Sample size approximate formula

For sample sizes beyond about 30 the multiplying constant for the 95 per cent confidence interval is approximately equal to two. Sometimes for reasonably large sample sizes we may not agonise over the value of the multiplying constant and simply use the value two as a good approximation. This gives us an approximate formula for the 95 per cent confidence interval as (3c — 2se, x + 2se). 

When sampling to determine particle size distribution, the sampling variation will generally be different for each class size. The formula must therefore be recalculated for each class. For this application, cl in (B.3) represents the true content fraction of the class of interest. The value c, of an individual particle is either 1 or 0 depending on whether the ith particle is in this class or not. As in the general case discussed above, size and density approximations are used to... 

A general, simplified formula for approximating the sample size necessary for detection of a given deviation A of slope from unity or intercept from zero is ... 

Note that the sample size depends not only on the value of A, hut also on the individual proportions themselves. The implication of this is that the sponsor must make a reasonable estimate of the response in the placebo group (that is, PpLACEBo) then postulate a value of A that is clinically relevant. The corresponding value of p.pgg.p can be obtained by subtraction. This first sample size estimate ( ) can be improved through the use of a continuity correction, which gives more accurate results when a discrete distribution (in this case the binomial distribution) is used to approximate a continuous distribution (in this case the normal). The sample size formula with continuity correction is ... 

We shall present an approximate formula for sample size determination. An exact formula introduces complications which need not concern us. In discussing the sample size requirements we shall use the following conventions ... 

We are now in a position to consider the (approximate) formula for sample size, which is... 

The first complication is that the formula is only approximate. It is based on the assumption that the test of significance will be carried out using a known standard deviation. In practice we do not know the standard deviation and the tests which we employ are based upon using an estimate obtained from the sample under study. For large sample sizes, however, the formula is fairly accurate. In any case, using the correct, rather than the approximate, formula causes no particular difficulties in practice. 

The above formula assumes that the sample size n is much smaller than the total population size N, so that population can be considered infinite in size. When the sampling fraction is large (approximately 10 % or more) the estimate of the error must be corrected by multiplying by a finite population correction (FPC) [2]... 

This formula for estimating droplet size was determined experimentally. Of the various terms, the first is the most important for small values of V. As V becomes small, the second term gains in importance. Unless the density or viscosity of the sample solution changes markedly from the values for water, mean droplet size can be estimated approximately by using the corresponding values for water, as shown. 

Once the conditions for making a processible catalyst slurry were established, microspheres were generated using the reference formula with and without the different alumina powders. Table II summarizes particle size distributions for typical commercial FCC and our experimental catalysts. Two commercial samples (Catalysts V and VI) had particle sizes much higher than we could produce with our spray dryer. However, the size distributions of the experimental catalysts approximate those of the other four commercial grades. 

For solids, we can use a formula for the variance of the FE to design a sampling protocol to do this. Using experimental results and mathematical approximations, Gy (1992, p. 362) has derived a formula that estimates this variance, Var(FE). It is a function of the mass M, of the lot, the mass Ms of the sample, and the size, density, and content characteristics of the material ... 

Pitard (4) presents two formulas relating an estimate of the Var(FE) to the weight of the sample and the panicle size. One case is fairly general the other is for panicle size distribution. In each case, physical characteristics of the panicles are used a size factor and a density factor. By characterizing the type of material being sampled, we can determine if our sample weight is sufficient to get a desired low variance of the estimate, and if not, what we can to do reduce that variance. Because the characterizations are made on a preliminary examination of the material, these formulas are an order of approximation only. 

The basic description of a mesoporous sample requires two types of determinations X-ray diffraction and gas adsorption/dcsorption isotherm. The latter are usually represented as the amount of gas adsorbed by the sample as the function of relative pressure. This information characterizes pore size distribution. Nitrogen adsorption/desorplion isotherm at 77 K is most often used and relatively convenient to carry out. The adsorption of noble gases is used if accurate in-depth pore characterization is attempted, especially quantitative. The calculation of pore size distribution from the isotherms is carried out using appropriate formulas such as Kelvin and IIorwath-Kawazoe equations (e.g. as in Ref. 5 and [6]), which involve assumptions and approximations. A more detailed and rigorous treatments have been developed, as for example KJS (Kruk-Jaroniec-Sayari), which is relatively simple and accurate [42]. In practice, the diameter of mesopores can be quickly estimated directly from the position of the capillary condensation or, if not vertical, the p/p0 of the inflection point. The conversion table of p/po values to pore diameters can be found in Ref. [43] and is partially reproduced here in Table 2. 

The peak broadening can be attributed to nanosized ZrC crystallites. This ZrC phase was formed by the reaction between Zr and B C but also consumed by reaction (15) between 1100 and 1500 °C, so ZrC particles could not coarsen during this stage. The particle size of ZrC was estimated using the Scherrer formula (Equation 17) (Chamberlain et al, 2006), where p is the full-width athalf-maximum (radians), X the Cu Ka wavelength (1.54056 A), and 0 the peak position (radians). A ZrC particle size of approximately 20 nm was estimated from the broadening and position of the (111) diffraction peak in the XRD patterns of samples 5-8. 

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