Study Guides (238,586)
STA302H1 (10)

# tsolf07.pdf

7 Pages
168 Views

School
University of Toronto St. George
Department
Statistical Sciences
Course
STA302H1
Professor
Semester
Fall

Description
STA 302 H1F / 1001 HF { Fall 2007 Test 1 October 24, 2007 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: P P (X X)(Y ▯Y ) X Y ▯nXY b1= P i i = P ii 2 b0= Y ▯ b1X (Xi▯X)2 X inX ▯ ▯ ▯2 2 1 X2 Var(b1) =P 2 Var(b0) = ▯ n+ P 2 (Xi▯X) (Xi▯) 2 P Cov(b ;b ) = ▯ P ▯ X SSTO = (Y ▯ Y )2 0 1 (Xi▯X)2 i P ^ 2 2P 2 P ^ 2 SSE = (Yi▯ Yi) SSR = b 1 (X i X) = (Yi▯ Y ) ▯ ▯ ▯ ▯ 2 ^ ^ 2 1 P(Xh▯X) 2 2 ^ 2 1 PXh▯X) 2 ▯ fY h = Var(Y h = ▯ n + (Xi▯X)2 ▯ fpredg = Var(Yh▯ Y h = ▯ 1 + n+ (Xi▯)2 P (X ▯X)(Y ▯Y ) p r = p P i P i Working-Hotelling coe▯cient: W = 2F 2;n▯2; ▯ (Xi▯X)2 (Yi▯Y ) 1 2a 2bcdef 2ghi 2j 3 1 1. The following questions require derivations of results for the simple linear regression model. (a) (2 marks) In lecture we showed that e = 0 andP n e X = 0. Given these results, P i=1 i i=1 i i what is i=1ei i? Justify your answer. Xn X ^ ei i = ei(b0+ b1X i i=1 i=1 X Xn = b 0 ei+ b1 eiX i i=1 i=1 = 0 (b) (5 marks) Show that the total Sum of Squares in a regression can be decomposed as X ▯ ▯ Xn ▯ ▯ ^ 2 ^ 2 Yi▯ Y + Yi▯ Yi i=1 i=1 You may use any results that were derived in lecture. Xn SSTO = (Y ▯ Y ) i i=1 Xn ^ ^ 2 = (Yi▯ Yi+ Yi▯ Y ) i=1 Xn Xn Xn ^ 2 ^ 2 ^ ^ = (Yi▯ Y ) + (Yi▯ Yi) + 2 (Yi▯ Y )(i ▯ i ) i=1 i=1 i=1 and Xn X (Yi▯ Y )(i ▯ i ) = (Yi▯ Y )i i=1 i=1 X Xn = i i▯ Y ei= 0 i=1 i=1 ▯ ▯ ▯ ▯ Pn ^ 2 Pn ^ 2 So SSTO = i=1 Y i Y + i=1 Y i Y i (c) (5 marks) Assume that the X are non-random. Derive the formula for Cov(b ;b ) given i 0 1 on the ▯rst page. You may use any other formulae from the ▯rst page that you require except the formulae whose derivations require knowing0Co1(b ;b ). (Hint: You may want to start with the formula for the estimated intercept.) From the formula for0b := b0+ b1X So 2 Var(Y ) = Var(b 0 + X Var(b 1 + 2XCov(b 0b1) 2 ! ▯2 2 1 X 2 ▯2 = ▯ + + X + 2XCov(b 0b 1 n n SXX SXX ▯ X P n 2 Rearranging gives Cov0b1;b ) S ▯ where XX = i=1X i X) XX 2 2. The SAS output that follows was produced to examine the relationship between full-scale IQ (FSIQ) and brain size as measured by MRI (MRIcount). Measurements were taken on 20 university students chosen because their full-scale IQ was at least 130. The REG Procedure Descriptive Statistics Uncorrected Standard Variable Sum Mean SS Variance Deviation Intercept 20.00000 1.00000 20.00000 0 0 MRIcount 18518961 925948 1.725648E13 5730703420 75701 FSIQ 2728.00000 136.40000 372396 15.62105 3.95235 Dependent Variable: FSIQ Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 1 89.22306 89.22306 7.74 0.0123 Error 18 207.57694 11.53205 Corrected Total 19 296.80000 Root MSE 3.39589 R-Square 0.3006 Dependent Mean 136.40000 Adj R-Sq 0.2618 Coeff Var 2.48965 Parameter Estimates Parameter
More Less

Related notes for STA302H1

OR

Don't have an account?

Join OneClass

Access over 10 million pages of study
documents for 1.3 million courses.

Join to view

OR

By registering, I agree to the Terms and Privacy Policies
Just a few more details

So we can recommend you notes for your school.