Confidence interval limits

Determine confidence interval limits within which is the average measurement value at a=0.05. Five measurements were done (u=5). The arithmetic mean is X=31.2 and S=0.24. From Table C for a=0.05 and f=u-l=5-l=4 we obtain t0,o5=2.78 so that ... 

Hence, calculations from relation (2.26) facilitate determining the necessary number of measurements (u). Thereby, it is of course necessary to previously define the size of the random value that may be accepted and the coefficient or degree of measurement confidence. In practice, we are satisfied with the level that is not above 0.5%. Table 2.11 is used for practical determination of the necessary number of measurements, for known measurement confidence 1-a and for different confidence interval limits expressed by the error mean square of measurement AX/S. 

X 5.44/55.12) lower than those of analyst 2 using the assay method. Because the value 0 is not included in the 95%i confidence interval, we can conclude at the 5% level that there is a statistically significant difference between the two analysts, the same conclusion we had with ANOVA. However, a difference of 9.88%i, the maximum confidence interval limit, might not be large enough for us to reject that the two analysts are comparable in their performance of the assay method. This is a decision that was not possible based solely on ANOVA results. 

In this case example, the FDA s SRS + AERS database, through the end of the second quarter of 2005, was data mined to determine the lower 95% confidence interval limit of the EBGM scores (denoted as EB05), a measure of disproportional-ity, for rhabdomyolysis associated with the use of statins. The drugs of interest were atorvastatin, cerivastatin, fluvastatin, lovastatin, pravastatin, rosuvastatin and simvastatin. The event of interest was rhabdomyolysis. 

Figure 5.37 shows a 3D plot for Z = f X,Y), where Z is surface-fitted to both X and Y. In addition, the figure shows the nonlinear regression results estimated parameters, their 95% confidence interval limits, and parameters indicating model goodness R, Adj R , SSE, and RMSE. 

The distribution of the /-statistic (x — /ji)s is symmetrical about zero and is a function of the degrees of freedom. Limits assigned to the distance on either side of /x are called confidence limits. The percentage probability that /x lies within this interval is called the confidence level. The level of significance or error probability (100 — confidence level or 100 — a) is the percent probability that /X will lie outside the confidence interval, and represents the chances of being incorrect in stating that /X lies within the confidence interval. Values of t are in Table 2.27 for any desired degrees of freedom and various confidence levels. 

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. 

Confidence Intervals for Normal Distribution Curves Between the Limits p zo... 

In Section 4D.2 we introduced two probability distributions commonly encountered when studying populations. The construction of confidence intervals for a normally distributed population was the subject of Section 4D.3. We have yet to address, however, how we can identify the probability distribution for a given population. In Examples 4.11-4.14 we assumed that the amount of aspirin in analgesic tablets is normally distributed. We are justified in asking how this can be determined without analyzing every member of the population. When we cannot study the whole population, or when we cannot predict the mathematical form of a population s probability distribution, we must deduce the distribution from a limited sampling of its members. 

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. 

Confidence-Interval Estimates. Confidence-interval estimates for the expected hfe or rehabihty can be obtained easily in the case of the exponential. Here only the limits for failure-censored (Type II) and time-censored (Type I) life testing are given. It is possible to specify a test as either time- or failure-tmncated, whichever occurs first. The theory for such tests is explained in References 16 and 17. 

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... 

Confidence bounds or limits The end points of a confidence interval. 

Calculations of the confidence intervals about the least-squares regression line, using Eq. (2-100), reveal that the confidence limits are curved, the interval being smallest at Xj = x. 

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... 

When a small number of observations is made, the value of the standard deviation s, does not by itself give a measure of how close the sample mean x might be to the true mean. It is, however, possible to calculate a confidence interval to estimate the range within which the true mean may be found. The limits of this confidence interval, known as the confidence limits, are given by the expression ... 

Hence, on increasing the number of replicate determinations both the values of and s/yfn decrease with the result that the confidence interval is smaller. There is, however, often a limit to the number of replicate analyses that can be sensibly performed. A method for estimating the optimum number of replicate determinations is given in Section 4.15. 

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... 

Up to now (1971) only a limited number of reaction series have been completely worked out in our laboratories along the lines outlined in Sec. IV. In fact, there are rather few examples in the literature with a sufficient number of data, accuracy, and temperature range to be worth a thorough statistical treatment. Hence, the examples collected in Table III are mostly from recent experimental work and the previous ones (1) have been reexamined. When evaluating the results, the main attention should be paid to the question as to whether or not the isokinetic relationship holds i.e., to the comparison of standard deviations of So and Sqo The isokinetic temperature /J is viewed as a mere formal quantity and is given no confidence interval. Comparison with previous treatments is mostly restricted to this value, which has generally and improperly been given too much atention. 

Assuming for the moment that a large number of measurements went into a determination of a mean Xmean and a standard deviation s, what is the width of the 95% confidence interval, what are the 95% confidence limits ... 

A table of cumulative probabilities (CP) lists an area of 0.975002 for z -1.96, that is 0.025 (2.5%) of the total area under the curve is found between +1.96 standard deviations and +°°. Because of the symmetry of the normal distribution function, the same applies for negative z-values. Together p = 2 0.025 = 0.05 of the area, read probability of observation, is outside the 95% confidence limits (outside the 95% confidence interval of -1.96 Sx. .. + 1.96 Sx). The answer to the preceding questions is thus... 

Figure 1.18. The Student s ( resp. t/Vn for various confidence levels are plotted the curves for p = 0.05 are enhanced. The other curves are for p = 0.5 (bottom), 0.2, 0.1, 0.02, 0.01, 0.002, 0.001, and 0.0001 (top). By plotting a horizontal, the number of measurements necessary to obtain the same confidence intervals for different confidence levels can be estimated. While it takes n - 9 measurements (/ = 8) for a t-value of 7.12 and p = 0.0001, just n = 3 f - 2) will give the same limits on the population for p = 0.02 (line A - C). For the CL on the mean, in order to obtain the same t/ /n for p = 0.02 as for p = 0.0001, it will take n = 4 measurements (line B ) note the difference between points D and ...

In Section 1.3.2, confidence limits are calculated to define a confidence interval within which the true value p is expected with an error probability of p or less. 

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/ ... 

Figure 2.8. The slopes and residuals are the same as in Figure 2.4 (50,75,100, 125, and 150% of nominal black squares), but the A -values are more densely clustered 90, 95, 100, 105, and 110% of nominal (gray squares), respectively 96, 98, 100, 102, and 104% of nominal (white squares). The following figures of merit are found for the sequence bottom, middle, top the residual standard deviations +0.00363 in all cases the coefficients of determination 0.9996, 0.9909, 0.9455 the relative confidence intervals of b +3.5%, +17.6%, 44.1%. Obviously the extrapolation penalty increases with decreasing Sx.x, and can be readily influenced by the choice of the calibration concentrations. The difference in Sxx (6250, 250 resp. 40) exerts a very large influence on the estimated confidence limits associated with a, b, Y(x), and X( y ).

Figure 2.9. The confidence interval for an individual result CI( 3 ) and that of the regression line s CLj A are compared (schematic, left). The information can be combined as per Eq. (2.25), which yields curves B (and S, not shown). In the right panel curves A and B are depicted relative to the linear regression line. If e > 0 or d > 0, the probability of the point belonging to the population of the calibration measurements is smaller than alpha cf. Section 1.5.5. The distance e is the difference between a measurement y (error bars indicate 95% CL) and the appropriate tolerance limit B this is easy to calculate because the error is calculated using the calibration data set. The distance d is used for the same purpose, but the calculation is more difficult because both a CL(regression line) A and an estimate for the CL( y) have to be provided.

While it is useful to know X(y ), knowing the CL(A ) or, alternatively, whether X is within the preordained limits, given a certain confidence level, is a prerequisite to interpretation, see Figure 2.11. The variance and confidence intervals are calculated according to Eq. (2.18). 

Assuming the specification limits SL are given (regulations, market, etc.) postulate a tentative confidence interval CI(X) no larger than about SI SI = specification interval. (See Fig. 2.12.)... 

Figure 2.12. The relationship between specification interval, SI, and confidence intervals of the test result. Cl. The hatched bars denote the product specifications while the horizontal bars give the test results with confidence limits. A ratio SI/CI > 4 is required if differentiation between result categories is needed.

Conclusions the residual standard deviation is somewhat improved by the weighting scheme note that the coefficient of determination gives no clue as to the improvements discussed in the following. In this specific case, weighting improves the relative confidence interval associated with the slope b. However, because the smallest absolute standard deviations. v(v) are found near the origin, the center of mass Xmean/ymean moves toward the origin and the estimated limits of detection resp. quantitation, LOD resp. 

The Production Department was not amused, because lower values had been expected. Quality Control was blamed for using an insensitive, unse-lective, and imprecise test, and thereby unnecessarily frightening top management. This outcome had been anticipated, and a better method, namely polarography, was already being set up. The same samples were run, this time in duplicate, with much the same results. A relative confidence interval of 25% was assumed. Because of increased specificity, there were now less doubts as to the amounts of this particular heavy metal that were actually present. To rule out artifacts, the four samples were sent to outside laboratories to do repeat tests with different methods X-ray fluorescence (XRFi °) and inductively coupled plasma spectrometry (ICP). The confidence limits were determined to be 10% resp. 3%. Figure 4.23 summarizes the results. Because each method has its own specificity pattern, and is subject to intrinsic artifacts, a direct statistical comparison cannot be performed without first correcting the apparent concentrations in order to obtain presumably true... 

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