Distributions chi-square

In the previous section we discussed the ramifications of the uncertainty in estimating means from small samples and described how the sample mean, x, follows a f-distribution. In this section, we discuss the ramifications of the uncertainty in estimating s2 from small samples. The variable s2 is called the sample variance, which is an estimate of the population variance, a2. For simple random samples of size n selected from a normal population, the quantity in Equation 3.10 

FIGURE 3.3 Illustration of the chi-square distribution (a) for various degrees of freedom and (b) the area under the curve equal to 0.95 at 5 degrees of freedom. 

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Here again the quantity is the (1 — /3) percentile of a chi square distribution with V degrees of freedom. If only a 100(1 — a)% lower confidence limit is desired, it can be calculated from... 

Chi-Square Distribution For some industrial applications, produrt uniformity is of primary importance. The sample standard deviation. s is most often used to characterize uniformity. In dealing with this problem, the chi-square distribution can be used where = (.s /G ) (df). The chi-square distribution is a family of distributions which are defined by the degrees of freedom associated with the sample variance. For most applications, df is equal to the sample size minus 1. 

The chi-square distribution can be applied to other types of apph-catlon which are of an entirely different nature. These include apph-cations which are discussed under Goodness-of-Fit Test and Two-Way Test for Independence of Count Data. In these applications, the mathematical formulation and context are entirely different, but they do result in the same table of values. 

Confidence Interval for a Variance The chi-square distribution can be used to derive a confidence interval for a population variance <7 when the parent population is normally distributed. For a 100(1 — Ot) percent confidence intei val... 

The derivation will not be provided. Suffice it to say that the failures in a time interval may be modeled using the binomial distribution. As these intervals are reduced in size, this goes over to the Poisson distribution and the MTTF is chi-square distributed according to equation 2.9-31, where = 2 A N T and the degrees of freedom,/= 2(M+i). 

Confidence is calculated as the partial integral over the chi-squared distribution, i.e., the partial integral over equation 2.5-31 which is equation 2.5-32. where is the cumulative... 

Table 2.5-1 Values of the Inverse Cumulative Chi-Squared Distribution in Irrms of percentage confidence with M fuitun ...

This says that the failure rate is less than or equal to the inverse cumulative chi-squared distribution with confidence a and degrees of freedom equal to twice the number of failures including pseudo- failures divided by twice the time including psuedo-time. 

Selection 2 is a similar calculation using the F-Number method (Section 2.5.3.2) 3 calculates the integral over the Chi-Squared distribution. When selected i nput the upper limit of integration... 

The chi-square distribution gives the probability for a continuous random variable bounded on the left tail. The probability function has a shape parameter... 

Second, the probability that the assigned analytical errors would yield at least the observed amount of scatter (usually referred to as the probability of fif ) can be calculated from the chi-squared distribution of v x MSWD about v. For example. 

CHIDIST is the Excel function for the one-tailed probability of the chi-squared distribution. 

One critical limitation of MAXSLOPE is that the method locates the hitmax correctly only for certain types of latent distributions (e.g., normal distribution). For other kinds of distributions (e.g., chi-square distribution), the estimated location of the hitmax and the base rate may be substantially off the mark. In other words, when the underlying distributions are of the difficult kind, MAXSLOPE will detect taxonicity, but the estimated taxon base rate may be incorrect. Moreover, MAXSLOPE may fail to detect taxonicity under certain circumstances. Specifically, this will happen if ... 

As was indicated in Section 7.2, the vector of measurement adjustments, e, has a multivariate normal distribution with zero mean and covariance matrix V. Thus, the objective function value of the least square estimation problem (7.21), ofv = eT l> 1 e, has a central chi-square distribution with a number of degrees of freedom equal to the rank of A. 

This statistic has chi-square distribution with / degrees of freedom under the null hypothesis, where / is the number of elements of 7. If T > xl /, Ho is rejected, otherwise Ho is accepted, a is the significance level of the test. 

A number of types of distributions have been fully studied, because they, or at least close approximations to them, frequently arise in practice. In connection with the theory of measurement errors and least squares adjustments, the normal and chi-square distributions are often used, so they are briefly discussed in the following paragraphs. 

As stated in Section 4.1.4, Y=X2 is distributed as a chi-squared distribution with one degree of freedom. 

The statistic X2/T will be large when there is evidence of a dose-related increase or decrease in the tumor incidence rates, and small when there is little difference in the tumor incidence between groups or when group differences are not dose related. Under the null hypothesis of no differences between groups, X2lT has approximately a chi-squared distribution with one degree of freedom. 

For rather obscure reasons, x2 is known as the log-rank statistic. An approximate significance test of the null hypothesis of identical distributions of survival time in the two groups is obtained by x2 to a chi-square distribution on 1 degree of freedom. 

The Wilcoxon Rank-Sum Test could be used to analyze the event times in the absence of censoring. A Generalized Wilcoxon Test, sometimes called the Gehan Test, based on an approximate chi-square distribution, has been developed for use in the presence of censored observations. 

Xi2( ) The 100(1 -ac)% point of the chi-squared distribution with 1 degree of freedom iji A parameter setting the relative contributions of the linearization and steepest descent methods in determining the correction vector b of Eq. (45)... 

Other important, mathematically defined distributions are the f-distribution (Figure 1.9, left), the chi-square distribution (Figure 1.9, right), and the F-distribution (Figure 1.10). These distributions are used in various statistical tests (Section 1.6.5) no details are given here, but only information about their use within R. The form of these distributions depends on one or two parameters, called degrees of... 

FIGURE 1.9 -Distributions with 3 and 20 DF, respectively, and standard normal distribution corresponding to a -distribution with DF = oo (left). Chi-square distribution with 3, 10, and 30 DF, respectively (right). 

If it can be assumed that the multivariate data follow a multivariate normal distribution with a certain mean and covariance matrix, then it can be shown that the squared Mahalanobis distance approximately follows a chi-square distribution... 

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