Sec. 5.5 x 2 WITH OVER ONE DEGREE OF FREEDOM 145 



the total number of Republicans in the sample. A similar statement 

 holds for the Democrats, but this is not an independent requirement 

 because the six expected numbers have been forced to total 401 by- 

 making the column totals add to the observed numbers 320, 70, and 

 11. Hence the number of chance differences between the observed 

 numbers and those expected mathematically upon the basis of H is 

 reduced to 2. This, then, is the number of degrees of freedom. 



In general, the number of degrees of freedom for a chi-square cal- 

 culated for an r X c contingency table is (r — 1) (c — 1). In the 

 example above, r = 2 and c = 3; hence, (r — 1) (c — 1) =2. 



Having decided that the x 2 of 2.35 has 2 D/F, it remains to deter- 

 mine from Table V if this is an unusual size for a sample chi-square. 

 Table V shows that P( x 2 ^ 2.35, 2 D/F) = .31, approximately; there- 

 fore it is entirely reasonable that this x 2 occurred while sampling 

 from a population for which the hypothesis, H , is true. With this 

 sampling result at hand, we would accept the proposed hypothesis. 



PROBLEMS 



1. Suppose that of 300 salmon which went up a fish ladder in a certain river 

 185 were chinooks, 65 were silver salmon, and 50 were humpbacks. At another 

 ladder farther south suppose that the following numbers were recorded: 

 chinooks, 150; silvers, 80; and humpbacks, 20. Do these samples (if satisfac- 

 torily random, and such is assumed) support the belief that the proportions of 

 these three species are the same at the two locations which were sampled? 



2. Referring to problem 1, what matters would cause you to consider them 

 as truly random samples? What factors might cause you to think they were 

 not? 



3. Suppose that three independent samplings at one fish ladder led to the 

 following records: 



Number Which Were 

 Not 

 Sampling Chinook Chinook Sum 



Sum 182 73 255 



Is the hypothesis that the percentage of chinooks stayed the same during the 

 time of the sampling an acceptable one according to these data? 



4. Suppose that some entomologists investigated yellow, short-leaved, and 

 spruce pines in a certain forest to see how many were being seriously attacked 



