Confidence Intervals sample characteristics

Figure 12.10 Confidence intervals in dependence of sample size (a) fora and (b) forthe Weibull modulus. The confidence intervals forthe characteristic strength depend on the modulus.

The use of confidence intervals is one way to state the required precision. Confidence limits provide a measure of the variability associated with an estimate, such as the average of a characteristic. Table I is an example of using confidence intervals in planning a sampling study. This table shows the interrelationships of variability (coefficient of variation), the distribution of the characteristic (normal or lognormal models), and the sample frequency (sample sizes from 4 to 365) for a monitoring program. 

As can be seen from Table 1, the estimated coefficients b[0] are not equal to zero for different samples, whereas the estimated coefficients b[l] are close to 1 within confidence interval. That means that coefficients b[0] estimated for different points of the territory are generalized relative characteristics of elements abundance at the chosen sampling points. Statistical analysis has confirmed that hypotheses Hi and H2 are true with 95% confidence level for the data obtained by any of the analytical groups involved. This conclusion allowed us to verify hypothesis H3 considering that the estimated average variances of the correlation equation (1) are homogeneous for all snow samples in each analytical group. Hypothesis H3... 

The frequency interpretation of the interval estimates on the unknown amounts is given by the following ( 27 ) With at least 1- a confidence, based on the sampling characteristics of the observations on the standards, at least P proportion of the interval estimates made from a particular calibration will contain the true amounts. The Bonferroni inequality insures the 1-a confidence since the confidence interval about the regression line and the upper bound on cr are each performed using a 1- a/2 confidence coefficient. Hence, the frequency interpretation states that at least (1-a) proportion of the standard calibrations are such that at least P proportion of the intervals produced by the method cover the true unknown amounts. For the remaining a proportion of standard calibrations the proportion of intervals which cover the true unknown values may be less than P. 

From the formula for a confidence interval, its width is determined by three parameters the sample size, population variability and the degree of confidence. Plainly, if the sample size is increased then we have seen the standard error will be reduced and hence the width of the interval will also be reduced. If we can reduce the variability of the characteristic being studied then... 

Another important characteristic of the normal distribution is that 95% of the data values he within 2s, and 99.7% in the range 3s, as shown in Figure 16.2. This distribution of error in a normal distribution allows the calculation of confidence intervals for x. The confidence interval is the range of concentration within which the real sample concentration is expected to occur, for a given degree of confidence. Therefore, the interval size depends of the degree of confidence (for greater... 

The second reason for potential deviations between analysis and bulk mass is the inevitable statistical fluctuation of any analytical value. Even in an ideal mixture the characteristic to be determined (e.g. the percentage passing at a particular dimension) will result in different values for each sample. The results of measurements will fluctuate around a mean value whereby results near the mean value are most frequently obtained. There is a probability smaller than unity that a result will be within a certain range around the mean value. Often, the confidence interval for 95% probability is used, which means that 95 out of 100 measurements will produce values within the range. The relative width of this interval depends critically on sample size and the nature of the characteristic to be measured—the larger the sample the narrower the interval. Detailed information is provided in textbooks on statistics and quality control.For example, for the determination of number populations a rule of thumb requires that the sample must consist of at least 1000 or better 10 000 particles. 

Output analysis assistance Automation of output analysis is certainly a characteristic of the new generation of simulation systems. Features that compare or relate data from replications or probable scenarios provide significant support to the user in reaching conclusions. Assistance in steady state analysis, confidence interval estimation, automatic stopping rules, sample size determination, and so on is very desirable. 

This chapter discusses several statistical principles that are used in pharmaceutical quality decisions, such as normal distribution, rounding, confidence interval, standard deviation, outliers, operating characteristic curves, acceptance sampling. Examples have been embedded in a pharmaceutical context. 

Before extracting statistical characteristics from a measurement series, such as mean value, standard deviation, confidence interval, etc., it has to be checked whether the data are proper with respect to certain criteria. For instance, the sample may contain extremely deviating measurements owing to a gross error (e.g., simply a typing error or improper measurement conditions). Such values are called outliers and must be removed form further consideration as they strongly falsify the characteristics of the sample. The same argumentation... 

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