1. The data on national track records for women is saved on speed.txt. Here you have 8 columns: 1: column 1, the time (in second) for 100m 2: column 2, the time (in second) for 200m 3: column 3, the time (in second) for 400m T4: column 4, the time (in minute) for 800m 15: column 5, the time (in minute) for 1500m 16: column 6, the time (in minute) for 3000m 7: column 7, the time (in minute) for Marathon The last column is for the country who belongs to. (a) Find the sample correlation matrix for this data (1 to 27), and determine its eigenvalues and eigenvectors. (b) Conduct a principal component analysis for the standardized variables. Select the right components (you will need to determine how many components to be selected). (c) Prepare a table to show the correlations of the standardized variables with the selected components, and the cumulative percentage of the total sample variance explained by the these components. (d) Interpret those components. (e) Convert the data into same measurement (seconds). Notice that the records for 800m, 1500m, 3000m and the marathon are given in minutes. Perform a principal component analysis using the covariance matrix S. Compare the new results with the results in (b). Do your interpretations of the component differ? Which analysis do you prefer? Why?