Class Notes (808,754)
Tony Quon (67)
Lecture 20

4 Pages
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School
University of Ottawa
Department
Course
Professor
Tony Quon
Semester
Fall

Description
Two Key Intervals 1.C.I. for the mean of Y at a given value of X 2. Prediction interval (P.I.) for a future observation of Y at a given value of X For a given x*, which interval will be bigger? P.I. since the mean varies less than a given observation Same point estimate for both! * y =b +b x x* 0 1 Two Key Intervals If the extrapolation penalty is small and n is reasonably large, then an approximate 95% P.I. is: y *2s x Why do both the C.I. and the P.I. increase in size as you move away from the mean of X? Summary Model of population: y = β +β x +ε i 0 1 i i 4 Assumptions: Linearity, constant variance, normality and independence Unfortunately, we don’t know 0, 1 or 2 so we take a sample (x1,y1),…,(xn,yn) and get sample estimates b0, b1 and s2 from which we can get C.I. and perform hypothesis tests to determine if there is a linear relationship What we are really interested in is estimating Y so we look at: 1. C.I. for the mean of Y for a given x* 2. P.I. for a future observed Y at a given x* Inference for Simple Linear Regression (Cont’d) Two Key Intervals: 1. C.I. for the mean of Y at a given value of X 2. Prediction interval (P.I.) for a future observation of Y at a given value of X For a given x*, which interval will be bigger? P.I. since the mean varies less than a given observation Same point estimate for both: ! * y =* +b x x 0 1 C.I.for µ Y | x 2 * 1 (x − x) b0+b 1 ±t α /2,n−2 n+ 2 ∑ (xi− x) P.I.forY | x * 1 x − x)2 b0+b 1 ±t α /2,n−21+ + 2 n ∑ (xi− x) Two Key Intervals: If the extrapolation penalty is small and n is reasonably large, then an approximate 95% P.I. is: ⌢ y *2s x Why do both the C.I. and the P.I. increase in size as you move away from the mean of X? Example: Resurfacing Contract The director of a highway department wants to predict the cost of a resurfacing contract that is up for bids (6-km road) The cost could be predicted as a function of the kilometers of road to be resurfaced Past contracts from various bidders gives the following data: Cost (Y) in \$10,000 6 14 10 14 26 Distance (X) in km. 1 3 4 5 7 ∑ x = 20 ∑ y = 70 ∑ xy =340 x = 4 C.I. for the average cost of a
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