ORBS7250 Applied Multivariate Analysis
Assignment 2
Due date: February 5, 2024
1. Let X be N3 (μ , Σ) with μ' = [-3, 1, 4] and
Which of the following random variables are independent? Explain.
(a) X1 and X2.
(b) X2 and X3.
(c) (X1 , X2 ) and X3.
(d) 
and X3.
(e) X2 and X2 - 
X1 - X3 .
2. Let X be N3 (μ , Σ) with μ' = [2, -3, 1] and
(a) Find the distribution of 3X1 - 2X2 + X3 .
(b) Relabel the variables if necessary, and ind a 2 x 1 vector a such that X2 and 
are independent.
(c) Find the conditional distribution of X3 , given that X1 = x1 and X2 = x2 .
3. A manufacturer of sports equipment has developed a new synthetic ishing line that he claims has a mean breaking strength of 8 kilograms. If a random sample of 20 lines is tested and found to have a sample mean breaking strength of 7.8 kilograms with a sample variance of 0.25. By using hypothesis testing, does this suggest at a 0:01 level of signiicant that the mean breaking strength is not 8 kilograms? Assume the population of the breaking strength to be normal.
4. A wildlife ecologist measured x1 = taillength(inmillimeters) and x2 = winglength(inmillimeters) for a sample of n = 45 female hook-billed kites. The sample mean and covariance ma-trix are
(FYI: The original data set is in Table 5.12 of the text book.)
(a) Suppose it is known that μ1 = 190 mm and μ2 = 275 mm for male hook-billed kites. Are these plausible values for the mean tail length and mean wing length for the female birds? Find and sketch the 95% conidence region (ellipse) for the population means μ1 and μ2 .
(b) Construct the simultaneous 95% T2-intervals for μ1 and μ2 .
(c) Construct the 95% Bonferroni intervals for μ1 and μ2. Compare with the result in (b) and what have you found?
5. Energy consumption in 2001, by state, from the major sources
x1 = petroleum x2 = naturalgas
x3 = hydroelectricpower x4 = nucliearelectricpower
is recorded in quadrillions (1015 ) of BTUs (Source: Statistical Abstract of the United States 2006 ). The sample size is n = 50.
The resulting mean and covariance matrix are
(a) Obtain the large sample 95% Bonferroni conidence intervals for the mean con- sumption of each of the four types of engergy.
(b) Obtain the large sample 95% simultaneous T2 intervals for the mean consumption of each of the four types of energy.