True slope, confidence interval

These standard deviations can be used to establish confidence intervals for the true slope and the true y-intercept... 

The following is an example of a mathematical/statistical calculation of a calibration curve to test for true slope, residual standard deviation, confidence interval and correlation coefficient of a curve for a fixed or relative bias. A fixed bias means that all measurements are exhibiting an error of constant value. A relative bias means that the systematic error is proportional to the concentration being measured i.e. a constant proportional increase with increasing concentration. 

These values can be used to calculate confidence intervals for the true intercept and true slope. Multiple repeats of this experiment would also report different values of slope (b) and intercept (a) but all results would be in close proximity to each other. 

The true intercept is 95% confidently between —0.098 and 0.042 and since this interval includes zero therefore it may be possible that no fixed bias is evident. The confidence interval, the true slope b of a regression line, is given by ... 

A 95% confidence interval of the true slope is between 1.017 and 0.997, therefore the relative bias will lie between 1.7% and 0.7%. In this case the interval is wider than the line through the centroid even with smaller t-value and residual standard deviations, and the fact that Cx2 was used instead of ]C(x — x)2, and because of this it is expected that the line through the origin would give a narrower confidence interval. There are many reasons for this and they are beyond the scope of this book. 

The question of whether the variance of (K /K 0) is underestimated by using the regression procedure of Equation 11 was studied in the following way. For each of the 1,000 estimated slopes, (K /Ky0) the 95-percent confidence interval was calculated using Equation 12. If Equation 12 is a satisfactory estimator for the 95-percent confidence interval, then about 95 percent of the computed confidence intervals should contain the true slope, 0.71. When these calculations were performed on the simulation generated distributions of (Kvc/Kv°), only about 86 percent of the supposed 95 percent confidence intervals in each simulation contained the true slope, 0.71. Because the simulation generated estimates of average (K /Ky0) were so near the true value of 0.71, it is presumed that the principal reason fewer confidence intervals contained 0.71 than expected is that the variance is underestimated by Equation 12. 

The coefficients p, and p2 refer to the intercept and slope of the straight line fitting the data points (see Fig. 4 a). Being based on the random variables X and y, the coefficients p, and p2 are outcomes of random variables themselves. The true coefficients can be covered by the following 95 % confidence intervals ... 

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