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# Problem 1 (50 points) Consider a simple linear regression yi = ['30 + lm + 51- With a sample of size n = 10, Which yielded cc 1 1 1 1 1 1 1 1 1 1 y

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

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?

Problem 1 (50 points) Consider a simple linear regression yi = [‘30 + ﬁlm + 51- With a sample of size n = 10, Which yielded cc 1 1 1 1 1 1 1 1 1 1 y 2.5 15 1.5 15 1.5 —2.5 —1‘5 —1‘5 —1.5 —2.5 For your convenience, n Still/i 22:13” 211912 22:13”? 2211mm
10 —1 0 34.5 10 18 a Find the least-squares estimates [5‘0 and Bl. (
(b Find an unbiased estimate of variance 0’2. )
) (c) At 5% level7 test Whether each of ﬁg and ,81 are zero. Show your work.
) (d What proportion of variation in y is explained by the regression line?

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