Problem 1 (50 points)

Consider a simple linear regression yi = β0 + β1xi + εi with a sample of size n = 10, which yielded

x 1 1 1 1 1 −1 −1 −1 −1 −1 y 2.5 1.5 1.5 1.5 1.5 −2.5 −1.5 −1.5 −1.5 −2.5

For your convenience,

n ni=1 yi ni=1 xi ni=1 yi2 ni=1 x2i ni=1 xiyi

10 −1 0

34.5 10 18

(a) Find the least-squares estimates βˆ0 and βˆ1. (b) Find an unbiased estimate of variance σ2.

(c) At 5% level, test whether each of β0 and β1 are zero. Show your work. (d) What proportion of variation in y is explained by the regression line?

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