Critical values for

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 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. 

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. 

Master curves can be used to predict creep resistance, embrittlement, and other property changes over time at a given temperature, or the time it takes for the modulus or some other parameter to reach a critical value. For example, a mbber hose may burst or crack if its modulus exceeds a certain level, or an elastomeric mount may fail if creep is excessive. The time it takes to reach the critical value at a given temperature can be deduced from the master curve. Frequency-based master curves can be used to predict impact behavior or the damping abiUty of materials being considered for sound or vibration deadening. The theory, constmction, and use of master curves have been discussed (145,242,271,277,278,299,300). 

The assumption in step 1 would first he tested hy obtaining a random sample. Under the assumption that p <. 02, the distrihiition for a sample proportion would he defined hy the z distrihiition. This distrihiition would define an upper hound corresponding to the upper critical value for the sample proportion. It would he unlikely that the sample proportion would rise above that value if, in fact, p <. 02. If the observed sample proportion exceeds that limit, corresponding to what would he a very unlikely chance outcome, this would lead one to question the assumption that p <. 02. That is, one would conclude that the null hypothesis is false. To test, set... 

The following table summarizes the critical values for various assumed temperature rises ... 

To further clarify the role of magnetic effects on compressibility, a shock compression experiment was performed on an fee 28.5-at. % Ni sample whose initial temperature was raised to 130°C. As is shown in Table 5.1, the compressibility was found to decrease to a value consistent with the nonmagnetic compressibility. Thus, the sharp change in compressibility, the critical values for the transition, and the magnitudes of the compressibility under the various conditions give a clear demonstration that a second-order magnetic transition has been observed, and we will proceed with a quantitative analysis of the transition. 

The vertex of the complete separation region, evaluated for k = 0.5 s is characterized by Yu - -86 and Ym 4 -75. The corresponding minimum value of Yi is Yi -7.135 -H 3.86 - 4.75 = 6.245, which is still higher than the critical value 7 = 6.045. We conclude that, considering negligible mass transfer, within the whole separation region built under these conditions, the Yi value does not affect the SMB performance. It should be pointed out that the presence of mass transfer resistance can influence the critical value for 7 and the form of the separation region [32]. 

The low rates of rusting observed at Khartoum, Abisko, Delhi, Basrah and Singapore are primarily associated with the absence of serious pollution. Moreover, at most of them the relative humidity is low, e.g. at Khartoum the relative humidity lies below the critical value for rusting throughout the whole year. 

Fig. 8.80 Inception and critical values for valves including data from Tullis...

The phase space theory in its present form suffers from the usual computational difficulties and from the fact it has thus far been developed only for treating three-body processes and a limited number of output channels. Further, to treat dissociation as occurring only through excitation of rotational levels beyond a critical value for bound vibrational states is rather artificial. Nevertheless, it is a useful framework for discussing ion-molecule reaction rates and a powerful incentive for further work. 

Equation (12.17) postulates that spontaneous deaths occur throughout the batch cycle. This means that dXjdt is initially negative. Is it possible to lose the inoculum completely if the induction period is too long Long induction periods correspond to small values of ot in the lag phase model of Problem 12.6. Find the critical value for ot at which the inoculum is lost. 

Critical ( -values for p - 0.05 are available. " - In lieu of using these tables, the calculated -values can be divided by the appropriate Student s t(f, 0.05) and V2 and compared to the reduced critical -vdues (see Table 1.12), and data file QRED TBL.dat. A reduced -value that is smaller than the appropriate critical value signals that the tested means belong to the same population. A fully worked example is found in Chapter 4, Process Validation. Data file MOISTURE.dat used with program MULTI gives a good idea of how this concept is applied. MULTI uses Table 1.12 to interpolate the cutoff point for p = 0.05. With little risk of error, this table can also be used fo = 0.025 and 0.1 (divide q by t(/, 0.025) /2 respectively t f, 0.1) V 2, as appropriate. 

Next, each difference was transformed into a g-value according to Eq. (1.25). With 70 - 9 = 61 degrees of freedom, the critical -value for the longest diagonal (adjacent means) would be 2.83, that for the top right corner (eight interposed means) 3.33, see Table 1.11. For this evaluation sep-... 

Results The uncertainties associated with the slopes are very different and n = H2, so that the pooled variance is roughly estimated as (V + V2)/2, see case c in Table 1.10 this gives a pooled standard deviation of 0.020 a simple r-test is performed to determine whether the slopes can be distinguished. (0.831 - 0.673)/0.020 = 7.9 is definitely larger than the critical /-value for p - 0.05 and / = 3 (3.182). Only a test for H[ t > tc makes sense, so a one-sided test must be used to estimate the probability of error, most likely of the order p = 0.001 or smaller. 

Plot analogous critical values for the mean/SD before resp. after elimination of points (dotted lines). Since the standard deviation will decrease on elimination of suspected outliers, the dotted sensitivity curve for after elimination will be higher than the one for before . Huber s k changes, too, but to a lesser degree. (See Fig. 1.1.)... 

Harter, H. L., Critical Values for Duncan s New Multiple Range Test, Biometrics, December 1960, 671-685. 

One significant result from the studies of stretched premixed flames is that the flame temperature and the consequent burning intensity are critically affected by the combined effects of nonequidiffusion and aerodynamic stretch of the mixture (e.g.. Refs. [1-7]). These influences can be collectively quantified by a lumped parameter S (Le i-l)x, where Le is the mixture Lewis number and K the stretch rate experienced by the flame. Specifically, the flame temperature is increased if S > 0, and decreased otherwise. Since Le can be greater or smaller than unity, while K can be positive or negative, the flame response can reverse its trend when either Le or v crosses its respective critical value. For instance, in the case of the positively stretched, counterflow flame, with k>0, the burning intensity is increased over the corresponding unstretched, planar, one-dimensional flame for Le < 1 mixtures, but is decreased for Le > 1 mixtures. 

Figure 6. Expanded view of neutral stability diagram for System I with k = 0.865. Critical values for bifurcation of families of cells in a 2Xc sample size are marked.

In polyelectrolyte solutions, the counterion condensation on linear polyelectrolyte chains is known to occur when the charge density along the chain exceeds the critical value [40]. Our work indicates the existence of a critical value for the separation distance between chains, where the interchain interaction changes drastically, most likely due to the transition in the binding mode of the counterions (see Fig. 13). Many peculiar forms of behavior, which are often interpreted by the cluster formation or the interchain organization of polyelectrolytes, have been reported for high concentrations of aqueous polyelectrolytes... 

The critical value for Q is 1/z. There is a proportional increase in the number of free counterions, f/z, as Q increases from zero, reaching a plateau when Q = z. Also, below this value the degree of dissociation,) , increases as the concentration decreases, and tends to unity as v tends to zero. When Q> /z, p decreases with 0 and tends to /zQ as 0 tends to zero. The number of free ions caimot exceed njz Q. Note that this number is inversely proportional to the square of the valence. The condensation of ions is thus very sensitive to valence for multivalent counterions it takes place at a lower value of Q and the number of free ions is much smaller... 

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