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MATH 423 midterm - 2006

# MATH 423 midterm - 2006 - Student Name Student Number...

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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 two-sample test. One of the variants of the test assumes homogeneous (or the same) variance in both groups. The t-statistic for the two-sample t-test 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 t-statistic 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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MATH 423 midterm - 2006 - Student Name Student Number...

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