Observation: Click here for proofs of the above formulas.Įxample 1: Find the 95% confidence and prediction intervals for Poverty where Infant Mortality is 7.0, White = 80 and Crime = 400 based on the data in Example 2 of Multiple Regression Analysis using Excel, which is reproduced in Figure 1 (in two blocks to fit better on the page). Here, the square-root term is called the standard error of the prediction. The 1 – α prediction interval of ŷ 0 is therefore The prediction interval is calculated in a similar way, except that now the variance is the variance of the residual y − ŷ, which is Here t crit is the critical value of the t distribution with df Res = n − k − 1 degrees of freedom with significance level α/2, i.e. The 1 – α confidence interval for the true value of ŷ 0 is therefore
If X 0 is the column array with values 1, x 01, x 02, …, x 0 k, then an unbiased estimate of the standard error of ŷ 0, called the standard error of the fit, is given by the formula Where the regression coefficients b j are based on the n × ( k +1) data array X (with ones in the first column).įor any specific values of the x j, say x 01, x 02, …, x 0 k, we have the predicted value Let’s assume that we have a regression line
0 kommentar(er)
