Variance sample size

Chi-Square Distribution For some industrial applications, produrt uniformity is of primary importance. The sample standard deviation. s is most often used to characterize uniformity. In dealing with this problem, the chi-square distribution can be used where = (.s /G ) (df). The chi-square distribution is a family of distributions which are defined by the degrees of freedom associated with the sample variance. For most applications, df is equal to the sample size minus 1. 

The population of differences is normally distributed with a mean [L ansample size is 10 or greater in most situations. 

The procedure for testing the significance of a sample proportion follows that for a sample mean. In this case, however, owing to the nature of the problem the appropriate test statistic is Z. This follows from the fact that the null hypothesis requires the specification of the goal or reference quantity po, and since the distribution is a binomial proportion, the associated variance is [pdl — po)]n under the null hypothesis. The primary requirement is that the sample size n satisfy normal approximation criteria for a binomial proportion, roughly np > 5 and n(l — p) > 5. 

Although this is a complicated expression, the results can be given a simple interpretation. The data sample size is n, whereas the prior sample size is Ko, and therefore J. is the weighted average of the prior data and the actual data. al is the weighted average of the prior variance (Go), the data variance (5-), and a tenn from the difference in the prior... 

The most useful description of variance are confidence limits since these take into account the sample size. 

A slight modification of the pilot study is a double-sampling plan for estimation (9,10). Double-sampling plans were developed to provide estimates with a fixed precision using as few observations as possible. If the sources of variation and variance components are known prior to the study, then a fixed sample size plan is the best... 

Fig. 5.3. Comparison of different free energy estimators. Plotted are distributions of estimated free energies using sample sizes (i.e., number of independent simulation runs) of N = 100 simulations (solid lines), as well as N = 1, 000 (long dashed) and N = 10,000 simulations short dashed lines), (a) Exponential estimator, (5.44). (b) Cumulant estimator using averages from forward and backward paths, (5.47). (c) Cumulant estimator using averages and variances from forward and backward paths, (5.48). (d) Bennett s optimal estimator, (5.50)...

In the equations above, the mean square error, the sample variance, and the finite sampling bias are all explicitly written as functions of the sample size N. Both the variance and bias diminish as /V — oc (infinite sampling). However, the variance... 

A multivariate normal distribution data set was generated by the Monte Carlo method using the values of variances and true flowrates in order to simulate the process sampling data. The data, of sample size 1000, were used to investigate the performance of the robust approach in the two cases, with and without outliers. 

ANOVA is robust for moderate departures from equality of variances (as determined by Bartlett s test) if the sample sizes are approximately equal. 

The sample size and sample allocation scheme is obtained in one of two ways. Either the cost is fixed and the variance of the mean is minimized or the variance is fixed and the cost is minimized. The first approach is used when the budget for sampling and analysis is determined in advance. The objective in this case is to use that budget to obtain an estimate with maximum precision (equivalently minimum variance). The second approach is used when the required precision of the estimator is specified in advance. Then the objective is to derive an estimator with the desired level of precision at the lowest possible cost. 

Overall sampling quality — quantitative analysis. For dynamical trajectories, the "structural decorrelation time" analysis [10] can estimate the slowest timescale affecting significant configuration-space populations and hence yield the effective sample size. For non-dynamical simulations, a variance analysis based on multiple runs is called for [1]. Analyzing the variance in populations of approximate physical states appears to be promising as a benchmark metric. 

Consider statistical approaches to estimate variance on the basis of sample size. 

The size required of the sample to identify a meaningful economic difference is frequently problematic. Often those setting up clinical trials focus on the primary clinical question when developing sample-size estimates. They fail to consider the fact that the sample required to address the economic questions posed in the trial may differ from that needed for the primary clinical question. In some cases the sample size required for the economic analysis is smaller than that required to address the clinical question. More often, however, the opposite is true, in that the variances in cost and patient preference data are larger than those for clinical data. Then one needs to confront the question of whether it is either ethical... 

In tong term trials there will usually be an opportunity to check the assumptions which underlay the original design and sample size calculations. This may be particularly important if the trial specifications have been made on preliminary and/or uncertain information. An interim check conducted on the blinded data may reveal that overall response variances, event rates or survival experience are not as anticipated. A revised sample size may then be calculated using suitably modified assumptions... ... 

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