The data on national track records for men are listed in Table 8.6. (Sec also the data on… 1 answer below »

The data on national track records for men are listed in Table 8.6. (Sec also the data on national track records for men on the website www.prenhall.com/statistics) Repeat the principal component analysis outlined in Exercise 8.18 for the men. Are the results consistent with those obtained from the women's data?

Refer to Exercise 8.18. Convert the national track records for women in Table 1.9 to speeds measured in meters per second. Notice that the records for 800 m, 1500 m, 3000 m, and the marathon are given in minutes. The marathon is 26.2 miles, or 42.195 meters, long. Perform a principal components analysis wing the covariance matrix S of the speed data. Compare the results with the results in Exercise 8.18. Do your interpretations of the components differ? if the nations are ranked on the basis of their score on the first principal component, does the subsequent ranking differ from that in Exercise 8.18? Which analysis do you prefer? Why?

Exercise 8.18

The data on national track records for women are listed in Table 1.9.

(a) Obtain the sample correlation matrix R for these data, and determine its eigenvalues and eigenvectors.

(b) Determine the first two principal components for the standardized variables. Prepare a table showing the correlations of the standardized variables with the nents, and the cumulative percentage of the total (standardized) sample explained by the two components.

(c) Interpret the two principal components obtained in Part b. (Note that the first component is essentially a normalized unit vector and might measure the athletic excellence of a given nation. The second component might measure the relative strength of a nation at the various running distances.)

(d) Rank the nations based on their score on the first principal component. Does this ranking correspond with your inituitive notion of athletic excellence for the various countries?

 

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