536 



APPENDIX 



2. Distribution of the Means Expected 

 for Groups of Statistics 



The arithmetic sample mean or average of 

 a group of statistics comprising a sample is 

 denoted by X (read "X bar") and is the 



average obtained by adding all the values 

 of X and dividing by X. In more sym- 

 bolic term-, 



X - N -A 



Given a population described by mean n 

 and standard deviation j x , one can predict 

 something about the range of X's to be 

 expected from drawing a great many 

 samples of size N from this population. If 

 many samples are drawn, it will be found 

 that the X values fall into a distribution 

 which has a theoretical mean equal to m 

 and which will be normally distributed 

 with a standard deviation <fc. This 



on = 



n'Vn 



Since s x is smaller than cr x by a factor of 

 \/N, it permits greater discrimination in re- 

 gard to error than does a x . When N ^ 10, 

 X will be quite nearly normally distributed, 

 if the distribution of X values does not 

 differ too widely from that expected of 

 measurements drawn from a normal curve. 

 Accordingly, the distribution of X, as 

 measured by o^, is usually also known by 

 means of calculation whenever m and a x 

 are known. 



In scientific papers the standard error, s- 

 or a x , is often given in the form X ± s x or 

 X ± s x , and refers to the reliability of the 

 mean of a sample. On the other hand, s x 

 or a x are standard deviations; and when 

 given as X ± s, or X ± tr x , they refer to 

 the variability of a single observation X. 



3. Testing hypotheses regarding m 



a. The r test. Suppose one finds for 

 N = 100 that X = 68.03. One may wish 

 iicM to tesl the null hypothesis, at the 5% 



level of significance, that j x = 3 and 



m = 67.15. In the present case 



05 = \Tou = °- 3; T = 



x - 



68.03 - 67.15 0.88 



0.3 



0.3 



which is > 1 .96. Consequently, one rejects 

 the hypothesis. 



b. The t test. Frequently one may have 

 to test some hypothetical m when <r x and a x 

 are unknown. In this situation, one utilizes 

 the best available estimate of o x ; this useful 

 approximation is the standard deviation of 

 the sample, s x . The value for s x can be 

 determined from the following: 



;(x - xy 



n - 1 



Note that s^ = x j— 



-4. 



With s x substituted for a x , the expression 

 X _ M u X - /i 



becomes 



4- 



= t. 



N 



(When a x is used, the final value is r; when 

 s x is substituted, the final value is called t 

 by convention.) If the value of t is too 

 large, the hypothesis regarding n will be 

 rejected. The decision to accept or reject 

 the null hypothesis depends upon the num- 

 ber of degrees of freedom, which equals 

 N — 1 if one is estimating j x from a single 

 sample. Figure A-3 gives the probabilities 

 for various degrees of freedom that t differs 

 from zero in either direction by a value 

 equal to or greater than that observed. If 

 X = 68.03, the hypothesized m = 67.15, 

 s x = 3.24, and N = 9, then t = 0.81. With 

 8 degrees of freedom, p >0.05. The hy- 

 pothesized m is accepted. 



c. Confidence intervals for /z- Suppose 

 one chooses to work at the 5% level of 

 significance. If a x is known, the 95% con- 

 fidence interval for /x = X ± 1.96 c x , or 

 H = X ± 1.96 a x . If only s x is known, 



