228 LINEAR REGRESSION AND CORRELATION Ch. 7 



the reader again is invited to consult Kendall's book on this subject 

 if interested. 



PROBLEMS 



1. Solve problem 5 of section 7.4, using the Spearman coefficient, r g . 



2. Solve problem 6 of section 7.4, using the Spearman coefficient, r g . 



Ans. r s = —1. 



3. Compute r g for the data of problem 9, section 7.4, letting X = 1 for 1929, 

 X = 2 for 1930, etc., and setting Y = wage. 



4. Compute the rank-difference coefficient of linear correlation for the pairs 

 of observations in Table 7.22. Attn. r g = .68. 



5. Make up a problem for which r g = +1, also for r g = —1, and r g = + .5. 

 Then make up another set for each case different from each of the others. 



6. A sampling study in cereal chemistry gave the following product-moment 

 linear correlations: 



Sample 1: n\ = 44, r% = —0.93 



Sample 2: n% = 44, r 2 = -0.81. 



Test H (p 1 = p 2 ) and draw appropriate conclusions. 



Ans. y = 2.40; P^ .017; reject H . 



7. Referring to problem 6, how small could the sample size become and still 

 result in the rejection of H at the 5 per cent point if the r's stayed the same 

 size? 



8. If r x = —.93, as in problem 7, could r 2 = —.90 ever result in the rejection 

 of the H of problem 6 at the 5 per cent level for any sized sample? If so, 

 what size would ?i x and n 2 have to be if they were equal? Ans. n x = n 2 = 225. 



r 



9. It has been stated that each of the ratios (b — /3)/sb and / - 



V (1 — r-)/(n — 2) 



follows the ^-distribution with (n — 2) degrees of freedom under random sampling 

 with a given 7i. Show that these two quantities are algebraically identical if /3 

 = 0; and hence that testing H (0 = 0) is identical to testing H (p =0). 



10. Suppose that the following results were obtained from two samples (in- 

 volving different methods of some sort), each containing 20 observations: 



Method 1 : ri = .40, H (pi = 0) accepted. 



Method 2: r 2 = .60, H (p 2 = 0) rejected, P < .01. 



Yet Ho (pi = p 2 ) is accepted readily because P > .40. 



Explain how such results are not contradictory. Also, determine what sizes n 

 must have in order that each of these three hypotheses will be rejected at the 

 5 per cent point if the correlations stay as they are. 



Ans. First H , n x = 25; second H , n 2 = 11; third H , n x = n 2 = 109. 



You are given the following two-variable frequency distribution table as the 



basis for solving problems 11 to 15 below. These data are derived from records 



of heights and weights of 9-year-old Kansas girls in certain schools. These data 



