Upper confidence interval

The 95% CIs are 95% lower and upper confidence intervals on the survival probability. 

To use the worksheet, we must first collect a number of stockpile characterization samples based either on regulatory guidance or on a specific sampling scheme with a certain budget in mind. Analytical results for these samples are entered into the worksheet as the mean concentration the 95 percent upper confidence interval is calculated. A minimum number of samples is calculated in Worksheet Step 11 Step 12 allows determining if a sufficient number of samples have been collected. If not, additional sampling may be necessary. Examples 2.3 and 5.13 illustrate these calculations. 

A baseline risk assessment is conducted to assess the potential human health and environmental impacts associated with soil contamination. The primary exposure pathways evaluated for explosives contaminated surface soils are dust inhalation, soil ingestion, and dermal absorption. Reasonable Maximum Exposure (RME) concentrations are based on the 95% upper confidence interval (UCI) on the arithmetic mean of soil sampling data. The land use scenarios quantitatively evaluated may include industrial and residential use, utilizing EPA standard default exposure parameters. 

Figure 14.1 Compass plots for penetrants in propylene glycol mixtures in PSFT. The interactions noted in this plot at left for triazine and phenol were positive and outside of the upper confidence interval for significant interactions (p <.05). One cell Is expanded to illustrate upper and lower bounds.

Model Final Initial Lower Upper Confidence Interval 95% Standard... 

As to the mean value, the following limits were achieved minimum value = 258 Mh, lower confidence interval 2.5% = 315 Mh, upper confidence interval 97.5% = 480 Mh, mean value = 385 Mh, Median = 381 Mh, Maximum = 746 Mh. 

There will be incidences when the foregoing assumptions for a two-tailed test will not be true. Perhaps some physical situation prevents p from ever being less than the hypothesized value it can only be equal or greater. No results would ever fall below the low end of the confidence interval only the upper end of the distribution is operative. Now random samples will exceed the upper bound only 2.5% of the time, not the 5% specified in two-tail testing. Thus, where the possible values are restricted, what was supposed to be a hypothesis test at the 95% confidence level is actually being performed at a 97.5% confidence level. Stated in another way, 95% of the population data lie within the interval below p + 1.65cr and 5% lie above. Of course, the opposite situation might also occur and only the lower end of the distribution is operative. 

The confidence limits for the slope are given by fc where the r-value is taken at the desired confidence level and (A — 2) degrees of freedom. Similarly, the confidence limits for the intercept are given by a ts. The closeness of x to X is answered in terms of a confidence interval for that extends from an upper confidence (UCL) to a lower confidence (LCL) level. Let us choose 95% for the confidence interval. Then, remembering that this is a two-tailed test (UCL and LCL), we obtain from a table of Student s t distribution the critical value of L (U975) the appropriate number of degrees of freedom. 

Because exceeds the confidence interval s upper limit of 0.346, there is reason to believe that a 2 factorial design and a first-order empirical model are inappropriate for this system. A complete empirical model for this system is presented in problem 10 in the end-of-chapter problem set. 

Two solutes, A and B, with distribution ratios of 9 and 4, respectively, are to be separated by a countercurrent extraction in which the volumes of the upper and lower phases are equal. After 100 steps, determine the 99% confidence interval for the location of each solute. 

If these limits on the expected life are designated by L and U for the lower and upper, respectively, then the 100(1 — a)% confidence interval on the rehabihty is... 

For standard deviations, an analogous confidence interval CI(.9jr) can be derived via the F-test. In contrast to Cl(Xmean), ClCij ) is not symmetrical around the most probable value because by definition can only be positive. The concept is as follows an upper limit, on is sought that has the quality of a very precise measurement, that is, its uncertainty must be very small and therefore its number of degrees of freedom / must be very large. The same logic applies to the lower limit. s/ ... 

The target number of commodity samples to be obtained in the OPMBS was 500, as determined using statistical techniques. A sample size of 500 provided at least 95% confidence that the 99th percentile of the population of residues was less than the maximum residue value observed in the survey. In other words, a sample size of 500 was necessary to estimate the upper limit of the 95% confidence interval around the 99th percentile of the population of residues. 

One can also construct 95% confidence intervals using unequal tails (for example, using the upper 2% point and the lower 3% point). We usually want our confidence interval to be as short as possible, however, and with a symmetric distribution such as the normal or t, this is achieved using equal tails. The same procedure very nearly minimizes the confidence interval with other nonsymmetric distributions (for example, chi-square) and has the advantage of avoiding rather tedious computation. 

III the equivalence approach, which typically compares a statistical parameters confidence interval versus pre-defined acceptance limits (Schuirmann, 1987 Hartmann et al., 1995 Kringle et al., 2001 Hartmann et al., 1994). This approach assesses whether the true value of the parameter(s) are included in their respective acceptance limits, at each concentration level of the validation standards. The 90% 2-sided Cl of the relative bias is determined at each concentration level and compared to the 2% acceptance limits. For precision measurements, if the upper limit of the 95% Cl of the RSDn> is <3% then the method is acceptable (Bouabidi et al., 2010) or,... 

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. 

A second possibility consists of experimentally determining the SST limits from measurements at the worst-case conditions (n measurements with standard deviation 9,12,13 gg-p limit is defined as the lower or upper limit of the one-sided 95% confidence interval around the worst-case average result. For example, for resolution, the lower limit will be considered, while for migration time it would be the upper. The confidence intervals are defined as in Equations (16) and (17), when considering the lower or the upper limit, respectively. 

Table II displays numerical results for values of ag = 1.0 and 1.5 and =. 25,. 50, 1.00, 1.50 and 2.00. Values of J and K are determined for confidence intervals with upper error bound set at 50 percent and 100 percent and confidence coefficients at 95 percent and 99 percent. In general, the table shows that it is very expensive to go from 95 percent to 99 percent confidence. It is also significantly more expensive to obtain an estimate with a 50 percent error than a 100 percent error. For example, if Og = Ol = 1.00, to be 95 percent certain that the error in the estimated level of airborne asbestos is less than 50 percent requires J = 47 and K = 1 at a cost of 24,440. If a 100 percent error in the estimate is acceptable, J is reduced to 16, K remains at 1 and the cost is 8,320.

The 80% confidence bands for a given concentration level are constructed such that all the blocks within the band are those whose 80% confidence interval contains the given concentration level. That is, if we want to estimate the 80% confidence band for 250 ppm lead, all those blocks whose lower limits are greater than 250 ppm lead are classified as blocks whose concentrations are greater than 250 ppm. Those blocks whose upper limits are less than 250 ppm are classified as blocks whose concentrations are less than 250 ppm. The blocks which are left over, those containing 250 ppm in the 80% confidence interval, constitute the confidence band about the 250 ppm concentration level. Figures 15 through 22 show the 80% confidence bands for 2500 ppm, 1000 ppm, and 500 ppm concentration levels for the RSR and DMC and 500 ppm and 250 ppm concentration levels for REF, respectively. 

The results of the measurements are shown as dots on the scale, the mean is marked as x. The vertical lines show the 95% confidence limits. In the upper case the certified value is outside these limits, therefore the bias is significant. In the lower case the certified value is inside the confidence interval, the bias is insignificant. 

In this figure, each individual confidence interval has been drawn as a vertical straight line joining the lower and upper limits. The horizontal line is positioned at the value 4.055 mmol/L- the population mean. This gives us 40 sample means that are not equal to one another, so they on their own like the original measurement show random... 

The upper end of the confidence interval, the upper confidence limit, is given by ... 

The quantity c is very closely related to the odds ratio in fact c is the log of the OR, adjusted for the covariates. The anti-log of c (given by e"") gives the adjusted OR. Confidence intervals in relation to this OR can be constructed initially by obtaining a confidence interval for c itself and then taking the anti-log of the lower and upper confidence limits for c. 

While thep-value allows us the ability to judge statistical significance, the clinical relevance of the finding is difficult to evaluate from the calculated confidence interval because this is now on the log scale. It is usual to back-transform the lower and upper confidence limits, together with the difference in the means on the log scale, to give us something on the original data scale which is more readily interpretable. The back-transform for the log transformation is the anti-log. 

In order to demonstrate non-inferiority, it is only one end of the confidence interval that matters in our example it is simply the lower end that needs to be above —2 mmHg. It is therefore not really necessary to calculate the upper end of the interval and sometimes we leave this unspecified. The resulting confidence interval with just the lower end is called a one-sided 97.5per cent confidence interval the two-sided 95 per cent confidence interval cuts off 2.5 per cent at each of the lower and upper ends, having the upper end undefined leaves just 2.5 per cent cut off at the lower end. The whole of this confidence interval must be entirely to the right of the non-inferiority margin for non-inferiority to be established. 

If this confidence interval is on the log scale, for example with both the odds ratio and the hazard ratio, then both the lower and upper confidence limits should be converted by using the anti-log to give a confidence interval on the original odds ratio or hazard ratio scale. 

In the third step, the 8th saA29th ordered individual ratios are the lower and upper limits, respectively, of the 90% confidence interval for the ratio of the median in vitro release rate (slope) for T over the median in vitro release rate for R. In the example, this confidence interval is 1.0343 to 1.2863, or in percentage terms, 103.43% to 128.63%. 

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