STOR 664 FALL 2022 Midterm Exam October 6 2022 Open book in class exam time limit 75 minutes This is a single multi part question but each part will be graded independently of the other parts You are allowed to consult course notes printed or e read homework assignments and any personal notes you have made during the course Other outside materials are not permitted Computers or ipads may be used only for the purpose of accessing pre stored course notes they are not to be used for computations during the exam A hand held calculator is permitted Answers should preferably be written in a university examination book blue book You may consult the teaching assistant in class or the instructor email text or phone if the wording is unclear or if you think there might be an error but the teaching assistant or instructor will not give hints how to solve the exam The university Honor Code is in e ect at all times Consider the linear regression model yi 0 x2 i 1 xi 2 x3 i 3 cid 15 i i 1 n Sk cid 80 n where the cid 15 i are independent normally distributed random variables with mean 0 and a common unknown variance 2 Please note the order of covariates x2 i in that order De ning i for any k 0 we assume that the xi s are symmetric about 0 to i Tk cid 80 n i xi and x3 i 1 yixk i 1 xk guarantee that Sk 0 for all odd values of k a Find explicit expressions for the least squares estimators 0 3 and their variances You should express the answer in terms of the values of Sk and Tk for k 0 or any expressions derived from them 25 points b We would like to test the hypotheses H0 2 3 0 versus the alternative H1 that at least one of 2 or 3 is not 0 How would the estimates in a and their variances change under the assumption that H0 is true 10 points c Now suppose we are setting up the formal F test of H0 against H1 Write SSE0 and SSE1 for the residual sum of squares under H0 and H1 respectively Show that SSE0 SSE1 A 2 2 B 2 3 C 2 3 where A B and C are constants that you should identify functions of n S2 S4 etc Hence write down the formal test of H0 against H1 and de ne the rejection region for a test of signi cance level 0 01 You are not expected to make an explicit numerical calculation but describe how to calculate it for example you may use R notation to de ne the needed percentage point of the F distribution which will be one component of your answer 25 points Turn the page for the last two parts of the question d Now consider the case where xi goes from 3 to 3 in steps of 0 5 so n 13 You can assume no need to check this S2 45 5 S4 284 375 S6 2099 094 the last to three decimal places Also assume 2 40 2 1 3 0 5 What in that case will be the power of the test in part c 20 points You should give detailed numerical calculations as far as you are able to go but the nal answer will depend on the non central F distribution and you should give the formula for calculating that as an R function or any equivalent notation that makes clear how to do the numerical calculation If you cannot do explicit numerical calculations at least state the formulas on which they may be based e Now suppose that the real purpose of the experiment is to determine a 95 prediction interval for a new observation taken at a new value x x Show i how to calculate such a prediction interval under the assumption H1 ii how the calculations in i would change if the experimenter did indeed assume H0 to be true 20 points Since it s not possible for you to give a numerical answer here you should describe precisely the seqeence of steps including any formulas for percentage points of relevant probability distributions For ii you do not need to repeat the full calculation of i but indicate at which steps the calculation would change 1 x2 1 x1 x3 1 1 x2 2 x2 x3 2 1 x2 1 x1 x3 1 a X form so X T X 1 Solutions and hence X T X S4 1 S2 1 S2 1 n 1 0 0 0 0 n S2 S2 S4 0 0 0 0 0 0 0 S2 S4 0 S4 S6 0 0 0 0 S6 2 S4 2 S4 2 S2 2 This is of block diagonal where 1 nS4 S2 T0 2 T2 T1 T3 2 S2S6 S2 4 just invert both 2 2 submatrices We also have X T Y so 0 S4T0 S2T2 1 respectively S4 2 1 n 2 1 S6 2 2 1 S2 2 2 1 S2T0 nT2 2 S6T0 S4T2 3 S4T0 S2T2 2 2 with variances b If the model is re tted under H0 the estimates for 0 and 1 are the same and their variances are the same as well This is because of the block diagonal structure of X T X implying that if you just take the rst two rows and columns of X T X you get the same inverse elements the result would not be true without this Of course under H0 we don t consider 2 and 3 because these are assumed to be 0 c Various ways to do this but I think the following argument is the simplest First we note that SSE0 SSE1 SSR1 SSR0 where SSR denotes the regression sum of squares Second recall the formula under either H0 or H1 that says cid 80 yi y 2 cid 80 yi yi 2 cid 80 yi y 2 where the rst term is SSE and the second term is SSR Therefore using the fact that 0 and 1 are the same in both models we can write SSR0 cid 80 0 1x2 and SSR1 cid 80 0 1x2 i y 2 cid 80 2xi 3x3 i y 2 cid 80 0 1x2 i y 2 cross product cid 80 0 1x2 i 2 the i is 0 because every term in the cross product Therefore SSE0 SSE1 SSR1 SSR0 cid 80 2xi 3x3 i 2xi 3x3 i y 2xi 3x3 includes Sk for some odd k i 2 which expands to S2 2 2 3 This is of the given form with A S2 B 2S4 C S6 2S4 2 3 S6 2 The F statistic is then F SSE0 SSE1 2 SSE1 n 4 and has the distribution F2 n 4 under H0 The degrees of freedom arise because the original model H1 has p 4 unknown parameters which H0 has p q 2 unknown parameters therefore p 4 q 2 The test will reject H0 when F c …