Sampling error random

With small sample sizes the uncertainty due to random sampling error usually is large and may become the dominant source of uncertainty in the output. This uncertainty could be reduced if there is relevant prior information, for example, reasonable estimates for distribution parameters from well-described datasets (Aldenberg and Luttik 2002). 

An early theory for description of the random sampling error in well mixed particulate materials with particle-size-dependent composition is described by BENEDETTI-PICH-LER [1956],... 

Sample uncertainty is also referred to as statistical random sampling error. This type of uncertainty is often estimated assuming that data are sampled randomly and without replacement and that the data are random samples from an unknown population distribution. For example, when measuring body weights of different individuals, one might randomly sample a particular number of individuals and use the data to make an estimate of the interindividual variability in body weight for the entire population of similar individuals (e.g. for a similar age and sex cohort). 

Probability distribution models can be used to represent frequency distributions of variability or uncertainty distributions. When the data set represents variability for a model parameter, there can be uncertainty in any non-parametric statistic associated with the empirical data. For situations in which the data are a random, representative sample from an unbiased measurement or estimation technique, the uncertainty in a statistic could arise because of random sampling error (and thus be dependent on factors such as the sample size and range of variability within the data) and random measurement or estimation errors. The observed data can be corrected to remove the effect of known random measurement error to produce an error-free data set (Zheng Frey, 2005). 

If a parametric distribution (e.g. normal, lognormal, loglogistic) is fit to empirical data, then additional uncertainty can be introduced in the parameters of the fitted distribution. If the selected parametric distribution model is an appropriate representation of the data, then the uncertainty in the parameters of the fitted distribution will be based mainly, if not solely, on random sampling error associated primarily with the sample size and variance of the empirical data. Each parameter of the fitted distribution will have its own sampling distribution. Furthermore, any other statistical parameter of the fitted distribution, such as a particular percentile, will also have a sampling distribution. However, if the selected model is an inappropriate choice for representing the data set, then substantial biases in estimates of some statistics of the distribution, such as upper percentiles, must be considered. 

From these results, it is clear that the random sampling error (the between-group variance) is statistically significantly different compared with the random analytical error (the within-groups variance). 

Describe the way in which sample size and the SD jointly influence random sampling error... 

Show howthe standard errorofthe mean (SEM) can be used to indicate the likely extent of random sampling error... 

WHAT FACTORS CONTROL THE EXTENT OF RANDOM SAMPLING ERROR ... 

In this world, nothing is certain but death, taxes and random sampling error. (Modified from Benjamin Franklin s original.)... 

Two factors control the extent of random sampling error. One is almost universally recognized, but the other is less widely appreciated. 

Most of us recognize that, the larger a sample is, the more likely it is to reflect the true underlying situation. However, it does not matter how big a sample is, we must always anticipate some random sampling error. 

A sample may mis-estimate the population mean as a result of bias or random sampling error. Bias is a predictable over- or under-estimation, arising from poor experimental design. Random error arises due to the unavoidable risk that any randomly selected sample may over-represent either low or high values. 

The extent of random sampling error is governed by the sample size and the SD of the data. Small samples are subject to greater random error than large ones. Data with a high SD are subject to greater sampling error than that with low variability. 

Null hypothesis - there is no real effect. The apparent difference arose from random sampling error. If we could investigate larger and larger numbers of subjects, the mean clearances for the two groups would eventually settle down to the same value. In stat speak the difference between the population mean clearances is zero . 

The statistical wet blanket. Whatever interesting feature has been noted in our sample (a change in an end-point or a relationship between two sets of data) is assumed not to be present in the general population. The apparent change or relationship is claimed to be due solely to random sampling error. 

Figure 6.2 Null hypothesis random sampling error produced the appearance of an effect, although none was really present...

We have met our first hypothesis test .The two-sample f-test is used to determine whether two samples have produced convincingly different mean values or whether the difference is small enough to be explained away as random sampling error. The data in each sample are assumed to be from populations that followed normal distributions and had equal SDs. 

We create a null hypothesis that there is no real experimental effect and that for large samples, the mean value of the end-point is exactly the same for both treatments. According to this hypothesis, random sampling error is responsible for any apparent difference and with extended sampling the apparent difference would eventually evaporate. 

Is the difference between these two sample means greater than can reasonably be accounted for by random sampling error ... 

The mechanism implied by the null hypothesis is also just an extension of that discussed in relation to the /-test. It is assumed that the five catalysts are in reality indistinguishable, but within these small samples, random sampling error has led to an illusion of variability in their effectiveness. Presumably, the effectiveness of some catalysts has been overestimated and/or that of others understated. 

Differences between the sample means (inter-group variation) - small differences may only reflect random sampling error, but if the sample means differ widely, a significant conclusion is more likely. 

The sample data may suggest that the new device is superior to the old one, but these are only samples and the apparent difference could have arisen as a result of random sampling error. We therefore need to set up and test a null hypothesis. 

The book s illustrative examples are all taken from the pharmaceutical sciences, so students (and staff) in the areas of pharmacy, pharmacology and pharmaceutical science should feel at home with all the material. However, the issues considered are of concern in most scientific disciplines and should be perfectly clear to anybody from a similar discipline, even if the examples are not immediately familiar. Material is arranged in a developmental manner. Initial chapters are aimed at level 1 students this material is fairly basic, with special emphasis on random sampling error. The next section then covers key concepts that may be introduced at levels 1 or 2. The final couple of chapters are most likely to be useful during final year research projects These include one on questionnaire design and analysis. 

The first term on the right-hand side of Eq. 4.56, the random-sampling error, reflects the variation in In CMD from sample to sample, assuming that each sample is taken from the same stable aerosol, without any experimental error. The term o e reflects the variation in In CMD that results from experimental error due to measurement. In most particle sizing situations N is large, so random sampling errors are small compared with experimental errors. The confidence interval (Cl) for In CMD is given by... 

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