Slope and Intercept

The confidence interval Cl(b) serves the same purpose as Cl(Xmean) in Section 1.3.2 the quality of these average values is described in a manner that is graphic and allows meaningful comparisons to be made. An example from photometry, see Table 2.2, is used to illustrate the calculations (see also data file UV.dat) further calculations, coiiunents, and interpretations are found in the appropriate Sections. Results in Table 2.3 are tabulated with more significant digits than is warranted, but this allows the reader to check 

BI- AND MULTIVARIATE DATA Table 2.1. Linear Regression Equations 

Correct formulation (sums E are taken over all measurements, / = 1... n)  

Algebraically equivalent formulation as used in pocket calculators (beware of numerical artifacts when using Eqs. (2.7-2.9) cf. Table 1.1)  

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A plot of P/n versus P should give a straight line, and the two constants and b may be evaluated from the slope and intercept. In turn, n may be related to the area of the solid ... 

Find the slope and intercept of a straight line not passing through the origin of the data set... 

If we go back and calculate the slope and intercept for the data set in Exercise 3-2 without the constraint that the line must pass through the origin, we get the solution vector 0.95, 0.09 for a line parallel to the line in Exercise 3-3 and 2.0 units distant from it, as expected. 

Plotting the left side of Eq. (3-22) as a function of gives a curve with as the slope and E° as the intercept. Ionic interference causes this function to deviate from lineality at m 0, but the limiting (ideal) slope and intercept are approached as OT 0. Table 3-1 gives values of the left side of Eq. (3-22) as a function of The eoneentration axis is given as in the corresponding Fig. 3-1 beeause there are two ions present for each mole of a 1 -1 electrolyte and the concentration variable for one ion is simply the square root of the concentration of both ions taken together. 

Fit a linear equation to the following data set and give the uncertainties of the slope and intercept (Fig. 3-2). 

The scatter of the points around the calibration line or random errors are of importance since the best-fit line will be used to estimate the concentration of test samples by interpolation. The method used to calculate the random errors in the values for the slope and intercept is now considered. We must first calculate the standard deviation Sy/x, which is given by ... 

Example 14 For the best-fit line found in Example 13, express the result in terms of confidence intervals for the slope and intercept. We will choose 95% for the confidence interval. 

When the value of c exceeds unity, the value of n can be derived from the slope and intercept of the BET plot in the usual way but because of deviations at low relative pressures, it is sometimes more convenient to locate the BET monolayer point , the relative pressure (p/p°) at which n/n = 1. First, the value of c is found by matching the experimental isotherm against a set of ideal BET isotherms, calculated by insertion of a succession of values of c (1, 2, 3, etc., including nonintegral values if necessary) into the BET equation in the form ... 

A more useful representation of uncertainty is to consider the effect of indeterminate errors on the predicted slope and intercept. The standard deviation of the slope and intercept are given as... 

The standard deviation about the regression, Sr, suggests that the measured signals are precise to only the first decimal place. For this reason, we report the slope and intercept to only a single decimal place. 

Equations 5.13 for the slope, h, and 5.14 for the y-intercept, ho, assume that indeterminate errors equally affect each value of y. When this assumption is false, as shown in Figure 5.11b, the variance associated with each value of y must be included when estimating [3o and [3i. In this case the predicted slope and intercept are... 

This is the equation of a straight line, so rj and r2 can be evaluated fron the slope and intercept of an appropriate plot. 

What makes Eq. (8.87) especially important is that it is the equation of a straight line. It predicts that a plot of n/RTc2 versus C2 will be linear for dilute solutions and that the slope and intercept of the plot will have the following significance ... 

From plots of n/c2 versus C2, evaluate M for each of the four polymer fractions. Do the data collected from the two different solvents conform to expectations with respect to slope and intercept values ... 

Interpret the slope and intercept values of the line in Fig. 10.12 in terms of the molecular weight and radius of gyration of cellulose nitrate in this solution. At 436 nm the refractive index of acetone is 1.359. 

A plot of log([A]g j[A — 1) versus log[B], called a Schild plot, yields a straight line of unit slope and intercept of iCg, the latter often expressed on a scale analogous to that for pH, so that = logif (46—48). 

The variables that are combined hnearly are In / 17T, and In C, Multilinear regression software can be used to find the constants, or only three sets of the data smtably spaced can be used and the constants found by simultaneous solution of three linear equations. For a linearized Eq. (7-26) the variables are logarithms of / C, and Ci,. The logarithmic form of Eq. (7-24) has only two constants, so the data can be plotted and the constants read off the slope and intercept of the best straight line. 

TABLE 8-3 Tuning Rules Using Slope and Intercept... 

Then vkt is calculated from the vX values as (-ln(l-vX)). The independent function Temperature vx is expressed as 1000 K/vT for the Arrhenius function. Finally the independent variable vy is calculated as In(vkt). Next a linear regression is executed and results are presented as y plotted against Xi The results of regression are printed next. The slope and intercept values are given as a, and b. The multiple correlation coefficient is given as c. 

I.D. The data has been curve fitted to a linear function and thus the enthalpy and entropy contributions are extracted as the slope and intercept of each curve. 

Thus, if the slope and intercept of the curve relating the reciprocal of the corrected retention volume to the concentration of the moderator are (p) and (tj)) respectively. 

Thus, from the slopes and intercepts of these two plots, the constants ki, k, and A an can be evaluated. It was found that ki and A a were not significantly different from zero. 

A plot of l/k against 1/(H ] should be linear from the slope and intercept and k can be evaluated. 

Instead of the definition in Eq. (7-82), the selectivity is often written as log k,). Another way to consider a selectivity-reactivity relationship is to compare the relative effects of a series of substituents on a pair of reactions. This is what is done when Hammett plots are made for a pair of reactions and their p values are compared. The slope of an LEER is a function of the sensitivity of the process being correlated to structural or solvent changes. Thus, in a family of closely related LFERs, the one with the steepest slope is the most selective, and the one with the smallest slope is the least selective.Moreover, the intercept (or some arbitrarily selected abscissa value, usually log fco for fhe reference substituent) should be a measure of reactivity in each reaction series. Thus, a correlation should exist between the slopes (selectivity) and intercepts (reactivity) of a family of related LFERs. It has been suggested that the slopes and intercepts should be linearly related, but the conditions required for linearity are seldom met, and it is instead common to find only a rough correlation, indicative of normal selectivity-reactivity behavior. The Br nsted slopes, p, for the halogenation of a series of carbonyl compounds catalyzed by carboxylate ions show a smooth but nonlinear correlation with log... 

A series of experiments with different values of [B] is done. Then k and k- are obtained as the slope and intercept of a plot of ke versus [B], The quotient of slope divided by intercept gives the equilibrium constant. 

From Eqs. (3-71) and (3-73), it follows that plots of k versus [OH-] and of the product ratio versus l/[OH ] will be straight lines. Their slopes and intercepts allow calculation of the three rate constants (see Problem 3-13). 

This equation is given in a linearized form. The slopes and intercepts allow one to calculate the desired ratios of rate constants. 

Solution The classic way of fitting these data is to plot ln( /7 ) versus T" and to extract and Tact from the slope and intercept of the resulting (nearly) straight line. Special graph paper with a logarithmic j-axis and a l/T A-axis was made for this purpose. The currently preferred method is to use nonlinear regression to fit the data. The object is to find values for kQ and Tact that minimize the sum-of-squares ... 

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