STA 302/1001 Summer 2001 Assignment 4 - Quiz Preparation

Sep 17, 2026

STA 302/1001 Summer 2001 Assignment 4 is a quiz-preparation worksheet covering core topics in the general linear regression model and multivariate normal distributions. It instructs students to review Chapter 6 (especially Section 6.9) and complete specified exercises. The problems request distributions under linear transformations, matrix-form regression model expressions, least-squares and maximum-likelihood estimators, derivations of Y, b, fitted values, and residuals, confidence intervals and hypothesis tests when error variance is known, and proofs of key identities and independence conditions. It also includes an applied SAS multiple-regression task using the SMSA dataset, emphasizing labeled outputs and interpretation of fitted results.

LanguageEnglish

CategoryExam

FormatPDF

Pages2

Reading Time5 min

AccessFree to download

Quiz on June 13th

Pre-reading and preparation notes

Distribution and matrix regression theory questions

Estimation, confidence intervals, and hypothesis testing (σ² known)

Proofs for regression identities and rank conditions

MLE completion and SAS application with SMSA data

Required SAS labeling and interpretation questions

STA 302/1001 Summer 2001 Assignment 4 Quiz on June 13th. Do this assignment in preparation for the quiz. Bring a calculator. Bring printouts of the log and list ﬁles to the quiz. Note that there will be no formula sheet on Quiz 4 . Some of the questions below are intended to provide information about what facts and formulas you need to store in your brain (there are not many) . Questions of this sort say “Just write down the answer.” Please begin by reading Chapter 6 . You may skip or skim Section 6 .8 on diagnostic and remedial measures for now. We’ll come back to this material later. Section 6 .9 is especially useful because it’s based upon the data used in our ﬁrst SAS lesson on multiple regression. Therefore, you can tell exactly what SAS is doing by comparing the output with Section 6 .9. Incidentally, doing a textbook example is always a good idea when you’re getting acquainted with a new piece of statistical software. 1. Do questions 6 . 1 and 6 .4, and exercises 6 . 22 and 6 .25. 2. Let X ∼ N (µ , Σ), and let c be a vector of constants. What is the distribution of X + c? Just write down the answer. 3. Let X ∼ N (µ, Σ), and let A be a matrix of constants. What is the distribution of AX? Just write down the answer. 4. Let X1 be Normal(µ1 ,σ 21), and X2 be Normal(µ2 ,σ22), independent of X1 . What is the joint distribution of Y1 = X1 + X2 and Y2 = X1 − X2 ? What is required for Y1 and Y2 to be independent? 5. State the general linear regression model in matrix terms. Just write down the answer. You are being asked for expression (6.19) in the text, except please say 􀀂 ∼ N (0,σ2 In) . 6. Give the formula for the least squares (and maximum likelihood) estimator b. You are being asked for expression (6.25) in the text. Just write down the answer. 7. Give the formulas for ˆY and e. Just write down the answers. 8. For the general linear regression model in matrix terms, derive the distributions of Y, b, ˆY and e. These derivations are very short. 9. Assuming that the errror variance σ 2 in the general linear regression model is known, (a) Derive a (1 − α) ∗ 100% conﬁdence interval for c􀀁 β, where c is a p × 1 vector of constants. (b) State the statistic that you would use for testing H0 : c􀀁β = h. Remember, you know the value of σ 2 , so you’re not using the t-distribution. 10. Prove (a) e􀀁 X = 0 (b) e􀀁ˆY = 0 (c) If the model has an intercept, the least squares plane goes through (x 1 ,..., xp−1 , Y ) . You may use 􀀁 ˆYi = 􀀁 Yi without proof. Hint: what is ~~1~~nX􀀁 1? 11. Prove that if the columns of X are linearly dependent, (X􀀁 X)−1 does not exist. 12. The following question guides you through the derivation of b for the general linear regression model. Much of it was done in lecture. (a) Here is a substitute for the “umbrella argument.” Show that for any σ 2 > 0, an estimator ˆβ minimizes Q = (Y − Xβ)􀀁 (Y − Xβ) over β if and only if it minimizes the likelihood function over ˆβ . Begin as follows. (Y − Xˆβ)􀀁 (Y − Xˆβ) ≤ (Y − Xβ)􀀁 (Y − Xβ) ⇔ ... (b) Show (Y − Xβ)􀀁 (Y − Xβ) = (Y − Xb)􀀁 (Y − Xb) + [X(b − β)]􀀁 [X(b − β)] . In showing the middle term zero, use e􀀁 X = 0, which you proved above. (c) How do you know the expression you just derived is minimized for β = b? (d) Why does the linear independence of the columns of X ensure that the minimum is unique? 13. In the last question, you showed that for every ﬁxed σ 2 > 0, b maximizes the likelihood function. Complete the derivation of the MLE by setting β = b in the likelihood function, and maximizing over σ 2 to obtain σˆ2. Don’t forget the second derivative test. 14. The SMSA data set (see p. 704 in your text for a description) is available on the data disk that came with your text. For your convenience, it is also on the course Website. On the data disk, it is called APC2 .DAT; on the Web, it’s smsa.dat. Please use SAS to ﬁt a multiple regression model in which the dependent variable is Total Serious Crimes, and th

FAQs

What preparation steps are required before the quiz?

Read Chapter 6, optionally skim Section 6.8, and focus on Section 6.9. Bring a calculator and printouts of the log and list files to the quiz. Note there is no formula sheet on Quiz 4.

How are the distribution questions 2–4 structured?

They ask for the distribution of transformed normal random vectors/matrices (X + c, AX), and the joint distribution of linear combinations Y1 = X1 + X2 and Y2 = X1 − X2, including the condition for Y1 and Y2 to be independent.

What must the SAS task using the SMSA dataset include?

Fit a multiple regression model with Total Serious Crimes as the dependent variable and Variables 2 through 10 as independent variables. Use SAS to produce simple descriptive statistics, ensure the SAS command file reads all data (including Geographic Region) and provides labels, and be ready to answer multiple interpretation and inference questions based on the output.

Sep 17, 2026

Uploaded by Sophia Brooks