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Practice Midterm II

Practice Midterm II - 5 units of the true slope 3 Let x 1 Y...

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STA131C, Spring 2007 Prof. W. Polonik Practice Exercises I 1. Consider the one-way ANOVA model Y ij = α i + ǫ ij , j = 1 , . . . , n i , i = 1 , . . ., I, with ǫ ij iid N (0 , σ 2 ) . (a) Show that h α i = Y i s = 1 n i n i i =1 Y ij , i = 1 , . . . , I are the LSEs of α i , i = 1 , . . ., I. (b) Find the MLE h σ 2 of σ 2 and state its sampling distribution. Based on the data given in Problem 3 of HW#4, (c) construct a 95%-con±dence interval for σ 2 , and (d) ±nd the value of the BLUE for α 1 1 2 ( α 2 + α 3 ) , i.e., ±nd the BLUE for the di²erence of the mean of group 1 and the average between the means of group 2 and 3. 2. Consider the linear regression model Y i = a + b x i + ǫ i , i = 1 , . . ., n, where ǫ i iid N (0 , σ 2 ) and not all the x i -values are the same. Assume the variance σ 2 is known . (a) Find a 95%-con±dence interval for the slope b based on the LSE h b. (b) Construct a test for H 0 : b = 0 versus H 1 : b n = 0 at a given level α. (HINT: Use what you did in part (a).) (c) Suppose that n = 9 , the x i ’s are set to be 1 , 2 , . . ., 9 , and we know that σ 2 = 45 . Find the probability that the estimated slope lies within 1
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Unformatted text preview: . 5 units of the true slope. 3. Let ( x 1 , Y 1 ) , . . ., ( x n , Y n ) and ( x * 1 , Y * 1 ) , . . . , ( x * n , Y * n ) denote two sets of observations (all independent), coming from a simple linear regression model Y i = a + bx i + ǫ i and Y * i = a * + b * x i + ǫ * i , respectively. Here ǫ i , ǫ * i ∼ N (0 , σ 2 ) all independent with σ 2 > 0 unknown. (a) Construct a 95%-con±dence interval for the di²erence between the slopes b − b * . (As always, begin with ±nding a pivot statistic. In this context it might be useful to recall that the sum of two independent χ 2-variables is also a χ 2-variables. More precisely, if X i ∼ χ 2 ν i , i = 1 , 2 then X 1 + X 2 ∼ χ 2 ν 1 + ν 2 . ) (b) Construct a test for the null-hypothesis b = b * versus the alternative hypothesis b n = b * . (HINT: Use what you did in part (a).)...
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