Confidence level, table

Both correlation and variance analysis results showed that the hypothesis on the linear correlation between inter-laboratory data and the homogeneity of the corresponding variances is true for all data sets, at the for 95% confidence level. Table 2 presents a typical example of such a comparison. Based on the detected property of homogeneous variances, root-mean-square standard deviation, S, for all melted snow samples was estimated S = 0.32 0.06 for 95% confidence level [3]. 

The value of q is multiplied by the tabulated values of the highest density regions for the degrees of freedom, and an appropriate confidence level (Table 26-4). The distribution curve (the posterior density of a) is skewed so that the interval is larger on one side of the standard deviation than on the other, with low values of the standard deviation less probable than high values. 

Kuhns et al compared nine methods for the determination of decaborane in commercial samples in the 90 - 97% purity range. Gas chromatographic, infra-red, ultra-violet, iodine titration and iodo-metric procedures gave comparable results at the 95% confidence level (Table 22). Decaborane was determined by these five methods without significant influence by the impurities present. 

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. 

The second complication is that the values of z shown in Table 4.11 are derived for a normal distribution curve that is a function of O, not s. Although is an unbiased estimator of O, the value of for any randomly selected sample may differ significantly from O. To account for the uncertainty in estimating O, the term z in equation 4.11 is replaced with the variable f, where f is defined such that f > z at all confidence levels. Thus, equation 4.11 becomes... 

Values for t at the 95% confidence level are shown in Table 4.14. Note that t becomes smaller as the number of the samples (or degrees of freedom) increase, approaching z as approaches infinity. Additional values of t for other confidence levels can be found in Appendix IB. 

The t-values in this table are for a two-tailed test. For a one-tailed test, the a values for each column are half of the stated value, column for a one-tailed test is for the 95% confidence level, a = 0.05. For example, the first... 

The precision of measurement does not appear to be very high. Confidence levels in the precision may be made by use of "Student f" Tables. 

From standard Student tables, the value for t l/fT= 1.049 at the 95% confidence level. Thus, mean value = 56.3 13.0 (95% confidence level) and one could be confident that 95% of measured values would fall in the range 69.3 - 43.3. This is a large range and is not very precise. 

From the t-tables, the value of t for the 95 per cent confidence level with (n— 1), i.e. three degrees of freedom, is 3.18. 

The most widely used test is that for detecting a deviation of a test object from a standard by comparison of the means, the so-called t-test. Note that before a f-test is decided upon, the confidence level must be declared and a decision made about whether a one- or a two-sided test is to be performed. For details, see shortly. Three levels of complexity, a, b, and c, and subcases are distinguishable. (The necessary equations are assembled in Table 1.10 and are all included in program TTEST.)... 

This table is used for the two-sided test, that is one simply asks are the two distributions different Approximations to tabulated x -values for different confidence levels can be made using the algorithm and the coefficients given in section 5.1.4. 

Table 5.1. Coefficients for Approximating -Values for Various Confidence Levels (The coefficients h and j are necessary only for p 0.0001)...

The results for the RF screening study are shown in Table 3. The most striking result to come out of this experiment was that there appears to be a strong correlation between the low level of catalyst concentration and gel formation. The low level was outside the range of what had previously been tried. This has been confirmed in many subsequent experiments. Another important conclusion was that the chemistry appears to dominate the process, so it was reasonable to proceed with an RSM which dealt only with the formulation variables. Although the oven time was significant at the 90% confidence level, it was decided to optimize the chemistry first and deal with this as part of the processing conditions in later experiments. 

The mouse bioassay for PSP, described in its original form by Sommer in 1937 (29), involves i.p. injection of a test solution, typically 1 mL, into a mouse weighing 17-23 g, and observing the time from injection to death. From the death time and mouse weight, the number of mouse units is obtained by reference to a standard table 1 mouse unit is defined as the amount of toxin that will kill a 20-g mouse in 15 min (77). The sensitivity of the mouse population used is calibrated using reference standard saxitoxin (70). In practice, the concentration of the test solution is adjusted to result in death times of approximately 6 min. Once the correct dilution has been established, 5 mice will generally provide a result differing by less than 20% from the true value at the 95% confidence level. The use of this method for the various saxitoxins and indeterminate mixtures of them would appear... 

The major components have been identified tentatively as phenolic and fatty acids. At this time, seven phenolics have been identified in only four of the fractions. These are shown in Table III. A measure of the magnitude of the confidence level (cc) with a spectrum of standards is given. The first three entries are from the sunflower the last, from the Jerusalem artichoke. In all fractions isolated, both from the sunflower and the Jerusalem artichoke, a homologous series of fatty acids ranging from Cjo to Ci8 have been identified also by GC-MS. Even-chain, Cj6 to Cjs saturated and Cxs mono- and di-unsaturated, predominated. This is not surprising, since fatty acids are major constituents of plant... 

One way to choose the value of p is as follows. Assume that the distribution of squared residuals is normal, as is often done in crystallography. Then tables are available [17] which give the probability p that a particular experiment will give a X2 less than p. The value of p can be chosen according to the desired confidence level, p. Of course, other ways to choose are possible. Indeed, other choices for the agreement of statistic are possible. 

In this chapter as a continuation of Chapters 58 and 59 [1, 2], the confidence limits for the correlation coefficient are calculated for a user-selected confidence level. The user selects the test correlation coefficient, the number of samples in the calibration set, and the confidence level. A MathCad Worksheet ( MathSoft Engineering Education, Inc., 101 Main Street, Cambridge, MA 02142-1521) is used to calculate the z-statistic for the lower and upper limits and computes the appropriate correlation for the z-statistic. The upper and lower confidence limits are displayed. The Worksheet also contains the tabular calculations for any set of correlation coefficients (given as p). A graphic showing the general case entered for the table is also displayed. 

For this chapter we continue to describe the use of confidence limits for comparison of X, Y data pairs. This subject has been addressed in Chapters 58-60 first published as a set of articles in Spectroscopy [1-3]. A MathCad Worksheet ( 1986-2001 MathSoft Engineering Education, Inc., 101 Main Street Cambridge, MA 02142-1521) provides the computations for interested readers. This will be covered in a subsequent chapter or can be obtained in MathCad format by contacting the authors with your e-mail address. The Worksheet allows the direct calculation of the f-statistic by entering the desired confidence levels. In addition the confidence limits for the calculated slope and intercept are computed from the original data table. The lower limits for the slope and the intercept are displayed using two different sets of equations (and are identical). The intercept confidence limits are also calculated and displayed. 

Using the g-Value Table (90% Confidence Level as Table 73-1 we note that if Qn < g-Value, then the measurement is NOT an Outlier. Conversely, if Qn > g-Value, then the measurement IS an outlier. 

Table 73-1 Q-Value table (at different confidence levels)...

A confidence interval is calculated from t x s/+fn (see Section 6.1.3). To obtain a standard uncertainty we need to calculate sl Jn. We therefore need to know the appropriate Student /-value (see Appendix, p. 253). However, statements of this type are generally given without specifying the degrees of freedom. Under these circumstances, if it can be assumed that the producer of the material carried out a reasonable number of measurements to determine the stated value, it is acceptable to use the value of t for infinite degrees of freedom, which is 1.96 at the 95% confidence level. If the degrees of freedom are known, then the appropriate /-value can be obtained from statistical tables. In this example, the standard uncertainty is 3/1.96 = 1.53 mg D1. 

F statistic A statistic which is used as a measure of the goodness of fit of a data set to a correlation equation. The larger the value of F, the better the fit. Confidence levels can be assigned by comparing the F value calculated with the values in an F table for the Np and DF values of the data set. 

---