Least-squares line

Carter, K. N., Scott, D. M., Salomon, J. K., and Zarcone, G. S., Confidence Limits for the Abscissa of Intersection of Two Least-Squares Lines Such as Linear Segmented Titration Curves, Anal. Chem. 63, 1991, 1270-1278. 

Any plot which is formed during data analysis can be accumulated and all accumulated plots can be printed out after the analysis is complete. An example of a standard run is shown in Figure 2. For fracture studies, the fracture data can also be plotted along with the least squares line which yielded the Fracture energy. 

In practical situations the absorbance of a sample is determined by making two measurements, the first to determine 70 and the second to determine I. The determination of I0 is used to cancel a large number of experimental factors that could affect the result. When measuring I0 the sample container must closely match the unknown container in all ways except for the analyte content. The cuvettes should be a matched pair if a double beam instrument is used and the same cuvette can be used for both the blank and sample with a single beam instrument. The blank solution filling the cuvette should be identical to the solvent that the sample is dissolved in, except for the sample itself. If done correctly, the least-squares line for the calibration graph will come very close to the 0,0 point on the graph. 

Enter data into a spreadsheet and obtain a graph of absorbance versus concentration of the standards. Obtain the least-squares line and its equation (Fig. 5.6). 

To determine the concentration using the graphical method draw a horizontal line from A — 0.443 to the least-squares line. Then construct a vertical line from that point to the concentration axis and read the value of the unknown. In this case it appears to be slightly less than 6.0 ppm. Using the least-squares equation we get... 

Calibration curves are plots of concentration (xt) versus some response of the instrument (yt). Concentration values are assumed to be the most precise and all of the error is associated with the response measurement. With those definitions, we write the equation for the least-squares line as y — mx + b and, omitting the derivation, find that... 

Preparation of a calibration curve has been described. From the fit of the least-squares line we can estimate the uncertainty of the results. Using similar equations we can determine the standard deviation of the calibration line (similar to the standard deviation of a group of replicate analyses) as... 

Also determine the slope and intercept of the least-squares line for this set of data. Determine the concentration and standard ... 

Fig. 10. Jump rates of the phenyl ring as a function of temperature obtained from Ti values (solid line) and line shape calculation (circle). The dashed line is the least-square line for the jump rates.

This suggests that a plot of r0 vs pA (Fig. 4) or r0/pA vs pA should be linear. It is very difficult to reject this model on the basis of data curvature, even though it is evident that some curvature could exist in Fig. 4. However, Eq. (16) demands that Fig. 4 also exhibit a zero intercept. In fact, the 99.99% confidence interval on the intercept of a least-square line through the data does not contain zero. Hence the model could be rejected with 99.99% certainty. 

Plackett-Burman design. (----) least-squares line through all effects (-) assumed line through... 

The values of a and b are then chosen to minimize Q which is the sum of the squares of yj - a - bXj) , hence the name Least Squares. Accordingly, the equation for the best fit least squares line is ... 

Figure 3. Plot of [a]D vs. the reciprocal of the molecular weight for chiral 3-substituted cyclopentenes48. The least-squares line shown corresponds to [ ]n = 12.61 x 10J (l/MW) + 17.41.

Fig. 2 Log A vs. AE of Scheme 37. The individual ions 75 are identified. The least-squares line is determined from all data points except those for 75a, 75b, 75f and 75p.

The activation parameters are subjected to errors, especially in the precise determination of temperatures such as Tc, the frequency separation (Av), line widths, coupling constants, variation of the concentration with the temperature etc. Although the experimental determination of the activation parameters could have been performed accurately, it should not be pretended to possess excessive accuracy. Caution is advised especially with entropies of activation, due to the inherent imprecision of the method (ordinates at the origin from least-squares line fitting) ... 

In the example given, the line coincides exactly with the least-squares line calculated for estimating y from x. 

Tests were run by two different laboratories on samples of emulsion polymer. Determine the least-squares line for estimating the expected result for laboratoty II, given the value obtained by laboratory I, and the expected result for laboratory I given the value obtained by laboratory II. Are these two lines the same ... 

Least significant difference, 22 Least squares line, 36,37 Linear programming, 65... 

The uppermost line is a least-square line drawn through the experimental data obtained on five separate dilatometer samples, each prepared by Procedure II and irradiated at three dose rates at least. The slope of this line is 0.48 zb 0.07, and again represents polymerization rates that are higher than any of the other values reported here or elsewhere (3, 16). 

The blank absorbance was 0.038 at 562 nm in a 1.000-cm cell. A serum sample had an absorbance of 0.129. After the blank was subtracted from each standard absorbance, the points in Figure 18-8 were obtained. The least-squares line through the standard points is... 

The complexes contributing to the least squares line in Figure 44 contain either two-coordinate donor atoms with very little steric hindrance (for example, acetylacetonate, oxalate, dithiocarbamates or xanthates), or tetrahedrally coordinated donor atoms (for example, 1,2-diaminoethane or 1,3-diaminopropane). The agreement between the theoretical curve and the experimental line is very good. The observation that the line of best fit lies 0.5° below the calculated line may be... 

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