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# t1sol05 - STA 302 1001 H Fall 2005 Test 1 LAST NAME...

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STA 302 / 1001 H - Fall 2005 Test 1 October 19, 2005 LAST NAME: SOLUTIONS FIRST NAME: STUDENT NUMBER: ENROLLED IN: (circle one) STA 302 STA 1001 INSTRUCTIONS: Time: 90 minutes Aids allowed: calculator. A table of values from the t distribution is on the last page (page 8). Total points: 50 Some formulae: b 1 = ( X i - X )( Y i - Y ) ( X i - X ) 2 = X i Y i - n X Y X 2 i - n X 2 b 0 = Y - b 1 X Var( b 1 ) = σ 2 ( X i - X ) 2 Var( b 0 ) = σ 2 ± 1 n + X 2 ( X i - X ) 2 ² Cov( b 0 , b 1 ) = - σ 2 X ( X i - X ) 2 SSTO = ( Y i - Y ) 2 SSE = ( Y i - ˆ Y i ) 2 SSR = b 2 1 ( X i - X ) 2 = ( ˆ Y i - Y ) 2 σ 2 { ˆ Y h } = Var( ˆ Y h ) = σ 2 ± 1 n + ( X h - X ) 2 ( X i - X ) 2 ² σ 2 { pred } = Var( Y h - ˆ Y h ) = σ 2 ± 1 + 1 n + ( X h - X ) 2 ( X i - X ) 2 ² r = ( X i - X )( Y i - Y ) p ( X i - X ) 2 ( Y i - Y ) 2 Working-Hotelling coe±cient: W = p 2 F 2 ,n - 2;1 - α 1 2 3 4abc 4def 4ghi 10 8 7 11 9 5 1

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1. (10 points) A simple linear regression model is ft on n observed data points. (a) What is the diFerence between β 1 and b 1 ? (3 marks) β 1 : slope of model, unobserved parameter b 1 : estimate of β 1 , calculated from data (b) What does it mean i± R 2 = 1? (1 mark) Data points Ft exactly on a line. (c) In lecture we showed n i =1 e i = 0 and n i =1 e i X i = 0. Show that n i =1 e i ˆ Y i = 0. (You may use the results shown in class i± they are help±ul.) (2 marks) X e i ˆ Y i = X e i ( b 0 + b 1 X i ) = b 0 X e i + b 1 X e i X i = 0 (d) Explain why the result in (c) implies that the residuals and predicted values are uncor- related and why this is use±ul. (4 marks) r = e i ˆ Y i - n e ˆ Y q ( e i - e ) 2 ( ˆ Y i - ˆ Y ) 2 = 0 using e = 0 since e i = 0 This is useful for residual plots since we then don’t expect a pattern in the plot of the e i ’s versus the ˆ Y i ’s.
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t1sol05 - STA 302 1001 H Fall 2005 Test 1 LAST NAME...

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