Constant Sample Size Research Articles

Theoretical study of system peaks and elution profiles for large concentration bands in the case of a binary eluent containing a strongly sorbed additive

The profile of system peaks when using high concentrations and a binary mobile phase was studied using the semi-ideal model of chromatography. The profile of the sample band depends on the concentration of the strong solvent and on the sample size. It is very strongly influenced by the relative adsorption strength of the strong solvent and the sample. This relative strength is measured by the ratio of (a) the origin slope of the equilibrium isotherm of the strong solvent between the stationary phase and the pure weak solvent to (b) the origin slope of the isotherm of the sample ( i.e., the ratio between their respective column capacity factors in the pure weak solvent). When this ratio is smaller than 0.2, the sample band profile depends only on the equilibrium isotherm of the sample in the binary mixture. If the ratio becomes larger, the competition between the strong solvent and the sample molecules for interaction with the stationary phase becomes more intense. For Langmuir-competitive isotherms of the strong solvent and the sample, if the concentration of the strong solvent and/or its strength are progressively increased at constant sample size, the retention time of the band decreases (as expected), whereas its typical Langmuirian asymmetry decreases and, past a narrow transition range, reverses. Bands traditionally associated with an anti-Langmuir isotherm are then obtained. Extremely unusual, broad, characteristic profiles, with a sharp front and a sharp rear, which sometimes exhibit two maxima, are obtained in the transition region. When the origin slope ratio is large, the profile of a high-concentration-sample band in the transition region has two shock layers. The occurrence of such band profiles, under experimental conditions similar to those predicted here, have been reported previously, but their origin has remained unexplained until now.

Experimental study of systm peaks and elution profiles for large concentration bands in the case of a binary eluent containing a strongly sorbed additive

Band profiles of high-concentrations samples of single compounds eluted from a silica column by a binary mobile phase were recorded under various sets of experimental conditions. The mobile phase was a dilute solution 0.02–2%, v/v) of lights alkanols in dichloromethane or n-hexane. When the concentration of the strong solvent increases at a constant sample size, or the sample size increases at a constant strong solvent concentration, the elution profile, which is first of the type associated with a Langmuir adsorption isotherm with a sharp front and a slanted tail, changes the direction of its asymmetry and becomes similar in shape to the profiles associated with an anti-Langmuir isotherm, with a slanted front and a sharp tail. The shape of the profiles in the transition range is unusual and appears similar to the composite of two or three different bands. Finally, at large sample sizes and high strong solvent concentrations, the profile develops a hump on its front, which eventually may turn into a second peak. These experimental profiles are in excellent qualitative agreement with those predicted by theory and described in the accompanying paper. Their variations with the experimental parameters follow the trends predicted in that study. The competition between the molecules of the strong solvent (or modifier) and those of the samples is entirely responsible for these profiles.

Sampling of Powders and Bulk Materials

1 Introduction.- 2 Fundamentals of Statistics.- 2.1 Frequency distribution of discrete attributes.- 2.2 Cumulative frequency of discrete attributes.- 2.3 Arithmetic mean and sample variance for discrete distributions.- 2.4 Frequency distribution of continuous attributes.- 2.5 Expectation and variance of one-dimensional probability functions.- 2.6 Two-dimensional distribution functions.- 2.7 Expectations and variances of a two-dimensional distribution function.- 2.8 Conditional expectations.- 2.9 Rules for calculating expectations and variances.- 2.9.1 One-dimensional distribution functions.- 2.9.2 Two-dimensional distribution functions.- 2.9.3 Conditional distribution functions.- 3 Random Sampling Distributions.- 3.1 Concept of the random sample.- 3.2 Sampling without replacement.- 3.3 Random sampling with replacement.- 3.3.1 Binomial replacement.- 3.3.2 Poisson distribution.- 3.3.3 Normal distribution.- 4 Sampling from a Population Having an Arbitrary Distribution of the Attribute.- 4.1 Sampling with replacement.- 4.2 Sampling without replacement.- 5 Inference from the Sample About the Population (Confidence Intervals).- 5.1 Sampling when the variance of the population is known.- 5.2 Normally distributed populations with known variance ?2.- 5.3 Sampling from normally distributed populations with unknown variance.- 5.3.1 Chi-square distribution (confidence interval for sample variances.- 5.3.2 The t distribution (confidence intervals for means when the variance is unknown.- 6 Sampling Procedures.- 6.1 Random and systematic sampling.- 6.2 Stratified sample.- 6.3 Proportional subdivision a stratified sample.- 6.4 Cost-optimizing selection for a stratified sample.- 6.5 Example of stratified sampling.- 6.6 Multi-stage sampling.- 6.7 Cost-optimizing selection in multi-stage sampling.- 6.8 Example of multi-stage sampling.- 6.9 Double sampling for bulk material.- 7 Sampling from a Random Mixture.- 7.1 Sampling with a constant number n of particles from a two-material mixture.- 7.1.1 Variance of the numerical concentration.- 7.1.2 Variance of the concentration by mass.- 7.2 Sampling with a constant sample size n from a multi-material mixture.- 7.3 Sampling with a constant mass (or volume) from a two-component mixture.- 7.3.1 The problem of a random mixture formed from two even-grained fractions of any size.- 7.3.2 Random mixture of two even-grained fractions.- 7.4 Constant sample mass and the random mixture of several components of unequal grain size.- 7.5 Random mixture of two particle-size distributions.- 7.6 Random mixture for suspensions.- 8 Sampling in a Sample Divider.- 8.1 Numerical concentration in the case of two or more components.- 8.2 Variance of the concentration by mass of two or more components having differing even-grain fractions.- 8.3 Variance of the concentrations by number and by mass for a two-material mixture.- 8.4 Experiments on a rotary sample-divider from Messrs. Retsch.- 9 Sampling for Grain-Size Analysis.- 9.1 Object and method of sampling.- 9.2 Sampling error for sampling with a constant particle number n from a numerical distribution Q0.- 9.3 Sampling error for sampling with a constant particle number n from a cumulative volume distribution Q3.- 9.4 Sampling error of a cumulative volume distribution Q3 for constant sample masses of solid in the sample.- 9.5 Sampling error of a numerical distribution Q0 or of a cumulative volume distribution Q3 in sampling with a constant suspension volume VSus from a suspension.- 9.6 Sampling error in sample-dividing.- 9.6.1 Cumulative numerical distribution Q0.- 9.6.2 Cumulative volume distribution Q3.- 10 Sampling Error when Sampling from Ores and Fuels.- 11 Investigations of Random Packings.- 11.1 Definitions and formulation of the problem.- 11.2 Relationship between area porosity and volume porosity.- 11.3 Test for random packing.- 11.4 Grain size analysis by determination of the diameters of sectioned solid particles.- 11.4.1 Determination of all moments Mn,o of the sphere diameter distribution from the moments Mk,o of the section circle diameter distribution.- 11.4.2 Solution of the integral equation.- 11.4.3 Determination of sphere diameter distributions qo (d) with the aid of a histogram of section circle diameters.- 12 Sampling from Non-Random Mixtures.- 12.1 Systematic variance.- 12.2 Correlation length, correlogram.- 13 Sampling from a Convevor Belt.- 13.1 Variance within and between belt sections of lenght L.- 13.2 Inspection of given quantities when sampling randomly or systematically from a conveyor belt.- 14 Sampling Devices.- 14.1 Devices for sampling liquids.- 14.2 Devices for sampling gases and dusts.- 14.3 Devices for sampling granular and lumpy material.- 14.4 Sample splitters.- Notes.- Literature.

Two classes of models for magnitude estimation

One class of models assumes that presentation of a signal results in an internal representation as a random variable. Depending on whether the signal is close to or far from the preceding signal, the variance of the representation is smaller or larger. Responses are determined largely by this random variable; however, when the signal is close to the preceding one, the response is generated by modifying the representation multiplicatively by some function of the ratio of the previous response to its representation. Power and linear functions are explored. The form of the random variable is assumed to be that arising from either the timing or the counting model operating on a Poisson process. Detailed analyses are carried out successfully only for the timing model with neural sample sizes independent of intensity; however, the data require the sample to increase with intensity. The linear response function coupled with the constant sample size counting model appears somewhat viable, but detailed calculations are very difficult to carry out. The second class of models postulates a power function relation between magnitude estimates and signals intensity for which the exponent is a Gaussian distributed random variable and the unit is the product of two log normal random variables. Again we assume an attention band such that succesive stimuli that are widely separated in intensity lead to independent samples of the random variables while a variety of assumptions is explored for successive stimuli that are near each other in intensity. Although they each give rise to the qualitative features of the data, estimates of parameters are sufficiently inconsistent that we are led to reject all of the submodels studied.

On the asymptotic sampling distribution of sediment weight fractions

A commonly measured variable in sediment studies is that part of the total sample weight that is contributed by grains possessing some particular attribute. This would include, for example, the sample weight of sediment within a given size fraction, or the sample weight contributed by a mineral distributed through the sediment. Despite the widespread use of such variables, little appears to be known of their sampling distributions. The purpose of this pai~er is to establish conditions for the asymptotic normality of the fraction weights with respect to a number of sediment sampling procedures. By "asymptotic" is meant either (1) sample size tending to infinity as in the case of constant sample size, or (2) expected sample size tending to infinity for variable sample size. The size,N, of a given sample refers here to the total number of sediment particles in that sample. Whether N is a constant or a variable will depend on the nature of the sampling procedure.

Sampling by Variables to Control the Fraction Defective: Part II

The paper, as a sequel to the historical and theoretical approach of Part I, provides the practical information on the matching of variables sampling plans to those in the existing attributes standard. Of the many possible types of matching possible, two types are tabulated here. The first is a match on two points of the OC curve, namely at the AQL and P.10 points. The second is a one-point match at the AQL with constant sample size for each Code Letter. The k values obtained in this second method are compared with those of MIL-STD-414.

Visual detection in relation to display size and redundancy of critical elements I

Visual detection was studied in relation to displays of discrete elements, randomly selected consonant letters, distributed in random subsets of cells of a matrix, the S being required on each trial to indicate only which member of a predesignated pair of critical elements was present in a given display. Experimental variables were number of elements per display and number of redundant critical elements per display. Estimates of the number of elements effectively processed by a S during a 50 ms. exposure increased with display size, but not in the manner that would be expected if the S sampled a fixed proportion of the elements present in a display of given area. Test-retest data indicated substantial correlations over long intervals of time in the particular elements sampled by a S from a particular display. Efficiencies of detection with redundant critical elements were very close to those expected on the hypothesis of constant sample size over trials for any given display size and were relatively invariant with respect to distance between critical elements.

Attributenkeuring in theorie en praktijk*

SummaryTheory and practice of sampling by attributes.This paper brings a survey of present day theory and practice of sampling by attributes. In section 2 the main points in the theory of OC‐curves are recapitulated. Section 3 lists the various features which people have attempted to incorporate in sampling tables, and the various factors which influence our choice of a sampling plan. Subsequent sections discuss economic theories, the importance of the distribution of the precentage of defective items over the inspection lots, the relation between sample size and lot size, and some of the existing sampling tables. In a final section the possibilities for further improvements in existing sampling practices is considered. One conclusion is that in many situations a constant sample size regardless of lot size would have very definite advantages.

On the Bulk Compression Characteristics of Wool Fibers

A method has been found to prepare bulk samples of wool fibers in such a way that reproducible compression tests may be performed upon them. An evaluation of the bulk compression characteristics of 29 widely different wool samples shows that compressive load, rather than resilience, serves to bring out differences among them. This finding suggests that quality differences among wools, as determined by handling, are related to differences in the wools' resistance to compression rather than to differences in compressional resilience. For these 29 wool samples there is an inverse relationship between compressive load and mean fiber diameter. Although this finding is in agreement with similar results re ported for cotton fibers, it conflicts with the predictions from a theoretical model that has been proposed to explain the compressional behavior of wool. The theoretical model, which was based on a consideration of bending forces only during compression, has been claimed to fit results found for 310 samples of Merino wool. There is, therefore, an implication either that there is a different dependency of resistance to compression on fiber diameter within a wool breed as compared to that among different breeds or that the proposed theory is inadequate. When the compressing piston size is varied at a constant sample size for a Targhee 60's wool card sliver, it is found that the effective volume of fibers being compressed is greater than the volume of fibers beneath the piston, probably because of fiber-to-fiber entanglements. The experimental results indicate that a constant area should be added to the compressing piston areas in order to achieve a constant compressive stress. This area increment is independent of sample diameter, providing the sample diameter is suf ficiently greater than that of the compressing piston, and this area increment decreases with increasing degree of compression.