State the multivariable linear regression model with all the necessary assumptions.

MATH 4630 / 6632 3.0 – Fall 2021 Assignment 3 (Due Date: December 2, 2021)
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 = 195 and X2 = 150.
f. Find a 95% woking Hotelling prediction region for a new response when X1 = 195 and X2 = 150.
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.
d. Based on your reduction in dimension, do you think the principle components obtained from sample variance and the principle components obtained from sample correlation are different? Why?
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Question 3: Elston and Grizzle (1962) measured the ramus bone of 20 boys at four different ages and the data are recorded in the EXCEL file tab “q3”.
a. Extract loadings by the principal component method using two factors.
b. Do a varimax rotation for your answer in part (a).
c. Extract loadings by the mle method using two factors.
Question 4: • Find the eigenvalues and eigenvectors R = 1 r r 1 !
where r is the sample correlation of X and Y . • Let the sample variance matrix be S = 1 4 4 25 !

1. Obtain the sample correlation matrix, R. 2. Compare the principal components obtained from S and R. Are they the same? 3. What is the advantage of the principal components obtained using R.
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   Question 5: Use the data in the EXCEL file.
   a. Find the canonical correlation between (y1,y2) and (x1,x2).
   b. Find the standardized coefficients for the canonical variates.
   c. Test the significance of each canonical correlation.