Randomized samples

A random number (between 0 and 1) is picked, and the associated value of gross reservoir thickness (T) is read from within the range described by the above distribution. The value of T close to the mean will be randomly sampled more frequently than those values away from the mean. The same process is repeated (using a different random number) for the net-to-gross ratio (N/G). The two values are multiplied to obtain one value of net sand thickness. This is repeated some 1,000-10,000 times, with each outcome being equally likely. The outcomes are used to generate a distribution of values of net sand thickness. This can be performed simultaneously for more than two variables. 

Selection of film systems for the random sample control measurements... 

Do we expect this model to be accurate for a dynamics dictated by Tsallis statistics A jump diffusion process that randomly samples the equilibrium canonical Tsallis distribution has been shown to lead to anomalous diffusion and Levy flights in the 5/3 < q < 3 regime. [3] Due to the delocalized nature of the equilibrium distributions, we might find that the microstates of our master equation are not well defined. Even at low temperatures, it may be difficult to identify distinct microstates of the system. The same delocalization can lead to large transition probabilities for states that are not adjacent ill configuration space. This would be a violation of the assumptions of the transition state theory - that once the system crosses the transition state from the reactant microstate it will be deactivated and equilibrated in the product state. Concerted transitions between spatially far-separated states may be common. This would lead to a highly connected master equation where each state is connected to a significant fraction of all other microstates of the system. [9, 10]... 

The coarse-graining approach is commonly used for thermodynamic properties whereas the systematic or random sampling methods are appropriate for static structural properties such as the radial distribution function. 

There will be incidences when the foregoing assumptions for a two-tailed test will not be true. Perhaps some physical situation prevents p from ever being less than the hypothesized value it can only be equal or greater. No results would ever fall below the low end of the confidence interval only the upper end of the distribution is operative. Now random samples will exceed the upper bound only 2.5% of the time, not the 5% specified in two-tail testing. Thus, where the possible values are restricted, what was supposed to be a hypothesis test at the 95% confidence level is actually being performed at a 97.5% confidence level. Stated in another way, 95% of the population data lie within the interval below p + 1.65cr and 5% lie above. Of course, the opposite situation might also occur and only the lower end of the distribution is operative. 

The control chart is set up to answer the question of whether the data are in statistical control, that is, whether the data may be retarded as random samples from a single population of data. Because of this feature of testing for randomness, the control chart may be useful in searching out systematic sources of error in laboratory research data as well as in evaluating plant-production or control-analysis data. ... 

It will be convenient to deal first with the distribution aspect of the problem. One of the clearest ways in which to represent the distribution of sizes is by means of a histogram. Suppose that the diameters of SOO small spherical particles, forming a random sample of a powder, have been measured and that they range from 2-7 to 5-3 pm. Let the range be divided into thirteen class intervals 2-7 to 2-9 pm, 2-9 to 3-1 pm, etc., and the number of particles within each class noted (Table 1.5). A histogram may then be drawn in which the number of particles with diameters within any given range is plotted as if they all had the diameter of the middle of the... 

The amount of aspirin in the analgesic tablets from a particular manufacturer is known to follow a normal distribution, with p, = 250 mg and = 25. In a random sampling of tablets from the production line, what percentage are expected to contain between 243 and 262 mg of aspirin ... 

A randomly collected sample makes no assumptions about the target population, making it the least biased approach to sampling. On the other hand, random sampling requires more time and expense than other sampling methods since a greater number of samples are needed to characterize the target population. 

A sampling plan that divides the population into distinct strata from which random samples are collected. 

This experiment introduces random sampling. The experiment s overall variance is divided into that due to the instrument, that due to sample preparation, and that due to sampling. 

James L. Unmack, "A Comparison of Periodic Versus Random Sampling From an Information Theory Point of View," presented at CMA Exposure Assessment Workshop, Washington, D.C., 1986. 

The basic underlying assumption for the mathematical derivation of chi square is that a random sample was selected from a normal distribution with variance G. When the population is not normal but skewed, square probabihties could be substantially in error. 

If the null hypothesis is assumed to be true, say, in the case of a two-sided test, form 1, then the distribution of the test statistic t is known. Given a random sample, one can predict how far its sample value of t might be expected to deviate from zero (the midvalue of t) by chance alone. If the sample value oft does, in fact, deviate too far from zero, then this is defined to be sufficient evidence to refute the assumption of the null hypothesis. It is consequently rejected, and the converse or alternative hypothesis is accepted. 

Application. A company has received a very large shipment of rivets. One product specification required that no more than 2 percent of the rivets have diameters greater than 14.28 mm. Any rivet with a diameter greater than this would be classified as defective. A random sample of 600 was selected and... 

The assumption in step 1 would first he tested hy obtaining a random sample. Under the assumption that p <. 02, the distrihiition for a sample proportion would he defined hy the z distrihiition. This distrihiition would define an upper hound corresponding to the upper critical value for the sample proportion. It would he unlikely that the sample proportion would rise above that value if, in fact, p <. 02. If the observed sample proportion exceeds that limit, corresponding to what would he a very unlikely chance outcome, this would lead one to question the assumption that p <. 02. That is, one would conclude that the null hypothesis is false. To test, set... 

For all practical purposes, source testing can be considered as simple random sampling (2). The source may be considered to be composed of such a large population of samples that the populahon N is infinite. From this population, n units are selected in such a manner that each unit of the population has an equal chance of being chosen. For the sample, determine the sample mean, y ... 

Like XPS, the application of AES has been very widespread, particularly in the earlier years of its existence more recently, the technique has been applied increasingly to those problem areas that need the high spatial resolution that AES can provide and XPS, currently, cannot. Because data acquisition in AES is faster than in XPS, it is also employed widely in routine quality control by surface analysis of random samples from production lines of for example, integrated circuits. In the semiconductor industry, in particular, SIMS is a competing method. Note that AES and XPS on the one hand and SIMS/SNMS on the other, both in depth-profiling mode, are complementary, the former gaining signal from the sputter-modified surface and the latter from the flux of sputtered particles. 

Iman, R. L. and M. J. Shortencarrier, 1984, A FORTRAN 77 Program and User s Guide for the Generation of Latin Hypercube and Random Samples for Use with Computer Model, NUREG/CR-3624, March. 

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