Normal distribution assumption

The Wilcoxon s Rank-Sum Test (WRST) is a non-parametric alternative. The WRST is robust to the normal distribution assumption, but not to the assumption of equal variance. Furthermore, this test requires that the two groups of data under comparison have similarly shaped distributions. Non-parametric tests typically suffer from having less statistical power than their parametric counterparts. Similar to the /-test, the WRST will exhibit false positive rate inflation across a microarray dataset. It is possible to use the Wilcoxon test statistic as the single filtering mechanism however calculation of the false positive rate is challenging (48). 

Prescriptive approaches for group decision making, 2212-2214 Presentation language, 131-132 Presentation support software, 142 Present worth, probability distribution for, 2367-2369, 2371-2376 discrete distribution, 2372-2373 expected present worth, 2367-2368 mean and variance, using only, 2373-2374 normal distribution, assumption of, 2374-2376... 

Schmider, E., Ziegler, M., Danay, E., Beyer, L., Buhner, M., 2010. Is it Really Robust Reinvestigating the Robustness of ANOVA against Violations of the Normal Distribution Assumption. Schmider, Emanuel. 

The normal-distribution assumption on both D and L allows a well-defined convolution of demand over the lead time, which is therefore also normally distributed, with mean... 

A wide variety of pattern tests (also called zone rules) can be developed based on the IID and normal distribution assumptions and the properties of the normal distribution. For example, the following excerpts from the Western Electric Rules (Western Electric Company, 1956 Montgomery and Runger, 2007) indicate that the process is out of control if one or more of the following conditions occur ... 

We can now draw an important conclusion regarding the distribution of b. Eq. (7.128) shows that b is a linear combination of u. If is a multivariate normal distribution (assumption 5, Sec. 7.3), then b is also a multivariate normal distribution, that is. 

Understanding the distribution allows us to calculate the expected values of random variables that are normally and independently distributed. In least squares multiple regression, or in calibration work in general, there is a basic assumption that the error in the response variable is random and normally distributed, with a variance that follows a ) distribution. 

The most commonly used form of linear regression is based on three assumptions (1) that any difference between the experimental data and the calculated regression line is due to indeterminate errors affecting the values of y, (2) that these indeterminate errors are normally distributed, and (3) that the indeterminate errors in y do not depend on the value of x. Because we assume that indeterminate errors are the same for all standards, each standard contributes equally in estimating the slope and y-intercept. For this reason the result is considered an unweighted linear regression. 

Statistical Criteria. Sensitivity analysis does not consider the probabiUty of various levels of uncertainty or the risk involved (28). In order to treat probabiUty, statistical measures are employed to characterize the probabiUty distributions. Because most distributions in profitabiUty analysis are not accurately known, the common assumption is that normal distributions are adequate. The distribution of a quantity then can be characterized by two parameters, the expected value and the variance. These usually have to be estimated from meager data. 

The basic underlying assumption for the mathematical derivation of chi square is that a random sample was selected from a normal distribution with variance G. When the population is not normal but skewed, square probabihties could be substantially in error. 

The F distribution, similar to the chi square, is sensitive to the basic assumption that sample values were selected randomly from a normal distribution. 

The population of differences is normally distributed with a mean [L ansample size is 10 or greater in most situations. 

The above assumes that the measurement statistics are known. This is rarely the case. Typically a normal distribution is assumed for the plant and the measurements. Since these distributions are used in the analysis of the data, an incorrect assumption will lead to further bias in the resultant troubleshooting, model, and parameter estimation conclusions. 

The data used to generate the maps is taken from a simple statistical analysis of the manufacturing process and is based on an assumption that the result will follow a Normal distribution. A number of component characteristics (for example, a length or diameter) are measured and the achievable tolerance at different conformance levels is calculated. This is repeated at different characteristic sizes to build up a relationship between the characteristic dimension and achievable tolerance for the manufacture process. Both the material and geometry of the component to be manufactured are considered to be ideal, that is, the material properties are in specification, and there are no geometric features that create excessive variability or which are on the limit of processing feasibility. Standard practices should be used when manufacturing the test components and it is recommended that a number of different operators contribute to the results. 

Ullman (1992) argues that the assumption that stress and strength are of the Normal type is a reasonable one because there is not enough data available to warrant anything more sophisticated. A problem with this is that the Normal distribution... 

The varianee for any set of data ean be ealeulated without referenee to the prior distribution as diseussed in Appendix I. It follows that the varianee equation is also independent of a prior distribution. Here it is assumed that in all the eases the output funetion is adequately represented by the Normal distribution when the random variables involved are all represented by the Normal distribution. The assumption that the output funetion is robustly Normal in all eases does not strietly apply, partieularly when variables are in eertain eombination or when the Lognormal distribution is used. See Haugen (1980), Shigley and Misehke (1996) and Siddal (1983) for guidanee on using the varianee equation. 

Another consideration when using the approach is the assumption that stress and strength are statistically independent however, in practical applications it is to be expected that this is usually the case (Disney et al., 1968). The random variables in the design are assumed to be independent, linear and near-Normal to be used effectively in the variance equation. A high correlation of the random variables in some way, or the use of non-Normal distributions in the stress governing function are often sources of non-linearity and transformations methods should be considered. 

The allowable misalignment toleranee for the vertieal tie rod, tp = 1.5 , is also eonsidered to be normally distributed in praetiee. With the assumption that approximately 6 standard deviations are eovering this range, the standard deviation beeomes = 0.5 . The mean of the angle on whieh the prineipal plane lies is /i, and the loads must be resolved for this angle, but its standard deviation is the statistieal sum of cr and as given by equation 4.103 ... 

From the worked example (Example 1 in Section 4.8) for the analysis of an iron ore sample, the standard deviation is found to be +0.045 per cent. If the assumption is made that the results are normally distributed, then 68 per cent (approximately seven out of ten results) will be between +0.045 per cent and 95 per cent will be between +0.090 per cent of the mean value. It follows that there will be a 5 per cent probability (1 in 20 chance) of a result differing from the mean by more than +0.090 per cent, and a 1 in 40 chance of the result being 0.090 per cent higher than the mean. 

The validity of the results of this procedure depends on whether the assumption is valid that the steps are of uniform size in a system in which the frequency of expins is normally distributed. 

It is intended as a quick introduction under the tacit assumption of normally distributed values. 

Given that the assumption of normally distributed data (see Section 1.2.1) is valid, several useful and uncomplicated methods are available for finding the most probable value and its confidence interval, and for comparing such results. 

For this reason alone the tacit assumption of a normal distribution when contemplating analytical results is understandable, and excusable, if only because there is no practical alternative (alternative distribution models require more complex calculations, involve more stringent assumptions, or are more susceptible to violations of these basic assumptions than the relatively robust normal distribution). 

Use Calculate Student s t-values given p and df Student s t is used instead of the normal deviate z when the number of measurements that go into a mean is relatively small and the assumption of p and a being infinitely precise has to be replaced by the assumption of a normally distributed mean and a x -distributed s. ... 

The optimization of empirical correlations developed from the ASPEN-PLUS model yielded operating conditions which reduced the steam-to-slurry ratio by 33%, increased throughput by 20% while maintaining the solvent residual at the desired level. While very successful in this industrial application the approach is not without shortcomings. The main disadvantage is the inherent assumption that the data are normally distributed, which may or may not be valid. However, previous experience had shown the efficacy of the assumption in other similar situations. 

The standard way to answer the above question would be to compute the probability distribution of the parameter and, from it, to compute, for example, the 95% confidence region on the parameter estimate obtained. We would, in other words, find a set of values h such that the probability that we are correct in asserting that the true value 0 of the parameter lies in 7e is 95%. If we assumed that the parameter estimates are at least approximately normally distributed around the true parameter value (which is asymptotically true in the case of least squares under some mild regularity assumptions), then it would be sufficient to know the parameter dispersion (variance-covariance matrix) in order to be able to compute approximate ellipsoidal confidence regions. 

The principle of Maximum Likelihood is that the spectrum, y(jc), is calculated with the highest probability to yield the observed spectrum g(x) after convolution with h x). Therefore, assumptions about the noise n x) are made. For instance, the noise in each data point i is random and additive with a normal or any other distribution (e.g. Poisson, skewed, exponential,...) and a standard deviation s,. In case of a normal distribution the residual e, = g, - g, = g, - (/ /i), in each data point should be normally distributed with a standard deviation j,. The probability that (J h)i represents the measurement g- is then given by the conditional probability density function Pig, f) ... 

Equations (41.15) and (41.19) for the extrapolation and update of system states form the so-called state-space model. The solution of the state-space model has been derived by Kalman and is known as the Kalman filter. Assumptions are that the measurement noise v(j) and the system noise w(/) are random and independent, normally distributed, white and uncorrelated. This leads to the general formulation of a Kalman filter given in Table 41.10. Equations (41.15) and (41.19) account for the time dependence of the system. Eq. (41.15) is the system equation which tells us how the system behaves in time (here in j units). Equation (41.16) expresses how the uncertainty in the system state grows as a function of time (here in j units) if no observations would be made. Q(j - 1) is the variance-covariance matrix of the system noise which contains the variance of w. 

There are two statistical assumptions made regarding the valid application of mathematical models used to describe data. The first assumption is that row and column effects are additive. The first assumption is met by the nature of the smdy design, since the regression is a series of X, Y pairs distributed through time. The second assumption is that residuals are independent, random variables, and that they are normally distributed about the mean. Based on the literature, the second assumption is typically ignored when researchers apply equations to describe data. Rather, the correlation coefficient (r) is typically used to determine goodness of fit. However, this approach is not valid for determining whether the function or model properly described the data. 

The above implicit formulation of maximum likelihood estimation is valid only under the assumption that the residuals are normally distributed and the model is adequate. From our own experience we have found that implicit estimation provides the easiest and computationally the most efficient solution to many parameter estimation problems. 

Rumpf (R4) has derived an explicit relationship for the tensile strength as a function of porosity, coordination number, particle size, and bonding forces between the individual particles. The model is based on the following assumptions (1) particles are monosize spheres (2) fracture occurs through the particle-particle bonds only and their number in the cross section under stress is high (3) bonds are statistically distributed across the cross section and over all directions in space (4) particles are statistically distributed in the ensemble and hence in the cross section and (5) bond strength between the individual particles is normally distributed and a mean value can be used to represent each one. Rumpf s basic equation for the tensile strength is... 

The Log-Probit Model. The log-probit model has been utilized widely in the risk assessment literature, although it has no physiological justification. It was first proposed by Mantel and Bryan, and has been found to provide a good fit with a considerable amount of empirical data (10). The model rests on the assumption that the susceptibility of a population or organisms to a carcinogen has a lognormal distribution with respect to dose, i.e., the logarithm of the dose will produce a positive response if normally distributed. The functional form of the model is ... 

The F distribution, similar to the chi square, is sensitive to the basic assumption that sample values were selected randomly from a normal distribution. The Microsoft Excel function FDIST(X, dfh df2) gives the upper percent points of Table 3-8, where X is the tabular value. The function FINV(Percent, dfh df2) gives the table value. 

The population from which the observations were obtained is normally distributed with an unknown mean p and standard deviation G. In actual practice, this is a robust test, in the sense that in most types of problems it is not sensitive to the normality assumption when the sample size is 10 or greater. 

The first of these assumptions is the use of the Normal distribution. When we perform an experiment using a sequential design, we are implicitly using the experimentally determined value of s, the sample standard deviation, against which to compare the difference between the data and the hypothesis. As we have discussed previously, the use of the experimental value of s for the standard deviation, rather than the population value of a, means that we must use the f-distribution as the basis of our comparisons, rather than the Normal distribution. This, of course, causes a change in the critical value we must consider, especially at small values of n (which is where we want to be working, after all). 

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