12.6 Confidence Intervals
A confidence interval represents a closed interval where a certain percentage of the population is likely to lie. For example, a 90% confidence interval with a lower limit of \(A\) and an upper limit of \(B\) implies that 90% of the population lies between the values of \(A\) and \(B\). Out of the remaining 10% of the population, 5% is less than \(A\) and 5% is greater than \(B\).
12.6.1 Confidence Interval on Regression Coefficients
A \(100(1-\alpha)\) percent confidence interval on \(\beta_1\) is obtained as follows:
\[{{\hat{\beta }}_{1}}\pm {{t}_{\alpha /2,n-2}}\cdot se({{\hat{\beta }}_{1}})\,\]
Similarly, a \(100(1-\alpha)\) percent confidence interval on \(\beta_0\) is obtained as:
\[{{\hat{\beta }}_{0}}\pm {{t}_{\alpha /2,n-2}}\cdot se({{\hat{\beta }}_{0}})\,\]
12.6.2 Confidence Interval on Fitted Values
A \(100(1-\alpha)\) percent confidence interval on any fitted value, \(\hat{y_i}\), is obtained as follows:
\[{{\hat{y}}_{i}}\pm {{t}_{\alpha /2,n-2}}\sqrt{{{{\hat{\sigma }}}^{2}}\left[ \frac{1}{n}+\frac{{{({{x}_{i}}-\bar{x})}^{2}}}{\underset{i=1}{\overset{n}{\mathop \sum }}\,{{({{x}_{i}}-\bar{x})}^{2}}} \right]}\,\]
where:
- \(\hat{\sigma}^{2}\) is the mean square error (\(MSE\)).
- The term that multiplies \({t}_{\alpha /2,n-2}\) is the standard error of the fit
It can be seen that the width of the confidence interval depends on the value of \({x}_{i}\) and will be a minimum at \({{x}_{i}}=\bar{x}\,\) and will widen as \(\left| {{x}_{i}}-\bar{x} \right|\,\) increases.
The residual sum of squares (or error sum of squares) is defined as
\[ SS_E = \sum_{i=1}^n (y_i - \hat{y_i})^2 = \sum_{i=1}^n e_i^2\] and the estimate of \(\sigma^2\) is
\[\sigma^2 = \dfrac{SS_E}{n-2} \]