Statistical population parameter

We need to make a decision related to the disposition of soil that has been excavated from the subsurface at a site with lead contamination history. Excavated soil suspected of containing lead has been stockpiled. We may use this soil as backfill (i.e. place it back into the ground), if the mean lead concentration in it is below the action level of 100 milligram per kilogram (mg/kg). To decide whether the soil is acceptable as backfill, we will sample the soil and analyze it for lead. The mean concentration of lead in soil will represent the statistical population parameter. 

The mean x is the statistic corresponding to the population parameters I, which is the arithmetic average of all the items in the popiila-... 

Finite or infinite set of individuals (ob- Frank and Todeschini [1994] jects, items). A population implicitly con- —> Sample (in the statistical tains all the useful information for cal- sense) culating the true values of the population parameters , e.g., the mean p and the standard deviation o. 

Maximum (maximized) likelihood is a statistical term that refers to the probability of randomly drawing a particular sample from a population, maximized over the possible values of the population parameters. Selected entries from Methods in Enzymology [vol, page(s)] Theory, 210, 203 testing by simulations, 210, 225 computer applications for, 210, 233 fitting of sums of exponentials to dwell-time distributions, 207, 772 fluorescence data analysis, 210,... 

The calculation of mean and standard deviation only really makes sense when we are dealing with continuous, score or count data. These quantities have little relevance when we are looking at binary or ordinal data. In these situations we would tend to use proportions in the various categories as our summary statistics and population parameters of interest. 

The statistical methods discussed up to now have required certain assumptions about the populations from which the samples were obtained. Among these was that the population could be approximated by a normal distribution and that, when dealing with several populations, these have the same variance. There are many situations where these assumptions cannot be met, and methods have been developed that are not concerned with specific population parameters or the distribution of the population. These are referred to as non-parametric or distribution-free methods. They are the appropriate methods for ordinal data and for interval data where the requirements of normality cannot be assumed. A disadvantage of these methods is that they are less efficient than parametric methods. By less efficient is meant... 

Stereochemical equilibration of DCP in DMSO at 343 K in the presence of LiCI yields a mixture containing 36.4 I + 0.3) % of the meso isomer. The statistical weight parameters evaluated from this result are used for theoretical calculation of the proportions of various conformers in meso and racemic DCP, and also in the three diastereoisomers of TCH. Calculations for TCH are compared with estimates of others for NMR coupling constants. It is shown that the less-favoured conformations, often ignored, contribute appreciably to the conformer populations of... 

From the table of random numbers take 20 different sample data with 10 random numbers. Determine the sample mean and sample variance for each sample. Calculate the average of obtained statistics and compare them to population parameters. 

Estimation involves the calculation of numerical values for the various population parameters (mean, variance, and so on). These numerical values are only estimates of the actual parameters, but statistical procedures permit us to establish the accuracy of the estimate. 

A statistical hypothesis is simply a statement concerning the probability distribution of a random variable. Once the hypothesis is stated, statistical procedures are used to test it, so that it may be accepted or rejected. Before the hypothesis is formulated, it is almost always necessary to choose a model that we assume adequately describes the underlying population. The choice of a model requires the specification of the probability distribution of the population parameters of interest to us. When a statistical hypothesis is set up, then the corresponding statistical procedure is used to establish whether the proposed hypothesis should be accepted or rejected. Generally speaking, we are not able to answer the question whether a statistical hypothesis is right or wrong. If the information from the sample taken supports the hypothesis, we do not reject it. However, if those data do not back the statistical hypothesis set up, we reject it. 

Statistical estimation uses sample data to obtain the best possible estimate of population parameters. The p value of the Binomial distribution, the p value in Poison s distribution, or the p and a values in the normal distribution are called parameters. Accordingly, to stress it once again, the part of mathematical statistics dealing with parameter distribution estimate of the probabilities of population, based on sample statistics, is called estimation theory. In addition, estimation furnishes a quantitative measure of the probable error involved in the estimate. As a result, the engineer not only has made the best use of this data, but he has a numerical estimate of the accuracy of these results. 

The best point estimate depends upon the criteria by which we judge the estimate. Statistics provides many possible ways to estimate a given population parameter, and several properties of estimates have been defined to help us choose which is best for our purposes. 

The planning team specifies the parameter of interest, which is the statistical parameter that characterizes the population. The parameter of interest provides a reasonable estimate of the true contaminant concentration, and it may be the mean, median or percentile of a statistical population. 

Various methods are available to estimate population parameters, but today the nonlinear mixed effects modeling approach is the most common one employed. Population analyses have been performed for mAbs such as basiliximab, daclizu-mab and trastuzumab, as well as several others in development, including clenolixi-mab and sibrotuzumab. Population pharmacokinetic models comprise three submodels the structural the statistical and covariate submodels (Fig. 3.13). Their development and impact for mAbs will be discussed in the following section. 

On many occasions, sample statistics are used to provide an estimate of the population parameters. It is extremely useful to indicate the reliability of such estimates. This can be done by putting a confidence limit on the sample statistic. The most common application is to place confidence limits on the mean of a sample from a normally distributed population. This is done by working out the limits as F— ( />[ i] x SE) and F-I- (rr>[ - ij x SE) where //>[ ij is the tabulated critical value of Student s t statistic for a two-tailed test with n — 1 degrees of freedom and SE is the standard error of the mean (p. 268). A 95% confidence limit (i.e. P = 0.05) tells you that on average, 95 times out of 100, this limit will contain the population... 

The confidence interval of the slope can be estimated by the t test. It is important to realize that the quantities a and b in the regression line y = a + bx are statistics that are estimates of the population parameters a and p. As the number of observations is increased without Unfit, a and b approach a and fi. 

To the statistician the process of sampling consists of drawing from a population a finite number of units to be examined. From sample statistics, such as mean and standard deviation, estimates are made of the population parameters. By appropriate tests of significance, confidence limits are placed on the estimates. Sampling for chemical analysis is an example of statistical sampling in that conclusions are drawn about the composition of a much larger bulk of material from an analysis of a limited sample. 

In the mixed-effects context, the collection of population parameters is composed of a population-typical value (generally the mean) and of a population-variability value (generally the variance-covariance matrix). The mean and variance are the first two moments of a probability distribution. They build a minimal set of hyperparameters or population characteristics for it, which is sufficient (in a statistical sense) when F is taken as normal or log-normal. 

It is impossible to conduct an infinite number of extractions of a speciman to determine the accuracy of a method. As a result, we estimate the accuracy of an assay by performing a finite number of extractions (n) on the specimen. We report the accuracy as the mean (x- = Hxifn, i = 1,2,. ..,n) of the multiple determinations, expressed as a percent of the known concentration. The finite group of determinations is a sample from the population, and its mean is referred to as the sample mean. The sample mean is a statistic that estimates the population parameter p. If we could obtain the means from an infinite number of same-size samples, regardless of their size, then the mean of these infinite sample means would equal p. In statistical terminology, we say that the sample mean is an unbiased estimator of the population mean. Unbiasedness is a... 

The chi-square table (Table 5) is entered according to the significance level (e.g., p = 0.05) and the degrees of freedom (df) for the calculated statistic. The degrees of freedom for the goodness-of-fit test are the total number of intervals less one. If, as in our case, population parameters for the assumed... 

A data set is often considered as a sample from a population and the sample parameters calculated from the data set as estimates of the population parameters (-> statistical indices). Moreover, it is usually used to calculate statistical models such as quantitative -> structure/response correlations. In this case the data set is organized into a data matrix X with n rows and p columns, where each row corresponds to an object of the data set and each column to a variable therefore each element represent the value of the yth variable for the ith object (/ = 1,. .., n j = 1,. .., p). 

We can return to the data presented in Table 1 for the analysis of the mineral water. If the parent population parameters, a and po, are known to be 0.82 mg kg- and 10.8 mg kg" respectively, then can we answer the question of whether the analytical results given in Table 1 are likely to have come from a water sample with a mean sodium level similar to that providing the parent data. In statistic s terminology, we wish to test the null hypothesis that the means of the sample and the suggested parent population are similar. This is generally written as... 

The various quantities mentioned in Section 7.3.1 used to describe data sets are referred to as sample statistics. They describe various aspects of the sample data. In estimation, they are used to estimate the corresponding population value, which is known as the population parameter. Hence estimation is actually parameter estimation . 

However, in practice one has just the sample statistic - such as p or X The most likely value for p is H the most likely value for X is p. Thus, the sample statistic gives a best guess of the population parameter this... 

Such a data set has a mean and a SD. The mean of the data set of sample statistics will be 11, the population parameter on average the sample statistics will be g and hence p from any sample is an unbiased estimator of g. The variation of the sample statistics, p, can be described in the same way as for any data set the SD of the distribution of sample statistics is known as the standard error (SE) (of the estimate) - here it would be SEp. 

It follows from the fact that the sampling distribution is normally distributed that 95% of the sample statistics will be within 1-96 SDs of the mean (the population parameter). The SD of the sampling distribution (as mentioned above) is referred to as the SE of the estimate. Because only 5% of sample statistics will be more than 1-96 SEs from the population parameter, for any sample statistic taken at random it is 95% likely that the population parameter is within 1-96 SEs of the sample statistic. This is the rationale for the calculation of confidence intervals (Cis) in estimation. So, a 95% CI is found by the expression ... 

In addition, it is very useful to estimate the difference between the population parameters using the difference between the sample statistics. So, the actual difference seen in the data gives a point estimate of the difference. In the data shown in Table 7.12 in which P = 0-017, the difference in percentage success between the two samples is 16-0% (82-7% - 66-7%), which gives a point estimate of the true difference. 

One of the primary goals of Statistics is to use data from a sample to estimate an unknown quantity from an underlying population, called a population parameter. In general, we typically use the arithmetic mean as the measure of central tendency of choice because the sample mean is an unbiased estimator of the population mean, typically represented by the symbol p. The main conceptual point about unbiased estimators is that they come closer to estimating the true population parameter, in this case the population mean, than biased estimators. When extreme observations influence the value of the mean such that it really is not representative of a typical value, use of the median is recommended as a measure of central tendency. 

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