INFERENCES IN SIMPLE LINEAR REGRESSION

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Inferences on the Estimated Slope (b₁)

Original Model: Y_i= b₀ + b₁X_i + e_i,where e_i~ N(0,s²) and cov(e_i,e_j)=0 for i¹ j.

(estimate of the slope parameter)

E(b₁) = = b
Var(b₁) =
b₁~N(,)

The variance of b₁ is estimated via

, and
The variable t=

follows a t-distribution with (n-2) d.f. Thus a 100(1-a )% confidence interval for b₁ is given by

To test H₀: b₁=b₁₀we would use the test statistic t= and

reject H₀ in favor of H_a: b₁>b₁₀ if t ³t_a,n-2
reject H₀ in favor of H_a: b₁<b₁₀ if t £-t_a,n-2
reject H₀ in favor of H_a: b₁¹b₁₀ if |t| ³t_a/2,n-2

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Jayant Rajgopal