# Obtaining standard error on a data point obtained from linear regression

I have data with standard error, included below for clarity,

    X           Y           Error in Y
0.0105574   -28.831027  0.04422
0.0070382   -27.800385  0.04225
0.0052787   -27.314088  0.04209
0.0042229   -27.054207  0.04185
0.0035191   -27.000188  0.04143
0.0030164   -26.891275  0.04108


I have obtained parameters a and b of the expression y=a*x*x + b from a weighted least squares regression using this data (fit in gnuplot). The regression returned what was called "Asymptotic Standard Error" associated with these parameters. I believe this error was calculated using the deviation from fitted point to actual points (Equation 34/35 here) and is used to assess the quality of a fit. However, this is not the error that I'm interested in.

I'm looking to determine the value of the data point at X=0.0 from my fitted function with standard error like my other values. The output of the regression was:

    Final set of parameters            Asymptotic Standard Error
a              = -19389.1         +/- 752          (3.878%)
b              = -26.7951         +/- 0.03915      (0.1461%)


So, to be quite specific, how might I determine the standard error at the point (X,Y)=(0.0, -26.7951) using my fitted function? I expect the error in this calculated point to be much larger than the errors of the values reported in the Y values of my table above.

I can see how gnuplot is not the right tool for this, as it only weights my data points using the standard error in my input. What I need to do is propagate the error in my original data points to obtain the error on the regression line.

This seems like a pretty basic exercise, sorry for my statistics ignorance. Thanks!

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It sounds like you want a "prediction interval". The answer (with formulas) is available on this site in several places. – whuber May 17 '11 at 20:02