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tut2_10

# tut2_10 - -α prediction interval for a new response at x...

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The University of Western Ontario Department of Statistical and Actuarial Sciences Statistical Sciences 3859a Tutorial 2 1. Consider the regression through the origin model: y i = βx i + ε i for i = 1 , 2 , . . . , n and where the ε i ’s are independent normal random variables with mean 0 and variance σ 2 . (a) Derive the Least-Squares estimator for β : β . (b) Obtain the unbiased estimator for σ 2 : MSE. (c) Show that β is unbiased. (d) Show that the variance of β is σ 2 n i =1 x 2 i and give an estimator for the standard error of β . (e) Show that if x 0 is a given number, then y | x 0 = βx 0 is an unbiased estimator for E [ y | x 0 ] = βx 0 . (f) Find the variance of y | x 0 , and deduce that a 1 - α confidence interval for E [ y | x 0 ] is βx 0 ± t α/ 2 ,n - 1 MSE( x 0 ) 2 n i =1 x
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Unformatted text preview: -α prediction interval for a new response at x is b βx ± t α/ 2 ,n-1 v u u t MSE ± 1 + x 2 ∑ n i =1 x 2 i ! . 2. Do not use lm() in this question. Consider the following data: x y 1 2 2 5 3 6 4 8 5 9 (a) Fit the regression through the origin model and estimate the error variance. (b) Compute the standard error estimate of b β and obtain a 95% conﬁdence interval for β . (c) Compute a 95% conﬁdence interval for E [ Y | x = 4 . 5]. (d) Give a 95% prediction interval for a new response at x = 4 . 5. 3. Repeat the previous question using the R function lm() ....
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