Confidence intervals calculations

Values are means of the percentages of weights + the limits of the 95% confidence intervals calculated from the Z test. Data from more... 

Previously when we had calculated a confidence interval, for example for a difference in rates or for a difference in means, then the confidence interval was symmetric around the estimated difference in other words the estimated difference sat squarely in the middle of the interval and the endpoints were obtained by adding and subtracting the same amount (2 x standard error). When we calculate a confidence interval for the odds ratio, that interval is symmetric only on the log scale. Once we convert back to the odds ratio scale by taking anti-logs that symmetry is lost. This is not a problem, but it is something that you will notice. Also, it is a property of all standard confidence intervals calculated for ratios. 

When n becomes large the t value tends toward the standardized normal value of 1.96 (z = 1.96), which was approximated to 2 above. The 95% confidence interval, calculated by equation 2.13, is sometimes explained much like the expanded uncertainty, as a range in which the true value lies with 95% confidence. In fact, the situation is more complicated. The correct statistical statement is if the experiment of n measurements were repeated under identical conditions a large number of times, 95% of the 95% confidence intervals would contain the population mean. ... 

Confidence intervals may be used to define a stop criterion, i.e. they can be used to judge whether the optimization process should be continued or halted. If the predicted optimum falls within one of the existing confidence intervals (calculated for 5=0.025), then the experimental capacity factors will be within 2.5% of the predicted values. It should be noted that an error of 2.5% in k can make a big difference if the relative retention (a) of a pair of peaks is close to one. It may therefore be required to use a lower value for S in eqn.(5.19). 

It is because of these unavoidable, random, and indeterminate errors that replicates are run. The mean of a series of replicates should be more accurate than any single value because the indeterminate errors will tend to cancel (average themselves out) in the runs. It is statistically seldom worth running more than four replicates because the number of trials in the standard deviation, cr, and confidence interval calculations is a square-root term in the denominator. Figure 6-13 illustrates the term accuracy and precision. 

These and most other equations developed by statisticians assume that the experimental error is the same over the entire response surface there is no satisfactory agreement for how to incorporate heteroscedastic errors. Note that there are several different equations in the literature according to the specific aims of the confidence interval calculations, but for brevity we introduce only two which can be generally applied to most situations. 

TABLE 13.8 Rate Constants and F ab Values at 290 K Calculated from Laser Flash (LF) Photolysis Experiments According to the Model and Values from Steady-state (SS) Photolyses Conducted at 295 K. Error Limits are the <-test Confidence Intervals Calculated at the 95% Confidence Level... 

Statistics teaches that the deviation of data based on less than 30 measurements is not a normal distribution but Student s t-distribution. So it is suitable to express the binding constant K with 95% confidence interval calculated by applying by Student s t-distribution. Student s t-distribution includes the normal distribution. When the number of measurements is more than 30, Student s t-distribution and the normal distribution are practically the same. The actual function of Student s t-distribution is very complicated so that it is rarely used directly. A conventional way to apply Student s t-distribution is to pick up data from the critical value table of Student s t-distribution under consideration of degree of freedom , level of significance and measured data. It is troublesome to repeat this conventional way many times. Most spreadsheet software even for personal computers has the function of Student s t-distribution. Without any tedious work, namely, picking up data from the table, statistical treatment can be applied to experimental results based on Student s t-distribution with the aid of a computer. In Fig. 2.12, an example is shown. When the measurement data are input into the gray cells, answers can be obtained in the cell D18 and D19 instantaneously. 

This confidence interval is calculated when a method of measurement is tested with a sample for which the corresponding true value Xq is known, though it remains to be seen if the latter is situated in the confidence interval calculated. 

Figure 2.8 shows the confidence intervals calculated for the means of the random data used earlier. Figure 2.8(a) shows 95% confidence intervals based on the population standard deviation (which we know a= 1) and z-value (1.96), which is of course the same for each value. Figure 2.8(b) is the 95% confidence interval calculated using equation 2.11. For small values of n the Student t interval is much greater than the one based on a knowledge of a, because, as discussed above,... 

The results of the objective function corresponding to each one of the proposed kinetic models are summarized in Table 2. As is observed, model 4 is the one with the best fit, although the difference with model 5 is very small. As this latter model is simpler, we adopted it for subsequent studies. The kinetic parameters corresponding to the kinetic model 5 (with 90% confidence intervals calculated by means of the Mardquardt algorithm for non-linear regression) are ... 

The different confidence intervals calculated so far for the mean mass of a bean, in the text as well as in the exercises, are graphically compared in Fig. 2.11. We can see a narrowing of the interval as the number of beans in the sample increases. Since this effect varies with the square root of n, increases in sample size become less attractive after a while. For example, to reduce by half the intervals obtained from a 140-bean sample, we would have to weigh 420 more beans (to have a total of 4 x 140 = 560 beans). One wonders if so much extra work is warranted by the increase in precision. 

Table 5.4 Sample-specific confidence intervals calculated for the PLS predictions of Cu in unknown commercial lubricating oils and two certified reference materials (CS2 example), values in pg Cu 1 ...

Statistics teaches that the deviation of data based on fewer than 30 measurements is not a normal distribution but rather a Studeut s t-distributiou. It is thus suitable to express the binding constant K with a 95% confidence interval calculated by apply-iug Student s t-distribution. When the number of measuremeuts is more than 30, Student s t-distribution and the normal distribution are practically the same. The actual function of Student s l-distribution is complicated, and it is rarely used directly. 

Figures 7 and 9 show the posterior distributions of the unknown parameters = lpaA, Cpca) and = lpsA,

In most mathematical models, the covariance between parameters, as measured by Eq. (7.137), is nonzero, that is, the parameters are correlated with each other. Careful experimental design may reduce, but never completely eliminate, this correlation. The individual confidence intervals calculated by Eq. (7.145) do not reflect the covariance. To do so, it is necessary to construct the joint confidence region of parameters. Using the multivariate normal distribution of b [Eq. (7.138)], we form the standardized normal variable ... 

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