114 SAMPLING FROM BINOMIAL POPULATIONS Ch. 5 



of this sort include the estimation of true average breaking strengths, 

 and comparisons of the strengths of different materials. 



(5.04) Studies involving two variables such as -prices of selected 

 stocks and the volume of production of finished steel, ACE score 

 and grade average in college, stand counts of wheat and the yield of 

 a plot, etc. In such investigations it would be necessary to estimate 

 from sampling data the relationship between the two variables, ex- 

 press it mathematically, and then use it in accordance with the pur- 

 poses of the investigation. 



It can be seen from the examples above that two general types of 

 statistical problems must be considered in sampling studies. One is 

 to derive from the sample observations some numbers which can 

 be used satisfactorily in place of one or more unknown population 

 parameters. These numbers which will be derived from the sample 

 are called sampling estimates of the parameters. They are change- 

 able from sample to sample and, being dependent upon chance events, 

 are subject to the laws of probability. 



The other general problem is to test hypotheses regarding pop- 

 ulations against actual sample evidence. For example, if the popula- 

 tions of the breaking strengths of two types (different shapes, for 

 example) of concrete columns each follow a normal frequency dis- 

 tribution with the same variance, a 2 , these populations can differ 

 only in their means pi, and /a 2 . That is, it is supposed that the en- 

 gineers in charge are satisfied that the two types of columns have 

 the same uniformity of performance from test to test, but it is yet 

 to be decided whether they have the same average strength. If so 

 (that is, if //.I = fio) , the populations of breaking strengths are iden- 

 tical normal populations. It then becomes a problem of deciding 

 from samples taken from each population whether or not ^ is in 

 fact equal to /xo- It usually is convenient statistically to assume 

 that m does equal jx 2 , and then to see how reasonable this hypothesis 

 is in the light of sample observations. 



It should be clear — intuitively, at least — that decisions based on 

 samples may be in error, and that we do not know in any particular 

 case if our sample is so unusual that it is misleading us. How, then, 

 can sample evidence become a satisfactory basis for making decisions 

 about populations? The answer lies in the fact that, while no one 

 can say whether a particular decision is right or wrong, it is possible 

 to determine the relative frequency with which correct decisions will 

 be made over the long-run of much experience if we are following 



