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STAT3500/STAT7500 Assignment 3- Linear Regression
Due Date: 9th October 2023, 5 PM.
Weighting: 25%

●The assignment consists of 4 (four) problems, each problem is worth 25 marks, and  each mark is e qually weig hted.

●The mathematical elements of the assignment can be completed by hand, in LaTeX (preferably), or in Word (or other typesetting software). The mathematical derivations and mani pulations should be accompanied by clear expl anations in English regarding necessary information required to interpret the mathematical exposition.

●Computation problems should be answered using programs in the R language.

●Computer generated plots and hand drawn graphs should be included together with the text where problems are answered.

●Submission files should include the follow ing (which ever applies to you):

Scans of handwritten mathematical exposition.

Typeset mathematical exposition, outputted as a pdf file.

Typeset answers to computational problems, outputted as a pdf file.

Program code/ scripts that you wish to submit, outputted as a txt file.

●Mathematical problems should be answered with reference to results presented in the Course Notes (refer to page numbers), if required. If a mathematical result is used that is not presented in the Lecture Notes, then its common name (e.g., “Bayes' Theorem", "Intermediate Value Theorem”,“Borel-Canteli Lemma", etc.) should be cited, or else a reference to a text contai mi ng the result should be provided (preferably a textbook).

●All submission fles should be labeled with your name and student number and ar chived together in a zip fle and submitted at the TurnItIn link on Bla ckboard. We suggest naming using the convention:

[LastName_ FirstName/StudentNumber]_ STAT35 00A3_ [AnythingE1se] . [FileExtension] .

●As per https: //my . uq. edu. au/ inf or mat ion- and- ser vices/manage- my- progr am/ student- int egr ity- and- c onduct/ ac ad emic- int egri ty - and- stud ent- conduct, what you submit should be your own work. Even where working from sources, you should endeavor to write in your own words. You should use consistent notation throughout your assi gnment and define whatever is required.

Problem 1 [25 Marks]

and together； we call  (X (1) ,..., X (k) ) the dummg uaTiable TepTesentation of X.  Treating X  as a categorical variable；we can model the relationship between Y and X via the regression relationship

where x is a realization of X；and

for each k∈[K].

(a) using (1),provide expressions for the expected value of Y；given X = ①:  E[Y |X = ①]； for each x ∈{0, 1,..., K},in terms of β0  and βT = (β1 ,...,βK ).  using the expectation expressions E[Y |X = x；provide an interpretation of the parameters β0  and the  elements β1 ,β2 ,...,βk of β, in regards to the relationship between Y and X.[5 Marks]

The chi ckwts dataset from the package datasets in R prov ides data regarding an experiment of weight gain in chicken when given different feed type. Here we can consider weight, in gr ams, as the response Y and the feed type as the r andom categorical variable X∈{0,1...,k},k=5. We can assume that the data consists of real izations (x1, y1)... (xn, yn), where (xi, yi) is a realization of the random pair (X;,Y:), IID for eachi∈[n], where each (X;, Y:) has the same DGP as (X,Y),  forn=71.