1.9. Consider the following eight pairs of measurements on two variables x and x2: x1 -6...

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1.9. Consider the following eight pairs of measurements on two variables x and x2: x1 -6 -3 -2 1 2 5 6 8 X2 -2 -3 1 -1 2 1 5 3 (a) Plot the data as a scatter diagram, and compute $11, $22, and 512- (b) Using (1-18), calculate the corresponding measurements on variables 1 and 2, as- suming that the original coordinate axes are rotated through an angle of 0 = 26 [given cos (26) = .899 and sin (26) = .438]. (c) Using the x and x2 measurements from (b), compute the sample variances 311 and $22. (d) Consider the new pair of measurements (x1, x2) = (4, -2). Transform these to measurements on 1 and 2 using (1-18), and calculate the distance d(O, P) of the new point P (x1, x2) from the origin 0 = (0, 0) using (1-17). Note: You will need 311 and 522 from (c). = (e) Calculate the distance from P = (4,-2) to the origin O = (0, 0) using (1-19) and the expressions for a11, a22, and a12 in footnote 2. Note: You will need $11, $22, and $12 from (a). Compare the distance calculated here with the distance calculated using the x and x values in (d). (Within rounding error, the numbers should be the same.) 1.9. Consider the following eight pairs of measurements on two variables x and x2: x1 -6 -3 -2 1 2 5 6 8 X2 -2 -3 1 -1 2 1 5 3 (a) Plot the data as a scatter diagram, and compute $11, $22, and 512- (b) Using (1-18), calculate the corresponding measurements on variables 1 and 2, as- suming that the original coordinate axes are rotated through an angle of 0 = 26 [given cos (26) = .899 and sin (26) = .438]. (c) Using the x and x2 measurements from (b), compute the sample variances 311 and $22. (d) Consider the new pair of measurements (x1, x2) = (4, -2). Transform these to measurements on 1 and 2 using (1-18), and calculate the distance d(O, P) of the new point P (x1, x2) from the origin 0 = (0, 0) using (1-17). Note: You will need 311 and 522 from (c). = (e) Calculate the distance from P = (4,-2) to the origin O = (0, 0) using (1-19) and the expressions for a11, a22, and a12 in footnote 2. Note: You will need $11, $22, and $12 from (a). Compare the distance calculated here with the distance calculated using the x and x values in (d). (Within rounding error, the numbers should be the same.)