Sec. 7.4 COEFFICIENTS OF LINEAR CORRELATION 221 



a large number of sample correlations, the z-curve would become 

 approximately normal in shape, as can be imagined from Figure 7.41. 



Problem 7.41. Finney and Barmore (Cereal Chemistry, Vol. 25 [1948], page 

 299) have reported that the linear correlation between the per cent of protein 

 in Nebred wheat flour and the loaf volume of bread baked therefrom was 

 r = .94 on a sample of 30 pairs of measurements. What useful information does 

 this provide? 



The mere fact that r 2 = (.94) 2 = .8836 tells us that 88.36 per cent 

 of the original sum of squares of the loaf volumes (Y) about their 

 mean, y, can be associated with the linear increase of that measure- 

 ment with increasing protein concentration in the flour (X). Loaf 

 volume is an important factor when the quality of bread is judged, and 

 it is important to know what affects it. 



It is inconceivable that such a large correlation coefficient would be 

 obtained accidentally on thirty random observations; but, to illustrate 

 the method, the hypothesis H Q (p = 0) will be tested. It is seen that 



.94 .94 



t = i = = 14.5, 28 D/F. 



'1 - .8836 0.065 



28 



Such a large t would occur by chance almost never; hence the hy- 

 pothesis H (p — 0) is decisively rejected. We know without even 

 seeing the scatter diagram that the sample points lie closely about 

 a linear regression line which has an upward trend. It also is ap- 

 parent that the loaf volume from Nebred flour meeting the condi- 

 tions of this experiment could be predicted quite accurately from a 

 knowledge of its protein concentration. 



There are some circumstances under which it is desirable to deter- 

 mine if two random samples probably were drawn from the same 

 bivariate population as regards one, or both, of (3 and p. For example, 

 it might be of interest to learn if one method of raising turkeys pro- 

 duces a more consistent relationship between the 16-week and the 

 28-week weights so that we could cull at 16 weeks of age with more 

 confidence. Such an improvement in the relationship between these 

 variables would indicate that the true coefficient of linear correla- 

 tion, p, had been increased by the new methods. It also might be 

 that superior poultry husbandry could increase the amount by which 

 a weight advantage at 16 weeks of age would be followed by a weight 

 advantage at 28 weeks of age. In the population considered earlier 

 in this chapter, a one-pound advantage in weight at 16 weeks of aj 



*&/f) 



