Course Notes - Department of Mathematics and Statistics

10.3 Confidence Intervals and Regression• We have looked at ANOVA to test the fit of the model.• It is also possible to get an idea of the fit of the model, by calculatinga 95% confidence interval for the slope of the model.• If β 1 is the true slope of the regression line then the standard errorof β 1 is :σ β1 =√ n∑√i=1σ e(x i − ¯x) 2• Where the variance of the errors σe 2 is estimated using the formula:n∑(y i − ŷ) 2• NOTE: s 2 e is just MSEs 2 e =i=1• The divisor is (n-2) rather than (n-1) because we are estimatingβ 0 and β 1 in ŷ.n−2• Therefore, the 95% confidence interval for β 1 isˆβ 1 ± t n−2√ n∑√i=1s e(x i − ¯x) 2• Easiest way to explain this concept is through an example, so letus return to our height correlation example.• Using ANOVA we found that the regression effect dominated theresidual effect.• Running a regression we get the line of least squares as:ŷ = 107.996 + 0.366x• Where ŷ is the height of the child and x is the height of the fatherand:218