STATISTICS 4M03/6M03: Test #1

29 October 2002

 

         1.      Consider the data matrix .

[11]           (a) Calculate the mean vector, the matrix of deviations, the sample covariance matrix (unbiased), the sample correlation matrix, the generalized sample variance, the generalized sample variance of the standardized variables, and the total sample variance.

[4]             (b) Assuming trivariate normality, find the conditional mean and variance of X₁, given X₂ = 10 and X₃ = 40.

 

[12]  2.      (a) Consider a bivariate distribution defined by

                                   

                  Find the mean vector and the covariance matrix. Find and graph the marginal and conditional distributions. [Note: If you recognize a distribution for which you already know the mean and variance, you do not have to show the integrals.]

[8]             (b) Consider a bivariate normal distribution with the same mean vector and the same covariance matrix as the distribution in (a). Draw the 95% ellipse of concentration.

 

[10]  3.      The attached scatterplot matrix and stars plot are for 7 national track records (100 m, 200 m, 400 m, 800 m, 1500 m, 3000 m, marathon) for women from 55 different countries. The stars begin on the right and wind counter-clockwise around the circle. Explain what the scatterplot matrix and the stars plot reveal about the data. Which plot is the more useful? How could the plots be improved?