206 STATISTICAL TREATMENTS 



Confidence Limits: In a statistical test of significance we select some 

 arbitrary probability limits. If a deviation as large as that observed occurs 

 by chance in only 5 per cent of the cases, it may be safe to conclude that 

 the deviation is the result of the experimental treatment. In another 

 experiment we might choose the 1 per cent level (probability = 0.01} 

 of significance. The level of statistical significance chosen tells the confi- 

 dence with which we can draw conclusions. 



We can never say absolutely "yes" or "no," but only "probably" or 

 "very probably." The actual choice of confidence limits depends upon 

 the seriousness of drawing the wrong conclusion. It is possible to accept 

 a false hypothesis or to reject a true hypothesis on the basis of chance vari- 

 ations alone. The rejection of a true hypothesis is called an error of the 

 first kind, and the acceptance of a false hypothesis is called an error of 

 the second kind. Depending upon the experiment, one error is more 

 serious than the other. 



As an example suppose we test the hypothesis, "Drug A is harmless 

 to human beings." The drug will be released for public use only after 

 statistical tests. If we select the P = 0.05 level of significance, we are 95 

 per cent certain that the drug is harmless. This is not sufficient confidence 

 because the 5 per cent chance that the drug is actually harmful is too 

 great a risk. 



Student's t Test: When the variance of the population is unknown 

 and cannot be reasonably estimated, as is true in experiments involving 

 small samples, the normal deviate test of significance cannot be used. An 

 English statistician who signed his name "Student" derived a statistical 

 test useful in such cases. A value of t is computed in a manner similar to 

 the computation of c: 



X — tn X — m 



cr/\/n s/\/n~ 



Here s is the standard deviation of the sample of which x is the 

 mean. If the sample is large, s is very nearly equal to cr. If the sample is 

 small, s is likely to be somewhat smaller than o". The probabilities of t 

 values arrived at in this way are thus slightly different from the distribu- 

 tion of c values. Tables of probability with which various values of t 

 occur are used in this test of significance. The actual probability of a 

 certain value of t depends upon the size of the sample; the test is said 

 to have a certain number of degrees of freedom, equal to n — 1, where 

 n is the sample size. 



To test whether a certain sample belongs to a population, we calculate 



