226 LINEAR REGRESSION AND CORRELATION Ch. 7 



12. Solve problem 1, and then test H (p = 0) and draw appropriate conclu- 

 sions. Ans. r — .999, y ^ 10, reject H decisively. 



13. For the data of problem 6 compute the CI 9 -, on /3, and then interpret this 

 interval in a practical way. 



14. For the data of problem 6 compute the CI 90 on p and interpret this in- 

 terval. Ans. -m^p^ - .80. 



15. Suppose that two random samples of 15 observations each have resulted 

 in the computation of i\ = .75 and r 2 = .65. Test H (p 1 = p 2 ) and draw appro- 

 priate conclusions. Also compute the CI 95 for each parameter. p x and p. 2 , and 

 interpret these intervals. Can these interpretations be related to the test of 

 tf ? 



16. Draw a random sample of 30 observations from Table 7.21, compute the 

 CI 90 on p, and discuss the meaning of this interval. 



17. Draw a random sample of 30 from Table 7.21 and test the hypothesis: 

 H (p — 0). How frequentfy would this procedure result in the rejection of i/ 

 when p9^ (as in this case) at the 5 per cent level of rejection? 



18. Draw two random samples of size 30 from Table 7.21 and test H^p^ = p 2 ), 

 assuming that the first sample is from a bivariate normal population with p = p v 

 and similarly for the second sample and p = p 2 . 



7.5 RANK CORRELATION 



Sometimes it is either necessary or convenient to correlate the 

 ranks of X's with those of their corresponding Y's. It may be that 

 the .Y's and the Y's are only ranks in the first place, or it may be 

 merely convenient to use ranks instead of four- or five-digit decimals, 

 for example. 



The practice of correlating ranks is both older and broader in its 

 applications than is sometimes realized. Karl Pearson apparently 

 was of the opinion that the idea of correlating ranks originated with 

 Francis Galton during his studies of inheritance. Sometimes C. 

 Spearman is credited with doing much to develop rank-correlation 

 methods, especially as applied in psychological studies. It is his 

 coefficient, r s , which will be discussed specifically below. The works 

 of M. G. Kendall, and others, recently have increased the use of 

 ranks in statistics to a considerable degree, but no attempt will be 

 made herein to give an exhaustive treatment of this subject. The 

 interested reader is referred to Kendall's book, Rank Correlation 

 Methods, published by Charles Griffin and Company, London. 



The calculation of the Spearman, or rank-difference, coefficient of 

 linear correlation (r s ) will be illustrated by means of the following 

 pairs of ranks of students in two mathematics courses. Each pair 

 gives the respective ranks of that student in statistics (X) and in 

 mathematics of finance (Y). For example, the first student listed 



