Confidence Intervals for the Coefficients of Regression Lines

For a regression linetheare themselves random variables. To find estimates for thewe form an expression for the sum of the error terms squared:We minimise this sum by allowing theto vary. Differentiating each with respect to eachleads to the following system of equations:

Because the regression lineis linear in the b-i the equations above are linear too. We can solve this system of linear equations to solve for thethese solutions are labelledTheare themselves random variables because they are functions of the random variablesBecause the equations are linear, theare normally distributed with corresponding standard deviationWe can then construct confidence intervals for each

Typically we want to test whether 0 is in the interval. If it is, then at the significance level of the test, there is no evidence of a correlation betweenand

Much of the time theandare found automatically with computer packages.

Example: The table below gives data on the amount of iron, aluminium and phosphate in soil.

 Observation =iron =aluminium =phosphate 1 61 13 4 2 175 21 18 3 111 24 14 4 124 23 18 5 130 64 26 6 173 38 26 7 169 33 21 8 169 61 30 9 160 39 28 10 244 71 36 11 257 112 65 12 333 88 62 13 199 54 40

A computer package returns the results:

 Parameter Estimate, Estimated standard deviation, -7.35100 3.48500 0.11273 0.02969 0.34900 0.07131

A 99% confidence interval foris then, with

A 99% confidence interval foris, with