5.4 The correlation matrix for a hypothetical data set is given in Table Q5.1. Table Q5.1 X1 Xz X3 X4 X L.OQO 0.690 2 0.280 0.350 L.oo0 0.255 0.195 1.O00 0.610 L.ooo The following estimated factor loadings were extracted by the principal axis factoring procedure: Variable F2 X Xa X; X 0.80 0.70 0.10 0.20 0.20 0.15 0.90 0.70 Compute and discuss the following: (a) Specific variances; (b) Communalities; (c) Propor- tion of variance explained by each factor; (d) Estimated Or reproduced correlation matrix; and (e) Residual matrix
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For example, the specific variance for X1 is 1 - 0.64 = 0.36. Show more…
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For the accompanying data set, (a) by hand, compute the correlation coefficient, and (b) determine whether there is a linear relation between x and y. Data set table: (x) 2, 4, 6, 6, 7 (y) 4, 8, 12, 14, 18 n 3 0.997 4 0.950 5 0.878 6 0.811 7 0.754 8 0.707 9 0.666 10 0.632 11 0.602 12 0.576 13 0.553 14 0.532 15 0.514 16 0.497 17 0.482 18 0.468 19 0.456 20 0.444 21 0.433 22 0.423 23 0.413 24 0.404 25 0.396 26 0.388 27 0.381 28 0.374 29 0.367 30 0.361 - By hand, compute the correlation coefficient. The correlation coefficient is r=___ - Determine whether there is a linear relation between x and y. Because the correlation coefficient is (negative or positive) and the absolute value of the correlation coefficient, __ is (greater or not greater) than the critical value for this data set, __ (no, a negative, a positive) linear relation exists between x and y.
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Question 16 The correlation matrix of the six variables, X1 to X6 for the crime related data in Table 8 of Question 15 is as follows: X1 X2 X3 X4 X5 X6 X1 1 -0.138 0.352 0.811 0.108 -0.281 X2 -0.138 1 -0.564 -0.509 0.405 0.773 X3 0.352 -0.564 1 0.574 -0.071 -0.798 X4 0.811 -0.509 0.574 1 -0.037 -0.686 X5 0.108 0.405 -0.071 -0.037 1 0.195 X6 -0.281 0.773 -0.798 -0.686 0.195 1 (a) Use this correlation matrix to identify the pair of variables the strongest correlation and the pair of variables with weakest correlation, and state the values of their correlation coefficient. (b) In contrast to the correlations, the sample covariance between X2 and X3 is larger in absolute value than the sample covariance between X2 and X4. Explain why this is the case.
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