Population confidence limit

The model of simple competitive antagonism predicts that the slope of the Schild regression should be unity. However, experimental data is a sample from the complete population of infinite DR values for infinite concentrations of the antagonist. Therefore, random sample variation may produce a slope that is not unity. Under these circumstances, a statistical estimation of the 95% confidence limits of the slope (available in most... 

The confidence interval for a given sample mean indicates the range of values within which the true population value can be expected to be found and the probability that this will occur. For example, the 95% confidence limits for a given mean are given by... 

Population confidence interval The limits on either side of a mean value of a group of observations which will, in a stated fraction or percent of the cases, include the... 

The mean, the standard deviation, and the confidence limits of the population at each concentration with multiple measurements are calculated and tabulated. 

In summary, for any stated value of the population correlation (p) the z statistic is denoted as Z(p), and the corresponding correlation confidence limits can be determined. For our example, the Z statistic of 0.6366 corresponding to the lower correlation coefficient confidence limit is shown in the graphic below (Graphic 60-6a) as having a p value of 0.562575 this represents the lower confidence limit for the correlation coefficient for this example. 

The question that must be asked is, given the total number of animals actually assayed was relatively small in comparison with the total number of animals produced, how significant are the results Based on the statistical design and the observation that the range (95% confidence limits) will reflect closely the realities of the animal population sampled, the estimate of the frequency of residues in any given slaughter class is a good estimate. 

Normally the population standard deviation a is not known, and has to be estimated from a sample standard deviation s. This will add an additional uncertainty and therefore will enlarge the confidence interval. This is reflected by using the Student-t-distribution instead of the normal distribution. The t value in the formula can be found in tables for the required confidence limit and n-1 degrees of freedom. 

Cost will be the primary controlling factor In the experimental design In essentially all cases. The cost of obtaining the entire population of possible measurements Is prohibitive. The field study design must select a subset which Is representative of the population within definable confidence limits and which can be obtained within the constraints of existing resources. 

For noncarcinogenic hazardous chemicals, NCRP believes that the threshold for deterministic effects in humans should be estimated using EPA s benchmark dose method, which is increasingly being used to establish allowable doses of noncarcinogens. A benchmark dose is a dose that corresponds to a specified level of effects in a study population (e.g., an increase in the number of effects of 10 percent) it is estimated by statistical fitting of a dose-response model to the dose-response data. A lower confidence limit of the benchmark dose (e.g., the lower 95 percent confidence limit of the dose that corresponds to a 10 percent increase in number of effects) then is used as a point of departure in establishing allowable doses. 

Although rarely presented in a dose-response assessment, in nearly all cases the lower bound on the incremental probability of a response will be zero or less (see Figure 3.7). That is, the statistical model that accounts for the uncertainty in the results of an animal study also accommodates the possibility that no response may occur at low doses and that, in fact, there may be fewer responses (e.g., cancers) than observed in the control population at some low doses. The possibility of reduced responses at low doses also is shown by the lower confidence limit of data on radiation-induced cancers in some organs of humans including, for example, the pancreas, prostate, and kidney (Thompson et al., 1994). 

Student s t-test is frequently used in statistical evaluations of environmental chemical data. It establishes a relationship between the mean (x) of normally distributed sample measurements, their sample standard deviation (,v), and the population mean (p). Confidence intervals may be calculated based on Student s t-test (Equation 10). The upper limit of the confidence interval is compared to the action level to determine whether the sampled medium contains a hazardous concentration of a pollutant. If the upper confidence limit is below the action level, the medium is not hazardous otherwise the opposite conclusion is reached. 

A less obvious (but far more common) abuse of statistics is their use to analyze health risks. For example, we know with high accuracy the average incidence of any of hundreds of different subtypes of cancer, based on reporting by doctors over the last several decades. Suppose I select 100 towns at random, analyze the incidence of 100 different types of cancer in each of these towns over a decade, and compare these incidences to the known averages with 95% confidence limits. Out of these 10,000 combinations, on average 500 will be outside the limits and will be statistically significant About 250 combinations of one town and one disease will be statistically high, and will terrify the local population when the... 

We could simply turn a blind eye to this non-normality and proceed to generate a 95 per cent Cl in the usual way and the results would be a mean of 95.9 with confidence limits of —23.1 to +215.0 ng/g. These are shown in Figure 5.12(a). The result is clearly nonsensical - the true population mean could not possibly take the sort of negative value implied by the lower limit of the confidence interval. 

The use of the chi-squared test in one case and the use of the F-test in another case to test variances is not surprising when it is realized that these two tests are interrelated. To find the confidence limits for the variance of a normal population, the following definition of the chi-squared is used ... 

The objective of statistical samphng is to establish likely values for the true error rate in the population of data being considered. If the tme error rate was known, the probabihties of given numbers of errors in samples could be obtained mathematically using standard statistical distributions. Statistical inference allows the reverse process — from an observed error rate in a sample likely and possible true error rates can be inferred. Likely data population error rates are defined by the 99% single upper confidence limit, and possible data population error rates by the 99.9% single upper confidence limit on the sample error rate. 

---