Z approximation

The sample size n is sufficiently large to meet the requirement for the Z approximation. 

H0 = assumption or null hypothesis regarding the population proportion H1 = alternative hypothesis a = significance level, usually set at. 10,. 05, or. 01 z = tabled Z value corresponding to the significance level a. The sample sizes required for the z approximation according to the magnitude of p0 are given in Table 3-6. 

To write down the electrostatic energy, we consider the case when the capacitance of the tube to the gate dominates those of the barriers (see Fig. 1), Cl, Cl unit length is c(z). Approximating the gate by an infinite plane at a distance R from the nanotube, we obtain... 

As the samples are large according to the definition given earlier, the test of the two proportions using the Z approximation is appropriate. For a two-sided test of size 0.05 the critical region is defined by Z < -1.96 or Z > 1.96. The value of the test statistic is calculated as ... 

Just as the hypothesis test using the Z approximation resulted in a rejection of the null hypothesis, so does this x test. We can also tell from the critical values in Table 10.5 that the p value must be < 0.001 because less than 0.001 of the area under the 1 df x distribution lies to the right of the value 10.38 and the calculated test statistic, 28.3646 lies to the right of that value. 

The two methods described earlier, the Z approximation and the y- test of homogeneity, are appropriate when the sample sizes are large enough. There are times, however, when the sample sizes in each group are not large enough or the proportion of events is low such that up < 5. In such cases another analysis method, one that does not require any approximation, is appropriate. 

The data from the study are represented as a contingency table as displayed in Table 10.8. As seen in Table 10.8, only four individuals had the event of interest. Neither the Z approximation nor the test would be appropriate given the small cell sizes of one and three. 

Chi-square test, Fisher s exact test, or z approximation... 

Experimental evidence for spin comes from an analysis of atomic line spectra, which show that states with orbital angular momentum (/>0) are split into two levels by a magnetic interaction known as spin-orbit coupling. It occurs in hydrogen but is very small there spin-orbit coupling increases with nuclear charge (Z) approximately as Z4 and so becomes more significant in heavy atoms. Dirac s equation, which incorporates the effects of relativity into quantum theory, provides a theoretical interpretation. 

In the large-Z approximation, we restrict the basis to those configurations that belong to a particular value of 1ZV, i.e. those configurations that correspond... 

Since only Coulomb potentials are involved, the matrix T v, v turns out to be energy independent. Its elements are pure numbers that depend only on N, the number of electrons, and are independent of the nuclear charge Z. The roots lK of the energy-independent interelectron repulsion matrix T v, v are also pure numbers (Table 1). In the large-Z approximation, the generalized Sturmian secular equation (41) reduces to the requirement ... 

Fig. 1 The ground state of the carbon-like isoelectronic series, calculated in the large-Z approximation. The energies divided by Z2 are shown as functions of Z. Experimental values are indicates by dots, while the energies calculated from (51) are shown as curves. The lower (solid) curve, which approaches the experimental values with increasing Z, has been corrected for relativistic effects. The upper (dashed) curve is uncorrected...

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