Normal approximation

The procedure for testing the significance of a sample proportion follows that for a sample mean. In this case, however, owing to the nature of the problem the appropriate test statistic is Z. This follows from the fact that the null hypothesis requires the specification of the goal or reference quantity po, and since the distribution is a binomial proportion, the associated variance is [pdl — po)]n under the null hypothesis. The primary requirement is that the sample size n satisfy normal approximation criteria for a binomial proportion, roughly np > 5 and n(l — p) > 5. 

Many distributions occurring in business situations are not symmetric but skewed, and the normal distribution cui ve is not a good fit. However, when data are based on estimates of future trends, the accuracy of the normal approximation is usually acceptable. This is particularly the case as the number of component variables Xi, Xo, etc., in Eq. (9-74) increases. Although distributions of the individual variables (xi, Xo, etc.) may be skewed, the distribution of the property or variable c tends to approach the normal distribution. 

Another common impurity of natural gas is nitrogen. Since nitrogen has essentially no calorific value, it lowers the heating value of gas, Gas purchasers may set a minimum limit of heating value (normally approximately 9.50 Biu/scl). In some cases it may be necessary to remove the nitrogen to satisfy this requirement. This is done in very low temperaliire plants or with permeable membranes. These proces [Pg.4]

Using the normal approximation to a binomial distribution, confidence intervals (CIs) for p(y =j X) can be established for a specific significance level, a ... 

If the normal approximation to the binomial distribution is valid (that is, not more than 20% of expected cell counts are less than 5) for drug therapy and symptom of headache, then you can use the Pearson chi-square test to test for a difference in proportions. To get the Pearson chi-square / -value for the preceding 2x2 table, you run SAS code like the following ... 

The output data set pvalue contains numerous Wilcoxon test statistics. Assuming that you want the two-sided normal approximation test p-value, the variable in the pvalue data set that you want is called P2 WIL. ... 

Fig. 1. Schematic for /zSR and fiLCR experiments. For pSR the muon spin polarization vector starts off in the x direction (open arrow). It then precesses about an effective field (the vector sum of the external field and the internal hyperfine field), which is normally approximately the z direction. The muons are detected in the M counter, and positrons from muon decay are detected in the L or R counters. For pLCR, the muon spin polarization is initially along the external field or t axis (solid arrow). The positron rates in the F and B counters are measured as a function of external field. A sharp decrease in the asymmetry of the F and B counting rates signifies a level crossing.

A widely used a = 5 percent significance level produces a 95 percent confidence interval extending over t91 confidence interval for a standard normal distribution. Therefore, the normal approximation of the t-distribution is correct to 12 percent for m> 10 and to 4 percent for m> 30. 

When k = 2, the Kruskal-Wallis chi-square value has 1 df This test is identical to the normal approximation used for the Wilcoxon Rank-Sum Test. As noted in previous sections, a chi square with 1 df can be represented by the square of a standardized normal random variable. In the case oik = 2, the //-statistic is the square of the Wilcoxon Rank-Sum Z-test (without the continuity correction). 

Hi) The curve, thus obtained, normally approximates to a straight line within the exclusion and total permeation limits,... 

If N is sufficiently large, the usual normal approximation applies with parameters... 

This expression would be very tedious to evaluate, so we use the normal approximation. By Eq. (1.26) we get ... 

The following examples illustrate the procedure for a statistical test [7]. In the first, we consider a very simple test on a single observation. The second applies the seven-step procedure to a test on the mean of a binomial population using a normal approximation. Here, and in the third example, we introduce the idea of one-sided and two-sided tests, while in the fourth example we illustrate the calculation of Type II error, and the power function of a test. 

This is a binomial distribution with two outcomes acceptance or rejection. However, since the sample size is large, we may use the normal approximation. Since p 0.048, we know that the population mean is p=np=1000 x 0.048=48... 

Since we are using a normal approximation, the test statistic is simply the standard normal variable ... 

The subtraction of 0.5 from the sample value improves the normal approximation. It is called the continuity correction . The numerator of the equation is the deviation of the sample value from the population mean. The denominator is simply the standard deviation of the presumed population. 

For a binomial distribution, the normal approximation can be used with good accuracy for sample sizes as low as 8, providing the binomial k is arbitrarily increased by 0.5 in calculating the approximate normal statistic. For values of the parameter p near 0 or 1, a larger sample must be used to obtain an accurate approximation. 

If the estimator is known (unlike CRB), it is possible to compute a criterion that considers the densities of the estimates themselves rather than the asymptotic normal approximation. A generic but computationally intensive approach is based on simulating a large number of estimates, as was done to generate Fig. 6. In the case of an LS estimator, an alternative is to approximate the densities of Gls using the analytical expression introduced by Pazman,42 which is much more complicated to... 

The inclusion of a primary radical termination process in radical polymerization schema usually leads to kinetic equations which cannot be reduced to a straightforward expression for the orders of reaction with respect to initiator and monomer concentration (48). It is interesting to note therdbre that, using only the normal approximations, such an expt ion can be derived from the above scheme which predicts that the polymerization will be half-or r in initiator and light intensity and first-order in monomer concentration, despite the participation of primary radical termination. Straightforward solution of the kinetics is made possiUe by the following assumptions, implicit in the scheme ... 

In the experiments with deuterium, a deuterium/l-butene ratio of 2 was employed. When deuterium oxide was used, the catalyst was allowed to pre-adsorb a small, controlled amount of D2O (normally approximately one molecule per cage) prior to the admission of the 1-butene. These reaction systems were examined over the least and the highest exchanged forms of the zinc and the cerium zeolites. 

Suxamethonium is destroyed by plasma pseudocholinesterase and so its persistence in the body is increased by neostigmine, which inactivates that enzyme, and in patients with hepatic disease or severe malnutrition whose plasma enzyme concentrations are lower than normal. Approximately 1 in 3000 of the European population have hereditary defects in amount or kind of enzyme, and cannot destroy the drug as rapidly as normal individuals. Paralysis can then last for hours and the individual requires ventilatory support and sedation until recovery occurs spontaneously. 

The one-tailed probability obtained using the normal approximation to the binomial probability is 0.0091, and the logarithm of the inverse of this probability value is used as the activity weight for the considered substructure, i.e. 

Normally, approximately 10% of glucose is catabolized through the hexose monophosphate pathway, but this fraction may be markedly increased when there is oxidative stress, as is the case when there is an infection or use of certain drugs. The principal function of the hexose monophosphate pathway is to reduce 2 moles of NADP to... 

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