Problem 1: A naturalist for the Alaska Fish and Game Department studies grizzly bears with the 3 goal of maintaining a healthy population. Measurements on \(n=61\) bears provided the following summary statistics:

1. Perform a principal component analysis using the covariance matrix. Can the data be effectively summarized in fewer than six dimensions?
2. Perform a principal component analysis using the correlation matrix.
3. Comment on the similarities and differences between the two analyses.

Problem 2 : Consider the data in Table 8.7 on Mali Family Farm data of problem 8.28.

1. Construct two-dimensional scatterplots of Family versus DistRd, and DistRd versus Cattle. Remove any obvious outliers from the data set.
2. Perform a principal component analysis using the correlation matrix R. Determine the number of components to effectively summarize the variability. Use the proportion of variation explained and a scree plot to aid in your determination. Interpret all nine principal components.

Problem 3 : Consider the data in Table 11.8 on Hemophilia data.

1. Evaluate whether each of the two groups are univariate normal in each of the two classification variables. Include the appropriate plots and a written paragraph describing the conclusions you draw from the plots.
2. Evaluate whether each of the two groups are bivariate normal in the two classification variables. Include the appropriate plots and a written paragraph describing the conclusions you draw from the plots.
3. Assume equal prior probabilities, equal costs of misclassification and that the covariance matrix for both groups are the same. Use Matlab or any Software to do a linear discrimination based on the estimated ECM rule for two normal populations (box on page 586) and calculate the apparent error rate (APER). Turn in your calculations and a paragraph describing your conclusions.
4. Assume equal prior probabilities, equal costs of misclassification and that the covariance matrix for both groups are not the same. Use Matlab or any Software to do a linear discrimination based on the quadratic classification rule for two normal populations with unequal covariance matrices (box on page 594) and calculate the apparent error rate (APER). Turn in your calculations and a paragraph describing your conclusions
5. Use SPSS to perform Fisher’s linear discriminant on these data. Calculate the APER and use the hold-one-out procedure in SPSS to calculate the est. AER. Turn in the important parts of the SPSS output and a paragraph describing your conclusions.
6. Write two to three paragraphs describing the similarities and difference in what you did in parts (c), (d), and (e).

Problem 4: Do problem 11.34 using Matlab or any Software for calculations.

Table 11.9 on pages 666-667 contains data on breakfast cereals produced by three different American manufacturers: General Mills (G), Kellogg (K), and Quaker $(Q)$. Assuming multivariate normal data with a common covariance matrix, equal costs, and equal priors, classify the cereal brands according to manufacturer. Compute the estimated \(E(\mathrm{AER})\) using the holdout procedure. Interpret the coefficients of the discriminant functions. Does it appear as if some manufacturers are associated with more "nutritional" cereals (high protein, low fat, high fiber, low sugar, and so forth) than others? Plot the cereals in the two-dimensional discriminant space, using different plotting symbols to identify the three manufacturers.

12.7. Sample correlations for five stocks were given in Example 8.5. These correlations, rounded to two decimal places, are reproduced as follows:

Treating the sample correlations as similarity measures, cluster the stocks using the single linkage and complete linkage hierarchical procedures. Draw the dendrograms and compare the results.

Problem 6 : Use cluster analysis on the data set in Table 11.9.

1. Find the best solution of three clusters (or 4 if 3 is not an option).
2. Find the dendogram for 2 to 10 clusters and turn it in.
3. Write a paragraph or two to describe the results of the cluster analysis and whether it was effective in terms of the pre-existing groups and which variables were the most important for the clustering.

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