8.2. Convert the covariance matrix in Exercise $8.1$ to a correlation matrix \(\boldsymbol{\rho}\).

1. Determine the principal components \(Y_{1}\) and \(Y_{2}\) from \(\boldsymbol{\rho}\) and compute the proportion of total population variance explained by \(Y_{1}\).
   \(\mathbf{\Sigma}=\left[\begin{array}{ll}
   5 & 2 \\
   2 & 2
   \end{array}\right]\)
2. Compare the components calculated in Part a with those obtained in Exercise 8.1. Are they the same? Should they be?
3. Compute the correlations \(\rho_{Y_{1}, z_{1}}, \rho_{Y_{1}, Z_{2}}\), and \(\rho_{Y_{2}, z_{1}}\).

8.7. Convert the covariance matrix \(\mathbf{S}\) in Exercise 8.6 to a sample correlation matrix \(\mathbf{R}\).

1. Find the sample principal components \(\hat{y}_{1}, \hat{y}_{2}\) and their variances.
2. Compute the proportion of the total sample variance explained by \(\hat{y}_{1}\).
3. Compute the correlation coefficients \(r_{\hat{y}_{1}, z k}, k=1,2\). Interpret \(\hat{y}_{1}\).
4. Compare the components obtained in Part a with those obtained in Exercise 8.6(a). Given the original data displayed in Exercise 1.4, do you feel that it is better to determine principal components from the sample covariance matrix or sample correlation matrix? Explain.

\(\overline{\mathbf{x}}=\left[\begin{array}{r}

155.60 \\

14.70

\end{array}\right], \quad \mathbf{S}=\left[\begin{array}{rr}

7476.45 & 303.62 \\

303.62 & 26.19

\end{array}\right]\)

8.8. Use the results in Example 8.5.

1. Compute the correlations \(r_{y_{i}, z_{k}}\) for \(i=1,2\) and \(k=1,2, \ldots, 5\). Do these correlations reinforce the interpretations given to the first two components? Explain.
2. Test the hypothesis

\(H_{0}: \boldsymbol{\rho}=\boldsymbol{\rho}_{0}=\left[\begin{array}{lllll}

1 & \rho & \rho & \rho & \rho \\

\rho & 1 & \rho & \rho & \rho \\

\rho & \rho & 1 & \rho & \rho \\

\rho & \rho & \rho & 1 & \rho \\

\rho & \rho & \rho^{\prime} & \rho & 1

\end{array}\right]\) versus

\(H_{1}: \boldsymbol{\rho} \neq \boldsymbol{\rho}_{0}\)

at the \(5 \%\) level of significance. List any assumptions required in carrying out this test.

The following exercises require the use of a computer.

8.10. The weekly rates of return for five stocks listed on the New York Stock Exchange are given in Table 8.4. (See the stock-price data on the following website: www.prenhall.com/statistics)

1. Construct the sample covariance matrix \(\mathbf{S}\), and find the sample principal components in (8-20). (Note that the sample mean vector \(\overline{\mathbf{x}}\) is displayed in Example 8.5.)
2. Determine the proportion of the total sample variance explained by the first three principal components. Interpret these components.
3. Construct Bonferroni simultaneous \(90 \%\) confidence intervals for the variances \(\lambda_{1}, \lambda_{2}\), and \(\lambda_{3}\) of the first three population components \(Y_{1}, Y_{2}\), and \(Y_{3}\).
4. Given the results in Parts a-c, do you feel that the stock rates-of-return data can be summarized in fewer than five dimensions? Explain.

8.12. Consider the air-pollution data listed in Table 1.5. Your job is to summarize these data in fewer than \(p=7\) dimensions if possible. Conduct a principal component analysis of the data using both the covariance matrix \(\mathbf{S}\) and the correlation matrix \(\mathbf{R}\). What have you learned? Does it make any difference which matrix is chosen for analysis? Can the data be summarized in three or fewer dimensions? Can you interpret the principal components?

8.26. Consider the psychological profile data in Table 4.6. Using the five variables, Indep, Supp, Benev, Conform and Leader, performs a principal component analysis using the covariance matrix \(\mathbf{S}\) and the correlation matrix \(\mathbf{R}\). Your analysis should include the following:

1. Determine the appropriate number of components to effectively summarize the variability. Construct a scree plot to aid in your determination.
2. Interpret the sample principal components.
3. Using the values for the first two principal components, plot the data in a two dimensional space with \(\hat{y}_{1}\) along the vertical axis and \(\hat{y}_{2}\) along the horizontal axis Can you distinguish groups representing the two socioeconomic levels and/or the two genders? Are there any outliers?
4. Construct a \(95 \%\) confidence interval for \(\lambda_{1}\), the variance of the first population principal component from the covariance matrix.

Price: $33.62

Solution: The downloadable solution consists of 22 pages, 1162 words and 7 charts.
Deliverable: Word Document