# week4 - Inference about the Slope and Intercept Recall we...

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STA302/1001 - week 4 1 Inference about the Slope and Intercept Recall, we have established that the least square estimates and are linear combinations of the Y i ’s. Further, we have showed that they are unbiased and have the following variances In order to make inference we assume that ε i ’s have a Normal distribution, that is ε i ~ N (0, σ 2 ). This in turn means that the Y i ’s are normally distributed. Since both and are linear combination of the Y i ’s they also have a Normal distribution. ( 29 ( 29 XX XX S S X n 2 1 2 2 0 ˆ Var and 1 ˆ Var σ β = + = 0 ˆ 1 ˆ 0 ˆ 1 ˆ

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STA302/1001 - week 4 2 Inference for β 1 in Normal Error Regression Model The least square estimate of β 1 is , because it is a linear combination of normally distributed random variables ( Y i ’s) we have the following result: We estimate the variance of by S 2 / S XX where S 2 is the MSE which has n -2 df. Claim: The distribution of is t with n -2 df. Proof: XX S N 2 1 1 , ~ ˆ σ β XX S S 2 1 1 ˆ β- 1 ˆ 1 ˆ
STA302/1001 - week 4 3 Tests and CIs for β 1 The hypothesis of interest about the slope in a Normal linear regression model is H 0 : β 1 = 0. The test statistic for this hypothesis is

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week4 - Inference about the Slope and Intercept Recall we...

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