Two-sample

The excess heat of solution of sample A of finely divided sodium chloride is 18 cal/g, and that of sample B is 12 cal/g. The area is estimated by making a microscopic count of the number of particles in a known weight of sample, and it is found that sample A contains 22 times more particles per gram than does sample B. Are the specific surface energies the same for the two samples If not, calculate their ratio. 

The ability to demonstrate that two samples have different amounts of analyte is an essential part of many analyses. A method s sensitivity is a measure of its ability to establish that such differences are significant. Sensitivity is often confused with a method s detection limit. The detection limit is the smallest amount of analyte that can be determined with confidence. The detection limit, therefore, is a statistical parameter and is discussed in Chapter 4. 

A measure of a method s ability to distinguish between two samples reported as the change in signal per unit change in the amount of analyte k). 

In most circumstances, populations are so large that it is not feasible to analyze every member of the population. This is certainly true for the population of circulating U.S. pennies. Instead, we select and analyze a limited subset, or sample, of the population. The data in Tables 4.1 and 4.10, for example, give results for two samples drawn at random from the larger population of all U.S. pennies currently in circulation. 

The result of an analysis is influenced by three factors the method, the sample, and the analyst. The influence of these factors can be studied by conducting a pair of experiments in which only one factor is changed. For example, two methods can be compared by having the same analyst apply both methods to the same sample and examining the resulting means. In a similar fashion, it is possible to compare two analysts or two samples. 

Unpaired Data Consider two samples, A and B, for which mean values, Xa and Ab, and standard deviations, sa and sb, have been measured. Confidence intervals for Pa and Pb can be written for both samples... 

A certain analytical method has a relative sampling variance of 0.40% and a relative method variance of 0.070%. Evaluate the relative error (a = 0.05) if (a) you collect five samples, analyzing each twice and, (b) you collect two samples, analyzing each five times. 

The design of a collaborative test must provide the additional information needed to separate the effect of random error from that due to systematic errors introduced by the analysts. One simple approach, which is accepted by the Association of Official Analytical Chemists, is to have each analyst analyze two samples, X and Y, that are similar in both matrix and concentration of analyte. The results obtained by each analyst are plotted as a single point on a two-sample chart, using the result for one sample as the x-coordinate and the value for the other sample as the -coordinate. ... 

A two-sample chart is divided into four quadrants, identified as (-P, -p), (-, -p), (-, -), and (-P, -), depending on whether the points in the quadrant have values for the two samples that are larger or smaller than the mean values for samples X and Y. Thus, the quadrant (-P, -) contains all points for which the result for sample X is larger than the mean for sample X, and for which the result for sample Y is less than the mean for sample Y. If the variation in results is dominated by random errors. 

Typical two-sample plot when (a) random errors are larger than systematic errors due to the analysts and (b) systematic errors due to the analysts are larger than the random errors. 

A visual inspection of a two-sample chart provides an effective means for qualitatively evaluating the results obtained by each analyst and of the capabilities of a proposed standard method. If no random errors are present, then all points will be found on the 45° line. The length of a perpendicular line from any point to the 45° line, therefore, is proportional to the effect of random error on that analyst s results (Figure 14.18). The distance from the intersection of the lines for the mean values of samples X and Y, to the perpendicular projection of a point on the 45° line, is proportional to the analyst s systematic error (Figure 14.18). An ideal standard method is characterized by small random errors and small systematic errors due to the analysts and should show a compact clustering of points that is more circular than elliptical. 

The data used to construct a two-sample chart can also be used to separate the total variation of the data, Otot> into contributions from random error. Grand) and systematic errors due to the analysts, Osys. Since an analyst s systematic errors should be present at the same level in the analysis of samples X and Y, the difference, D, between the results for the two samples... 

Relationship between point In a two-sample plot and the random error and systematic error due to the analyst. 

As part of a collaborative study of a new method for determining the amount of total cholesterol in blood, two samples were sent to ten analysts with instructions to analyze each sample one time. The following results, in milligrams of total cholesterol per 100 mb of serum, were obtained... 

A two-sample plot of the data is shown in Figure 14.19, with the average value for sample 1 shown by the vertical line at 245.9, and the average value for sample 2 shown by the horizontal line at 243.5. To estimate Grand and Gjys, it ... 

Two-sample plot for data In Example 14.7. The analyst responsible for each data point Is Indicated by the associated number. The true values for the two samples are Indicated by the. ... 

When the true values for the two samples are known, it is possible to test for the presence of systematic errors in the method. If no systematic method errors occur, then the sum of the true values for the samples, )J,tot... 

The two samples analyzed in Example 14.7 are known to contain the following concentrations of cholesterol. 

In the two-sample collaborative test, each analyst performs a single determination on two separate samples. The resulting data are reduced to a set of differences, D, and a set of totals, T, each characterized by a mean value and a standard deviation. Extracting values for random errors affecting precision and systematic differences between analysts is relatively straightforward for this experimental design. 

Collaborative testing provides a means for estimating the variability (or reproducibility) among analysts in different labs. If the variability is significant, we can determine that portion due to random errors traceable to the method (Orand) and that due to systematic differences between the analysts (Osys). In the previous two sections we saw how a two-sample collaborative test, or an analysis of variance can be used to estimate Grand and Osys (or oJand and Osys). We have not considered, however, what is a reasonable value for a method s reproducibility. 

Chichilo reports the following data for the determination of the %w/w A1 in two samples of limestone. ... 

The two-sample plot for the data in Example 14.7 is shown in Figure 14.19. Identify the analyst whose work is (a) the most accurate (b) the most precise (c) the least accurate and (d) the least precise. 

Construct a two-sample plot for these data, and estimate values for Grand and O ys assuming a = 0.05. 

Two samples taken from a single gross sample and used to evaluate an analytical method s precision. 

Figure 48.2 shows that a compares the ratio of atom or ion abundances for two isotopes in each of two samples. If a is not equal to 1, then the isotopes in one sample must have a different ratio from those in the other. If isotopes behave chemically almost identically on a universal stage,... 

For two samples A and B measured against the same reference standard X, the relationship between the a and 5 values can be approximated by the expression shown in Equation 48.1 (see also Figure 48.5). 

Dual viscous-flow reservoir inlet. An inlet having two reservoirs, used alternately, each having a leak that provides viscous flow. This inlet is used to obtain precise comparisons of isotope ratios in two samples. 

M is a small fractional order, this can amount to a considerable effect over a wide range of M values. For example, in the limit described in item (2), where a a, two samples of the same polymer showing a 1000-fold range of M will differ in a—as well as and (rg /M) -by a factor of 2. This... 

Fig. 9. Uptake curves for N2 in two samples of carbon molecular sieve showing conformity with diffusion model (eq. 24) for sample 1 (A), and with surface resistance model (eq. 26) for example 2 (0)j LDF = linear driving force. Data from ref. 18.

Flavor Description. TypicaHy, a sensory analyst determines if two samples differ, and attempts to explain their differences so that changes can be made. The Arthur D. Litde flavor profile (FP), quantitative descriptive analysis (QDA), and spectmm method are three of the most popular methods designed to answer these and more compHcated questions (30—33). AH three methods involve the training of people in the nominal scaling of the flavor quaHties present in the food being studied, but they differ in their method for quantitation. 

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