Elementary Biometrical Inferences 



535 



each of N = 20 were obtained, the exact 

 probabilities of obtaining different numbers 

 of successes would be expressed in the bi- 

 nomial distribution plotted in histogram 

 form in Figure A-7, where each class of 

 success is represented by a column whose 

 height is proportional to the frequency of 

 the class. Note that there are only 21 

 ways to score the outcome of a set of 20 of 

 these observations (from to 20 successes), 

 so we are dealing with 20 discrete variables. 

 The smooth curve shown is the normal 

 curve, which has the same mean and stand- 

 ard deviation as the histogram. The larger 

 the sample size, if p = %, the larger will 

 be the number of outcomes possible per 

 sample, and the closer the plot of the prob- 

 ability of successes will approach the nor- 

 mal curve. Therefore, as N increases 

 without bound, the number of possible 

 outcomes increases to provide us with an 

 example of a continuous variable, whose 

 values are said to be distributed normally. 



a x give absolute r values of 1.96 or greater 

 we reject the null hypothesis when X gives 

 a t value that equals or exceeds 1.96. 



This equation can be rearranged X = 

 M + t<j x . This expression means that any 

 given statistic is equal to the population 

 mean plus a distance off this mean as 

 measured by ra x , where r is the number 

 of Oj's that X is away from the population 

 mean. ([jJ is called the population vari- 

 ance.) 



Suppose one is concerned with height of 

 corn measured to the nearest inch; hy- 

 pothesize that m = 50 inches and a x = 4 

 inches. This information is completely 

 sufficient to describe the properties of 

 a normally distributed population. One 

 plant is 40 inches tall, its height being con- 

 sidered a quantitative trait. Calculation 

 of the value for t yields 



40 - 50 10 



B. Statistics Expected from a Normal 

 Curve 



1. Distribution of Individual Statistics 



If one obtains a very large number of 

 statistics having a normal curve as a pa- 

 rameter, they will be distributed in a curve 

 resembling the normal curve. The prob- 

 ability that any given statistic, X, is de- 

 rived from the hypothetical population 

 with mean m and standard deviation j x , 

 can be determined from the value r calcu- 

 lated from the following: 



r = • 



Ox 



which is > 1.96. Since p <0.05 one rejects 

 the null hypothesis and may conclude that 

 the plant measured cannot, at the 5% level 

 of significance, come from a theoretical 

 population where m = 50 and j x = 4. 



Number of Successes 



When the absolute value of t is 1.96, this 

 probability for X is 0.05. Since exactly 

 5% of the X values in a distribution 

 characterized by the hypothesized ju and 



figure A-7. Histogram of probabilities for 

 different numbers of successes for a binomial 

 distribution (N = 20, p = W), and a normal 

 curve with the same /x and a as the histogram. 



