7. One hundred students each took five exams, each of which scored on a percenta

Question

7. One hundred students each took five exams, each of which scored on a percentage was (0-100). The exams were: Physics (C), Biology (C), Geometry (o), Algebra (o), and Statistics (o) The letter in parentheses indicates whether the exam format was closed book or open book. PCA gave the coefficients and principal component variances listed in Display 17.20. (a) Which, if any, of the principal components measures a difference in performance on open-book versus closed-book exams? (b) Which, if any, of the principal components measures overall performance on all exams? (c) What percentage of the variability of all five exam scores is explained by the first two principal components?

Explanation / Answer

(a) We see that the correlation between principal component number 2 and the performance on Closed book is positive. And the correlation between principal component number 2 and the performance on open book is negative. So principal component number 2 measures the variability of performance on open and closed book factor. So principal component number 2, measures a difference in performance on open-book versus closed-book exams.

(b) We see that the correlation between principal component number 1 and performance on all exams is positive. So principal component number 1 measures the highest variability of performance on all exams. So principal component number 1, measures overall performance on all exams.

(c) Total variance = 679.2 + 199.8 + 102.6 + 83.7 + 31.8 = 1097.1

Total variance of principal component number 1 and 2 = 679.2 + 199.8 = 879

So, percentage of variability of all five exam scores explained by first two principal components = 879*100/1097.1

= 80.12 %

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