[ \fracS_xx\sigma_x^2 \sim \chi^2_(n-1) ]
Under the :
If (S_xx) is random (random (x)), the of (\hat\beta 1) involves the expectation of (1/S xx). But conditionally on (x), (S_xx) is constant. 5. Small-Sample Correction If (\sigma_x^2) is unknown, replace with (\hat\sigma x^2 = S xx/(n-1)): sxx variance formula
Therefore:
[ \mathrmVar(\hat\beta 1) = \frac\sigma^2S xx ] [ \fracS_xx\sigma_x^2 \sim \chi^2_(n-1) ] Under the :
[ \frac(n-1)s_x^2\sigma_x^2 \sim \chi^2_n-1 ] sxx variance formula