Simple linear regression: Difference between revisions

Revision as of 21:41, 31 July 2007

Revision as of 22:01, 31 July 2007

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Under the assumption that the error term is normally distributed, the estimate of the slope coefficient has a normal distribution with mean equal to '''b''' and standard error given by:

<math> SE_s_ \hat{b} = \sqrt { \frac {\sum_{i=1}^N \hat{\varepsilon_i}^2 /(N-2)} {\sum_{i=1}^N (x_i - \bar{x})^2} }</math>.

A confidence interval for ''b'' can be created using a t-distribution with N-2 degrees of freedom:

<math> [ \hat{b} - s_ \hat{b} t_{N-2}^*,\hat{b} + s_ \hat{b} t_{N-2}^*] </math>

== Numerical Example ==

Suppose we have the sample of points {(1,-1),(2,4),(6,3)}. The mean of X is 3 and the mean of Y is 2. The slope coefficient estimate is given by:

<math> \hat{b} = \frac {(1 - 3)((-1) - 2) + (2 - 3)(4 - 2) + (6 - 3)(3 - 2)} {(1 - 3)^2 + (2 - 3)^2 + (6 - 3)^2 } = 7/14 = 0.5 </math>

The standard error of the coefficient is 0.866. A 95% confidence interval is given by:

[0.5 - 0.866 x 12.7062, 0.5 + 0.866 x 12.7062] = [-10.504, 11.504].