182 SAMPLING NORMAL POPULATIONS Ch. 6 



from a mountain stream and are measured for length. The rainbows averaged 

 9.2 inches, with s = 2 inches; the brook trout averaged 8.7 inches, with standard 

 deviation = 2.1 inches. Test the hypothesis // (a' 1 = ^ an( ^ draw appropriate 

 conclusions. 



9. If from a certain study, x\ = 32.7 and x% = 35.9, and the pooled estimate of 

 a is s = 7.5. Both samples contained 12 observations. Test Hq \ m — ^ \ = 1. 



10. Suppose that 15 samples of each of two varieties of tomatoes have been 

 analyzed for vitamin C, with these results: 



Variety 1 Variety 2 



xi = 28.5 x 2 = 30.4 



2(z! 2 ) = 50 2(x 2 2 ) = 60 



Test the hypothesis that the true average ascorbic acid concentration in these 

 two varieties is the same. 



6.6 USE OF THE SAMPLE RANGE INSTEAD OF THE 



STANDARD DEVIATION IN CERTAIN TESTS OF 



STATISTICAL HYPOTHESES 



The most difficult computational part of the t-test is the determina- 

 tion of either s^ or sj, as the case may be. Another method of testing 

 hypotheses can be used in some situations without the need to com- 

 pute these standard deviations at all. It uses the sample range as its 

 measure of variation. The loss of precision is not serious for small 

 samples, becomes greater as the size of the sample is increased, and 

 renders the method useless for large samples. The trouble is that the 

 sampling variability of the range is almost as low as that of the stand- 

 ard deviation for small samples but increases quite rapidly with n. 

 The ratio 



(6.61) G = (x - n)/R, 



where R = sample range can be used in a manner analogous to the 

 f-test procedure. When G has been calculated, Table IX gives the 

 probability that such a sampling | G | will occur by chance for samples 

 of size n if the hypothesis regarding fx is exactly right. Thereafter the 

 reasoning is just as it was in section 6.4. 



When two random samples, each of size n, have been drawn from 

 what is assumed to be the same normal population, the ratio 



x\ - x 2 



(6.62) G = 



mean range 



where mean range = arithmetic mean of the ranges of the two samples 

 can be used on problems like those in section 6.5. Table X now is used 



