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Unformatted text preview: Student Name: Student Number: Faculty of Science MIDTERM EXAMINATION Mathematics 423 Regression and Analysis of Variance Sample Midterm Choose THREE of FIVE questions. Calculators are allowed. One 8.5” × 11” sheet of notes is allowed. Language dictionaries are allowed. There are 10 pages to this exam and the total number of marks for the exam is 100. 1 Question 1: (a) The standard test for testing the difference in mean between two groups is the twosample test. One of the variants of the test assumes homogeneous (or the same) variance in both groups. The tstatistic for the twosample ttest is: t = ¯ y 1 ¯ y 2 radicalbigg ∑ n 1 i =1 ( y i 1 ¯ y 1 ) 2 + ∑ n 2 i =1 ( y i 2 ¯ y 2 ) 2 n 2 and if the two groups are normally distributed with the same mean (i.e. H : μ 1 μ 2 = 0), then the tstatistic has a t distribution with n 1 + n 2 2 degrees of freedom Show that this test statistic is equivalent to the test statistic for β 1 = 0 for a simple linear regression model y i = β + β 1 x i where x i = 0 if the i th observation is in group 1 and 1 if it is in group 2. Question 2: (a) In the simple linear model, Y i = β + β 1 x i + ǫ i , prove that V ar ( ˆ β ) is a minimum if the x i are chosen so that ¯ x = 0. (b) If the x i from above can be selected anywhere in the interval [ a, b ], and if T is an even integer, prove that the variance of ˆ β 1 is minimized if T/ 2 values of x i are selected equal to a and T/ 2 values are selected equal to b . (c) What are the practical difficulties of an experimental design such as in (b) (i.e. why might that particular choice of covariate values not be a good idea, despite the fact that it minimizes the variance of ˆ β 1 )? 2 Question 3: In this question, we’ll show the unbiasedness of the residual variance via matrix manipulations for the general linear model y = X β + ǫ where X is n × p . (a) First, show that we can write s 2 = ∑ n i =1 ( y i ˆ y i ) 2 n p as s 2 = y t ( I X ( X t X ) 1 X t ) y n p ....
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 Spring '06
 STEELE
 Math, Linear Regression, Regression Analysis, Variance, Yi, Gestational age

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