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STOR 664 FALL 2020 Midterm Exam September 30 2020 This is an open book remote learning exam Time limit 2 hours Access to course materials and standard computational tools in particular R is allowed communication with other students or with anybody via the internet other than the instructor is not The university Honor Code is in e ect at all times Answers may either be typed using Word or Latex or handwritten and scanned or photographed if handwritten it is recommended you use blue or black ink on plain sheets of white paper They should then be uploaded in sakai The exam is worth 100 points total 35 for question 1 65 for question 2 points for each part question are stated below Although the questions are intended to be answered in sequence you may write out your answers in any order and errors in one part question will not prevent you gaining full credit in other parts of the same question Attempt all questions 1 Three objects are to be weighed in a scale whose true weights are written 1 2 3 a Consider the following weighing scheme i Each of objects 1 2 3 is placed on its own in the scale The observed weights are Y1 Y2 Y3 ii Each pair of objects 1 and 2 then 1 and 3 then 2 and 3 is placed in the scale together resulting in observed weights Y4 Y5 Y6 respectively iii The three objects are weighed together with observed total weight Y7 Assume each of the weighings has a random error of mean 0 and variance 2 and that all the random errors are uncorrelated Without using the computer nd formulas for the least squares estimates 1 2 3 as linear combinations of Y1 Y7 You should state explicitly the coe cients of Y1 Y7 in these estimators and show that the common variance of the three estimators is 3 2 8 18 points b Consider an alternative weighing scheme where object 1 is weighed n1 times object 2 is weighed n2 times object 3 is weighed n3 times where n1 n2 n3 7 Show that however n1 n2 n3 are chosen the mean variance of 1 2 3 is greater than in a 7 points c Now suppose that the main quantity of interest is not any of 1 2 3 but 2 3 1 2 This might be of interest for instance if we were substituting objects of types 1 and 2 in a system with objects of type 3 or the other way round and we wanted to know whether this would result in an increase or a decrease in the total weight of the system Find the variance of the estimator 2 3 1 2 from part a and show that there is at least one estimator derived from the scheme of part b that would have smaller variance 10 points 2 Consider the model yi 1 2xi cid 15 i yn i 1 3xi cid 15 n i 1 for i 1 n Here xi i 1 n is a covariate with cid 80 n i 1 xi 0 and cid 15 i i 1 2n are independent N 0 2 random variables Our ultimate interest is in testing the null hypothesis H0 2 3 against the alternative H1 2 cid 54 3 a Assuming H1 nd explicit formulas for the least squares estimators 1 2 3 in terms of y1 y2n and x1 xn and calculate their variances 10 points b Recall that there is a general formula that expresses the total sum of squares SST cid 80 yi y 2 as the sum of squares due SST SSR1 SSE1 to regression and the sum of squared residuals which in this case is given by cid 110 yi y 2xi 2 yn i y 3xi 2 cid 111 n cid 88 i 1 SSE1 The subscript 1 here is to denote that these calculations are made under H1 Under the above assumptions nd an explicit formula for SSR1 in terms of 2 3 and x1 xn 10 points c Repeat the calculations of parts a and b under H0 In particular assuming 3 2 nd least squares estimators for 1 and 2 their variances and formulas for SSR0 and SSE0 in this case To distinguish them from the estimators in a and b write 1 and 2 for these estimators 8 points d What are the relationships between 1 and 2 from part c and 1 2 3 in a 4 points e Prove the formula SSE0 SSE1 SSR1 SSR0 2 3 2 2 n cid 88 1 x2 i Note If you didn t succeed in proving this formula nevertheless you should assume it is correct for the remaining parts of the question 6 points f Show how to formally test H0 against H1 at signi cance level 0 05 Speci cally you should de ne a relevant test statistic which may be expressed in terms of SSE0 SSE1 or any quantities developed in previous parts of the question and explain how to de ne the rejection region so that the test has the desired signi cance level 8 points g Explain how you would calculate the power of this test for given values of 2 3 and 2 cid 12 cid 12 cid 12 1 n 10 and cid 80 n h In practice the main design of the experiment issue may well be the value of cid 80 x2 Assuming the other parameters are as in part g what value of cid 80 x2 i which the experimenter may be able to adjust by choosing suitable values of the xi s i would be needed x1 xn To illustrate your answer calculate the power of the test when i is any of i 20 ii 40 iii 60 iv 80 11 points cid 12 cid 12 cid 12 2 3 1 x2 to achieve power 0 8 8 points Note The last two parts are the only places on the exam where you are expected to use the computer if you use R you should indicate clearly which functions are being used and how 2


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UNC-Chapel Hill STOR 664 - Midterm Exam

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