Question 1
Consider the data given in the EXCEL file tab “q1”.
(a) State the multivariable linear regression model with all the necessary assumptions.
(b) Find the predicted model.
(c) Test the significance of the model.
(d) Regarless of your result in part (c), test if X1 is significant? How about X2?
(e) Find a 95% woking Hotelling confidence region for the mean response when X1 = 192 and X2 = 152.
(f) Find a 95% woking Hotelling prediction region for a new response when X1 = 192 and X2 = 152.
Question 2
Timm (1975) reported the results of an experiment in which subjects’ respond time to “probe words” at five position (Y1 is at the beginning of the sentence, Y2 is in the first quartile of the sentence, Y3 is in the middle of the sentence, Y4 is in the third quartile of the sentence, and Y5 is at end of the sentence). The data are recorded in the EXCEL file tab “q2”.
(a) Use the sample variance and obtain all the principle components.
(b) Timm specifically required the reduction in dimension should cover at least 90% of the total variance. How many principle components are needed? Why?
(c) Repeat parts (a) and (b) using the sample correlation matrix.
Question 3
Use the data in the EXCEL file tab “q2”.
(a) Find the canonical correlation between (x1, x2, x3) and (x4, x5).
(b) Test the significance canonical correlations.
(c) Regardless of your answer in part (b), is each canonical correlation individually significant?
Question 4
(a) Show that
≠12( ̃x ≠ μ ̃1)Õ ≠1( ̃x ≠ μ ̃1) + 12( ̃x ≠ μ ̃2)Õ ≠1( ̃x ≠ μ ̃2) = (μ ̃1 ≠ μ ̃2)Õ ≠1 ̃x ≠ 12(μ ̃1 ≠ μ ̃2)Õ (μ ̃1 + μ ̃2)
b) Let
f1(x)=(1≠|x|) f2(x) = (1 ≠ |x ≠ 0.5|)
for |x|Æ1
for ≠ 0.5 Æ x Æ 1
Question 5
Use the data in the EXCEL file tab “q5”. Assume the data is a sample from two multivariate normal distributions.
(a) Identify the classification rule for the case p1 = p2 and c(1|2) = c(2|1).
(b) Identify the classification rule for the case p1 = 0.25 and c(1|2) is half of c(2|1).
(c) Identify the Bayesian rule using p1 = 0.6.
(d) Based on the rules given in part (a), which population will a new data point (50, 48, 47, 49) be classified
into? How about using the rules in part (b)? Using the rule in part (c)?
