# Is the mean square error s 2 ² n x i 1 y i ˆ y i 2

This preview shows pages 3–6. Sign up to view the full content.

is the mean square error s 2 ² = n X i =1 ( y i - ˆ y i ) 2 / ( n - 2) = SSE n - 2 , s ² = r SSE n - 2 ( S ² is called the residual standard deviation) 11.6 Partitioning the variability y i - ¯ y = y i - ˆ y i + ˆ y i - ¯ y -→ & \$ % n X i =1 ( y i - ¯ y ) 2 = n X i =1 ( y i - ˆ y i ) 2 + n X i =1 y i - ¯ y ) 2 SSTotal = SSError + SSREG (Note: SS stands for ”Sum of Squares”). 3

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
A useful measure of model fit is the Coefficient of determination ( R 2 ). R 2 = SSREG SSTotal , 0 R 2 1 The larger the R 2 , the closer the fit. The Sample correlation coefficient between X and Y is r X,Y = S XY S XX · S Y Y , - 1 r X,Y 1 . It measures the linear relationship between X and Y . r X,Y = +1 ⇐⇒ Y = a + bx, b > 0 , r X,Y = - 1 ⇐⇒ Y = a - bx, b > 0 . For the simple linear regression ˆ y = ˆ β 0 + ˆ β 1 x ; we have (prove!) r 2 X,Y = r 2 Y, ˆ Y and r 2 X,Y = R 2 11.7 Distributions of the estimated model parameters In order to construct the CI’s for the unknown parameters β 0 and β 1 , or to do hypothesis test such as H 0 : β 0 = - v.s. H 1 : β 1 6 = 0 . We need to know the distributions of ˆ β 0 and ˆ β 1 . To do this, we assume the distribution of the random error ² to be normal, i.e. ² N (0 , σ 2 ). Under this normality assumption, T 1 = ˆ β 1 - β 1 s ² / S XX t n - 2 ; T 0 = ˆ β 0 - β 0 s ² · r X 2 i n · S XX t n - 2 . Under H 0 : β 1 = 0, T 1 = ˆ β 1 - 0 S ² / S XX . 4
( T 1 ) 2 = ( ˆ β 1 ) 2 · S XX s 2 ² = SSREG s 2 ² F 1 ,n - 2 . 11.8 Checking the model assumptions The constant variance assumption can be checked via a scatter plot of the residuals ( y i - ˆ y i ) versus x i (or ˆ y i ). This plot is often called the residual plot.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern