normal population would you 

 expect to lie beyond X = 1.1 if fi = 0.5 and a = 0.25? 



4. What proportion of the data described in problem 3 lies at least 0.15 unit 

 from the arithmetic mean if the numbers are in an array? Ans. .55. 



5. Certain frost data collected in the neighborhood of Manhattan, Kansas, 

 over a 69-year period indicates that the average date of the last killing frost 

 in the spring is April 24, with a standard deviation of 10 days. Assuming a 

 normal frequency distribution and assuming that the date of the last killing 

 frost cannot be predicted from a current year's weather, what is the probability 

 that the last killing frost next spring will come on or after May 1? 



4.5 STUDYING THE NORMALITY OF A FREQUENCY 

 DISTRIBUTION BY RECTIFYING THE r.c.f. CURVE 



A method was given in a preceding section for detecting gross non- 

 normality by calculating the proportions of a population lying within 

 such intervals as /x ± ka and comparing these with those proportions 

 which are typical of a perfectly normal population of measurements. 

 A graphic method will be presented in this section which will make 

 it quite easy to compare the whole of a population with a standard 

 normal population. The graphic method has these advantages: (1) 

 Like other graphs, it utilizes the eye-mindedness of many persons. 

 (2) It compares all the distribution with a standard normal instead 

 of comparing a few segments such as /x ± 0.5o-, fx ± la, etc. How- 

 ever, this graphic procedure has the disadvantage that it may encour- 

 age a hasty acceptance of the assumption that the given population 

 is sufficiently near normal for the purposes at hand. More rigorous 

 tests of normality exist in more mathematical textbooks, which 

 can be consulted if the situation demands that additional care. It 

 will be seen when the Central Limit Theorem is discussed in a later 

 chapter that a considerable amount of non-normality can be toler- 

 ated in sampling studies; hence a precise — and laborious — test for 

 normality is not often employed. In such situations a graphic test 

 may be sufficiently reliable. 



The process of rectifying a curve y = f(x), which is the basic 

 procedure of this section, is one of changing the scale of measure- 



