Critical values

Based o the test data, the parameter a6 is correlating with the residual resistance (table 1). It is discovered that the less resistible samples have much higher value of a6. On the base of collected data it is possible to identify the critical value of the accumulation coefficient (which is a defective sign of the material (if aG> AiScR-the sample is defected if aG< a6cr - the sample is without defects). 

In an ideal situation we would like to have a balanced combination between evaluation of the welder s performance on one hand, and a fracture mechanics basis on the other hand in tenns of "being certain that a defect with dimensions exceeding a certain critical value is not present". The second aspect could be regarded as a safety net with a balanced conservatism. 

An interesting point is that AH itself varies with r [10].] As is the case when P is varied, the rate of nucleation increases so strongly with the degree of supercooling that a fairly sharp critical value exists for T. Analogous equations can be written for the supercooling of a melt, where the heat of fusion AH/ replaces AH . 

Show that the critical value of 0 in Eq. XVII-53 is 4, that is, the value of 0 above which a maximum and a minimum in bP appear. What is the critical value of 67... 

Figure Al.2.10. Birth of local modes in a bifurcation. In (a), before the bifiircation there are stable anhamionic symmetric and antisymmetric stretch modes, as in figure Al.2.6. At a critical value of the energy and polyad number, one of the modes, in this example the symmetric stretch, becomes unstable and new stable local modes are bom in a bifurcation the system is shown shortly after the bifiircation in (b), where the new modes have moved away from the unstable syimnetric stretch. In (c), the new modes clearly have taken the character of the anliamionic local modes.

As a increases, a critical value " i-.iiiis reached each time the th layer of target atoms moves out of the shadow cone allowing for large-angle backscattering (BS) or small-/i collisions as shown in figure Bl.23.3. If the BS intensity 1, is monitored as a fimction of a, steep rises [36] witli well defined maxima are observed when the... 

The fonn of the classical (equation C3.2.11) or semiclassical (equation C3.2.11) rate equations are energy gap laws . That is, the equations reflect a free energy dependent rate. In contrast with many physical organic reactivity indices, these rates are predicted to increase as -AG grows, and then to drop when -AG exceeds a critical value. In the classical limit, log(/cg.j.) has a parabolic dependence on -AG. Wlren high-frequency chemical bond vibrations couple to the ET process, the dependence on -AG becomes asymmetrical, as mentioned above. 

Consequently, when D /Dj exceeds the critical value, close to the bifurcation one expects to see the appearance of chemical patterns with characteristic lengtli i= In / k. Beyond the bifurcation point a band of wave numbers is unstable and the nature of the pattern selected (spots, stripes, etc.) depends on the nonlinearity and requires a more detailed analysis. Chemical Turing patterns were observed in the chlorite-iodide-malonic acid (CIMA) system in a gel reactor [M, 59 and 60]. Figure C3.6.12(a) shows an experimental CIMA Turing spot pattern [59]. 

Imposition of no-slip velocity conditions at solid walls is based on the assumption that the shear stress at these surfaces always remains below a critical value to allow a complete welting of the wall by the fluid. This iraplie.s that the fluid is constantly sticking to the wall and is moving with a velocity exactly equal to the wall velocity. It is well known that in polymer flow processes the shear stress at the domain walls frequently surpasses the critical threshold and fluid slippage at the solid surfaces occurs. Wall-slip phenomenon is described by Navier s slip condition, which is a relationship between the tangential component of the momentum flux at the wall and the local slip velocity (Sillrman and Scriven, 1980). In a two-dimensional domain this relationship is expressed as... 

G is a multiplier which is zero at locations where slip condition does not apply and is a sufficiently large number at the nodes where slip may occur. It is important to note that, when the shear stress at a wall exceeds the threshold of slip and the fluid slides over the solid surface, this may reduce the shearing to below the critical value resulting in a renewed stick. Therefore imposition of wall slip introduces a form of non-linearity into the flow model which should be handled via an iterative loop. The slip coefficient (i.e. /I in the Navier s slip condition given as Equation (3.59) is defined as... 

The abbreviated table on the next page, which gives critical values of z for both one-tailed and two-tailed tests at various levels of significance, will be found useful for purposes of reference. Critical values of z for other levels of significance are found by the use of Table 2.26b. For a small number of samples we replace z, obtained from above or from Table 2.26b, by t from Table 2.27, and we replace cr by ... 

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. 

Next, an equation for a test statistic is written, and the test statistic s critical value is found from an appropriate table. This critical value defines the breakpoint between values of the test statistic for which the null hypothesis will be retained or rejected. The test statistic is calculated from the data, compared with the critical value, and the null hypothesis is either rejected or retained. Finally, the result of the significance test is used to answer the original question. 

Relationship between confidence intervals and results of a significance test, (a) The shaded area under the normal distribution curves shows the apparent confidence intervals for the sample based on fexp. The solid bars in (b) and (c) show the actual confidence intervals that can be explained by indeterminate error using the critical value of (a,v). In part (b) the null hypothesis is rejected and the alternative hypothesis is accepted. In part (c) the null hypothesis is retained. 

The critical value for f(0.05,4), as found in Appendix IB, is 2.78. Since fexp is greater than f(0.05, 4), we must reject the null hypothesis and accept the alternative hypothesis. At the 95% confidence level the difference between X and p, is significant and cannot be explained by indeterminate sources of error. There is evidence, therefore, that the results are affected by a determinate source of error. 

The critical value for F(0.05, 6, 4) is 9.197. Since Fexp is less than F(0.05, 6, 4), the null hypothesis is retained. There is no evidence at the chosen significance level to suggest that the difference in precisions is significant. 

The value of fexp is compared with a critical value, f(a, v), as determined by the chosen significance level, a, the degrees of freedom for the sample, V, and whether the significance test is one-tailed or two-tailed. 

The critical value for f(0.05, 10), from Appendix IB, is 2.23. Since fexp is less than f(0.05, 10) the null hypothesis is retained, and there is no evidence that the two sets of pennies are significantly different at the chosen significance level. 

Since Fgxp is larger than the critical value of 7.15 for F(0.05, 5, 5), the null hypothesis is rejected and the alternative hypothesis that the variances are significantly different is accepted. As a result, a pooled standard deviation cannot be calculated. 

The value of fexp is then compared with a critical value, f(a, v), which is determined by the chosen significance level, a, the degrees of freedom for the sample, V, and whether the significance test is one-tailed or two-tailed. For paired data, the degrees of freedom is - 1. If fexp is greater than f(a, v), then the null hypothesis is rejected and the alternative hypothesis is accepted. If fexp is less than or equal to f(a, v), then the null hypothesis is retained, and a significant difference has not been demonstrated at the stated significance level. This is known as the paired f-test. 

Another important example of redox titrimetry that finds applications in both public health and environmental analyses is the determination of dissolved oxygen. In natural waters the level of dissolved O2 is important for two reasons it is the most readily available oxidant for the biological oxidation of inorganic and organic pollutants and it is necessary for the support of aquatic life. In wastewater treatment plants, the control of dissolved O2 is essential for the aerobic oxidation of waste materials. If the level of dissolved O2 falls below a critical value, aerobic bacteria are replaced by anaerobic bacteria, and the oxidation of organic waste produces undesirable gases such as CH4 and H2S. 

This value for fexp is smaller than the critical value of 2.26 for f(0.05, 9). Thus, there is no evidence for a systematic error in the method at the 95% confidence level. 

This value of fexp is compared with the critical value for f(a, v), where the significance level is the same as that used in the ANOVA calculation, and the degrees of freedom is the same as that for the within-sample variance. Because we are interested in whether the larger of the two means is significantly greater than the other mean, the value of f(a, v) is that for a one-tail significance test. 

Using equation 14.25, we can calculate values of fexp for each possible comparison. These values can then be compared with the one-tailed critical value of 1.73 for f(0.05, 18), as found in Appendix IB. For example, fexp when comparing the results for analysts A and B is... 

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