Critical regions/values

Under the null hypothesis of no difference in population means, and assuming somewhat symmetric distributions, the test statistic follows a t distribution with 298 (that is, 146 + 154 — 2) df. Therefore the critical region (values of the test statistic that lead to rejection) is defined as t < -1.968 and t > 1.968. Note that this particular entry is not in Appendix 2, but the closest is for 300 df. 

The decision rule for each of the three forms would be to reject the null hypothesis if the sample value oft fell in that area of the t distribution defined by Ot, which is called the critical region. Other wise, the alternative hypothesis would be accepted for lack of contrary evidence. 

I. Under the null hypothesis, it is assumed that the respective two samples have come from populations with equal proportions pi = po. Under this hypothesis, the sampling distribution of the corresponding Z statistic is known. On the basis of the observed data, if the resultant sample value of Z represents an unusual outcome, that is, if it falls within the critical region, this would cast doubt on the assumption of equal proportions. Therefore, it will have been demonstrated statistically that the population proportions are in fact not equal. The various hypotheses can be stated ... 

Scaling laws provide an improved estimate of critical exponents without a scheme for calculating their absolute values or elucidating the physical changes that occur in the critical region. 

The first step in the application of the concept was to determine the critical load values for the different regions of eastern Canada. This was done using historical measurements of lake acidity in concert with the Integrated Assessment Model (IAM) which links atmospheric transport and deposition models with water chemistry and empirical biological response models. Details of the method are given in Jeffries and Lam (1993). 

Erpenbeck Miller (Addnl Ref I) stated that the Dieterici equation better represents the fluid properties near the liquid region and it gives a value RTc/pcvc =3.69, nearly the average for all common gases. Subscript c means critical. Same value is given by Joffe (Ref 2b, p 1216) for eqs (1)... 

An efficient method of solving the Percus-Yevick and related equations is described. The method is applied to a Lennard-Jones fluid, and the solutions obtained are discussed. It is shown that the Percus-Yevick equation predicts a phase change with critical density close to 0.27 and with a critical temperature which is dependent upon the range at which the Lennard-Jones potential is truncated. At the phase change the compressibility becomes infinite although the virial equation of state (foes not show this behavior. Outside the critical region the PY equation is at least two-valued for all densities in the range (0, 0.6). 

Figure 5. Distributions, based upon various sample sizes, of the estimated 95th percentile (xgsgL645) of a limiting distribution (vg° = 2.0 fig° = 3.2 BQ = 10.0) shaded areas represent critical regions (a =. 05) for a lower one-sided test of the hypothesis that the sample values were derived from the limiting distribution.

Using these limiting values, the saturation pressures, pBq, at which the zeolite cavities are completely filled are determined by the saturated vapor pressure-temperature semi-logarithmic formula extrapolated into the super-critical region (8, , 10, ll). These pressures correspond to the filling of the weakly adsorbing sites-the anionic sites. 

To make a test of the hypothesis, sample data are used to calculate a test statistic. Depending upon the value of the test statistic, the primary hypothesis H0 is accepted or rejected. The critical region is defined as the range of values of the test statistic that requires a rejection H0. The test statistic is determined by the specific probability distribution and by the parameter selected for testing. 

Accept or reject H0 by comparing the calculated value of the test statistic with the critical region. 

The test statistic is distributed normally with p lO and crx=9, if H0 is true. A value of X that is too far above or below the mean should be rejected, so we select a critical region at each end of the normal distribution. As illustrated in Fig. 1.5, a fraction 0.025 of the total area under the curve is cut off at each end for a=0.05. From the tables (Table B), we determine that the limits corresponding to these areas are Z=-1.96 and Z=1.96 so that if our single observation falls between these values, we accept H0. The corresponding values for a=0.01 are Z= 2.58. 

To determine the critical region, we must know the distribution of the test statistic. In this case, Z is distributed as the standard normal distribution. With H0 [t<48 and a=0.05, we determine that the critical region will include 5% of the area on the high end of the standard normal curve Fig. 1.6. The Z-value that cuts off 5% of the curve is found to be 1.645, from a table of... 

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