Analytical Chem istry - DePauw University

176 Analytical Chemistry 2.0where y i is the i th experimental value, and ŷ iis the corresponding value predictedby the regression line in equation 5.15. Note that the denominatorof equation 5.19 indicates that our regression analysis has n–2 degrees offreedom—we lose two degree of freedom because we use two parameters,the slope and the y-intercept, to calculate ŷ i.A more useful representation of the uncertainty in our regression isto consider the effect of indeterminate errors on the slope, b 1 , and the y-intercept, b 0 , which we express as standard deviations.sb1=ns2n x − ⎛ ⎞∑xii⎝⎜∑⎠⎟i2ri2=2sr2∑( x − xi ) 5.20isb0=sx2 2r ii2n x − ⎛ ⎞∑xii⎝⎜∑⎠⎟i∑i2=sr∑x2 2ii∑( −i )n x xi2 5.21We use these standard deviations to establish confidence intervals for theexpected slope, b 1 , and the expected y-intercept, b 0β 1= b 1± ts b 5.221β 0= b 0± ts b 5.230You might contrast this with equation4.12 for the confidence interval around asample’s mean value.As you work through this example, rememberthat x corresponds to C std , andthat y corresponds to S std .where we select t for a significance level of a and for n–2 degrees of freedom.Note that equation 5.22 and equation 5.23 do not contain a factor of( n)−1 because the confidence interval is based on a single regression line.Again, many calculators, spreadsheets, and computer software packagesprovide the standard deviations and confidence intervals for the slope andy-intercept. Example 5.10 illustrates the calculations.Example 5.10Calculate the 95% confidence intervals for the slope and y-intercept fromExample 5.9.So l u t i o nWe begin by calculating the standard deviation about the regression. To dothis we must calculate the predicted signals, ŷ i, using the slope and y‐interceptfrom Example 5.9, and the squares of the residual error, ( y − yˆ )2 .i iUsing the last standard as an example, we find that the predicted signal isyˆ = b + bx = 0. 209 + 120. 706×0. 500 60.5626 0 1 6 ( )=and that the square of the residual error is