Question: Consider the following eight pairs of measurements on two variables x1 and x2:

1. Plot the data as a scatter diagram, and compute \(s_{11}, s_{22}\), and \(s_{12}\).
2. Using (1-18), calculate the corresponding measurements on variables \(\tilde{x}_{1}\) and \({{\tilde{x}}_{2}}\), assuming that the original coordinate axes are rotated through an angle of \(\theta=26^{\circ}\) [given \(\cos \left(26^{\circ}\right)=.899\) and \(\left.\sin \left(26^{\circ}\right)=.438\right]\)
3. Using the \(\tilde{x}_{1}\) and \(\tilde{x}_{2}\) measurements from (b), compute the sample variances \({{\tilde{s}}_{11}}\) and \({{\tilde{s}}_{22}}\).
4. Consider the new pair of measurements \(\left(x_{1}, x_{2}\right)=(4,-2)\). Transform these to measurements on \(\tilde{x}_{1}\) and \(\tilde{x}_{2}\) using \((1-18)\), and calculate the distance \(d(O, P)\) of the new point \(P=\left(\tilde{x}_{1}, \tilde{x}_{2}\right)\) from the origin \(O=(0,0)\) using \((1-17)\). Note: You will need \({{\tilde{s}}_{11}}\) and \(\tilde{s}_{22}\) from (c).
5. Calculate the distance from \(P=(4,-2)\) to the origin \(O=(0,0)\) using \((1-19)\) and the expressions for \(a_{11}, a_{22}\), and \(a_{12}\) in footnote 2 . Note: You will need \(s_{11}, s_{22}\), and \(s_{12}\) from (a). Compare the distance calculated here with the distance calculated using the \(\tilde{x}_{1}\) and \(\tilde{x}_{2}\) values in (d). (Within rounding error, the numbers should be the same.)

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