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Question:Suppose three tests are administered to a random sample of college students. Let \({X_1},................,{X_N}\) be observation vectors in \({\mathbb{R}^3}\) that list the three scores of each student, and for \(j = 1,2,3,\), let \({x_j}\) denote a student鈥檚 score on the \({j^{th}}\) exam. Suppose the covariance matrix of the data is

\(S = \left( {\begin{array}{*{20}{c}}5&2&0\\2&6&2\\0&2&7\end{array}} \right)\)

Let \(y\) be an 鈥渋ndex鈥� of student performance, with \(y = {c_1}{x_1} + {c_2}{x_2} + {c_3}{x_3}\), and\(c_1^2 + c_2^2 + c_3^2 = 1\),. Choose \({c_1},{c_2},{c_3}\) so that the variance of \(y\) over the data set is as large as possible. (Hint: The eigenvalues of the sample covariance matrix are \(\lambda = 3,6,{\rm{ and }}9\).)

Step by step solution

01

Mean Deviation form and Covariance Matrix

The Mean Deviation form of any \(p \times N\)is given by:

\(B = \left( {\begin{array}{*{20}{c}}{{{\hat X}_1}}&{{{\hat X}_2}}&{........}&{{{\hat X}_N}}\end{array}} \right)\)

Whose \(p \times p\)covariance matrix is:

\(S = \frac{1}{{N - 1}}B{B^T}\)

02

The Variance

From question, the matrix and the maximum eigenvalue we haveis:

\(\begin{array}{l}S = \left[ {\begin{array}{*{20}{c}}5&2&0\\2&6&2\\0&2&7\end{array}} \right]\\{\lambda _1} = 9\end{array}\)

The respectiveunit vector is:

The new variable will be:

Hence, this is the required answer.

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