Sampling distribution of the

I. Under the null hypothesis, it is assumed that the respective two samples have come from populations with equal proportions pi = po. Under this hypothesis, the sampling distribution of the corresponding Z statistic is known. On the basis of the observed data, if the resultant sample value of Z represents an unusual outcome, that is, if it falls within the critical region, this would cast doubt on the assumption of equal proportions. Therefore, it will have been demonstrated statistically that the population proportions are in fact not equal. The various hypotheses can be stated ... 

As ToF-SIMS also allows the mapping of the chemical species inside the sample, distributions of the different extractives in the cross-section of the wood are evaluated. They show clearly that hinokinin is predominantly localized in parenchyma cells other extractives are distributed randomly in both parenchyma and tracheid cells (Figure 15.8c). This could be very helpful in understanding the heartwood formation mechanism. 

Fig. 8.5 Comparison of population histogram and sampling distribution of the mean of blood glucose levels.

In general, bias refers to a tendency for parameter estimates to deviate systematically from the true parameter value, based on some measure of the central tendency of the sampling distribution. In other words, bias is imperfect accuracy. In statistics, what is most often meant is mean-unbiasedness. In this sense, an estimator is unbiased (UB) if the average value of estimates (averaging over the sampling distribution) is equal to the true value of the parameter. For example, the mean value of the sample mean (over the sampling distribution of the sample mean) equals the mean for the population. This chapter adheres to the statistical convention of using the term bias (without qualification) to mean mean-unbiasedness. 

Figure 2.6 Sampling distribution of the mean x data from N(80,16), sample size n...

The relevant path is constructed from sampling distributions of the general form prescribed in Eq. (34) with... 

In most analytical experiments where replicate measurements are made on the same matrix, it is assumed that the frequency distribution of the random error in the population follows the normal or Gaussian form (these terms are also used interchangeably, though neither is entirely appropriate). In such cases it may be shown readily that if samples of size n are taken from the population, and their means calculated, these means also follow the normal error distribution ( the sampling distribution of the mean ), but with standard deviation sj /n this is referred to as the standard deviation of the mean (sdm), or sometimes standard error of the mean (sem). It is obviously important to ensure that the sdm and the standard deviation s are carefully distinguished when expressing the results of an analysis. 

If the assumption of a Gaussian error distribution is considered valid, then an additional method of expressing random errors is available, based on confidence levels. The equation for this distribution can be manipulated to show that approximately 95% of all the data will lie within 2 5 of the mean, and 99.7% of the data will lie within 3i of the mean. Similarly, when the sampling distribution of the mean is considered, 95% of the sample means will lie within approximately 2sj /n of the population mean etc. (Figure 5). 

The Central-Limit Theorem states that the sampling distribution of the mean, for any set of independent and identically distributed random variables, will tend toward the normal distribution, equation (3.17), as the sample size becomes large. ... 

Thus, the sampling distribution of the mean becomes approximately normal regardless of the distribution of the original variable, and the sampling distribution of the mean is centered at the population mean of the original variable. In addition, the standard deviation of the sampling distribution of the mean approaches... 

For any continuous random variable X which has a distribution with population mean, p, and variance, the sampling distribution of the mean for samples of size n has a distribution with population mean, p, and variance. 

The assumption of a normal distribution for the random variable X is somewhat restrictive. However, for any random variable, as the sample size increases, the sampling distribution of the sample mean becomes approximately normally distributed according to a mathematical result called the central limit theorem. For a random variable X that has a population mean, p, and variance, ct-, the sampling distribution of the mean of samples of size n (where n is large, that is, > 200) will have an approximately normal distribution with population mean, p, and variance, cs-ln. Using the notation described earlier, this result can be summarized as ... 

The second component is the standard error of the mean, which quantifies the extent to which the process of sampling has mis-estimated the population mean. The standard error of the mean has the same meaning as in the case for normally distributed data - that is, the standard error describes the degree of uncertainty present in our assessment of the population mean on the basis of the sample mean. It is also the standard deviation of the sampling distribution of the mean for samples of size n. The smaller the standard error, the greater the certainty with which the sample mean estimates the population mean. When ri is very large the standard error is very small, and therefore the sample mean is a very precise estimate of the population mean. As we know the standard deviation of the sample, s, we can make use of the following formula to determine the standard error of the mean, SE ... 

To summarize, the computational aspects of confidence intervals involve a point estimate of the population parameter, some error attributed to sampling, and the amount of confidence (or reliability) required for interpretation. We have illustrated the general framework of the computation of confidence intervals using the case of the population mean. It is important to emphasize that interval estimates for other parameters of interest will require different reliability factors because these depend on the sampling distribution of the estimator itself and different calculations of standard errors. The calculated confidence interval has a statistical interpretation based on a probability statement. 

In Chapter 6 we described the basic components of hypothesis testing and interval estimation (that is, confidence intervals). One of the basic components of interval estimation is the standard error of the estimator, which quantifies how much the sample estimate would vary from sample to sample if (totally implausibly) we were to conduct the same clinical study over and over again. The larger the sample size in the trial, the smaller the standard error. Another component of an interval estimate is the reliability factor, which acts as a multiplier for the standard error. The more confidence that we require, the larger the reliability factor (multiplier). The reliability factor is determined by the shape of the sampling distribution of the statistic of interest and is the value that defines an area under the curve of (1 - a). In the case of a two-sided interval the reliability factor defines lower and upper tail areas of size a/2. 

This is also a confidence interval for the parameter p, probability of success, of the binomial distribution. The use of the Z distribution for this interval is made possible because of the Central Limit Theorem. Consider the random variable X taking on values of 0 or 1, such that the sampling distribution of the sample mean (the proportion) is approximately normally distributed. A table of the most commonly encountered values of the standard normal distribution is provided in Table 8.3 for quick reference. Others are provided in Appendix 1. 

An estimate of the sample distribution of this test statistic under the null hypothesis has to be derived to perform a test of the form described above. This can be achieved by using the bootstrap to obtain the sample distribution of the differences of the objective function given the observation. For this method bootstrap data sets are constructed, and for each bootstrap data set the parameters are estimated and the objective functions are reported for each of the competing models. The confidence interval for the differences of the objective functions is calculated and if this interval does not include 0 then the null hypothesis that the models are equal would be rejected. The percentile method for computing the bootstrap confidence interval as described by Efron (19) is used, and 1000 bootstrap replicates are required for this. 

To execute this, an estimate of the sample distribution of the LED under the null hypothesis must be derived to perform a test. The bootstrap method for estimating sample distribution of the difference of the objective function given the observations is used to solve the problem. This allows one to reject the null hypothesis of equal noncentrality parameters, that is, of equality of fit if zero is not contained in the confidence interval so derived. One thousand bootstrap pseudosamples were constructed, the nonhierarchical models of interest were applied, and the percentile method for computing the bootstrap confidence intervals was used. 

Wolfe, J. H. A Monte Carlo study of the sampling distribution of the likelihood ratio for mixtures of multinormal distributions, 1971. 

The adequacy of ( ) can be monitored by recording various distributions observed during the run. As an example, Figure 5 shows the overall sampled distribution of the sum Y.t an( also (coarse) histograms of... 

After obtaining an estimator of a population parameter (/a, o) or a parameter associated with a particular family of distributions such as A in an exponential family, the sampling distribution of the estimator is the distribution of the estimator as its possible realizations vary across all possible samples that may arise from a given population. For example, let 0 be a parameter for a population or for a family of distributions. Let Ti,..., y be i.i.d. r.v s with c.d.f. F completely unspecified or where 0... 

Thus, there were only minor differences in the mean and standard deviation for the sampling distribution of the median when comparing 200 bootstrap samples to 20,000 bootstrap samples. However, note the big discrepancies between the quantiles. When generating 20,000 samples of size 11 from the original dataset, samples were obtained in which the median of the bootstrap sample was equal to the minimum value (1600) in the original dataset. Because the bootstrap median equals 0 = X (6), this result implies that, in the bootstrap samples having median = 1600, at least 6 of the 11 data values must be equal to 1600. This seems very unlikely. However, if we calculate the expected number of samples in the 20,000 samples having exactly 6 of their 11 values equal to 1600, we find ... 

A plot of the quantile function, kernel density estimator of the p.d.f., a box plot, and a normal reference distribution plot for the sampling distribution of the sample quantile are given in Figures 2.13 and 2.14 for 200 and 20,000 bootstrap samples. We note that there are considerable differences in the plots. The plots for 20,000 bootstrap samples reveal the discreteness of the possible values for the median when the sample size (n = 11 in our case) is very small. Also, we note that n = 11 is too small for the sampling distribution for the median to achieve its asymptotic result (n large), an approximate normal distribution. 

To be able to calculate n, given some , requires a value for the standard deviation of the individual unit measurements under a specified sampling and testing condition. The relationship between n and E is derived from the sampling distribution of the mean and the t-distribution as discussed in Section 3.4. Thus rearranging the usual form of the equation. 

The decision on statistical significance is based on the sampling distribution of the ratio of the difference between two means to the standard deviation of such differences as given by Student s /-distribution for the general case of unequal number of values for each mean. 

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