True variance

Ten measurements give an estimate of cr that may be as much as 3.3 times or only about one-half the true variance. 

It is possible to compare the means of two relatively small sets of observations when the variances within the sets can be regarded as the same, as indicated by the F test. One can consider the distribution involving estimates of the true variance. With sj determined from a group of observations and S2 from a second group of N2 observations, the distribution of the ratio of the sample variances is given by the F statistic ... 

In the following section constructural details of the Sequenator will be described to the extent required for an understanding of its operation. A detailed technical description has been published in 1967 by Edman and Begg ). Fig. 3 represents a diagrammatic view of the essential components of the instrument. The central part of the Sequenator is the reaction vessel (A), a cylindrical cup of pyrex glass mounted on the shaft of an electric motor (5). Correct functioning of this device requires that the inside cylindrical surface of the cup runs absolutely true. Variance would cause untolerable turbulence within the liquid film spread on the wall of the cup. An additional requirement is constant rotational speed of the cup. Variance of the speed would cause movement of the film up or down the wall. The cup is housed in a bell jar Q) which... 

Consider the multiple regression of y on K variables, X and an additional variable, z. Prove that under the assumptions A1 through A6 of the classical regression model, the true variance of the least squares estimator of the slopes on X is larger when z is included in the regression than when it is not. Does the same hold for the sample estimate of this covariance matrix Why or why not Assume that X and z are nonstochastic and that the coefficient on z is nonzero. 

The appropriate variance of the least squares estimator is Var[ y ]= (o1/n1 fLlx2. which is, of course, precisely what we have been analyzing above. It follows that the conventional estimator of the variance of the OLS estimator in this model is an appropriate estimator of the true variance of the least squares estimator. This follows from the fact that the regressor in the model, i, is unrelated to the source of heteroscedasticity, as discussed in the text. 

Assuming a normal distribution, a confidence interval for squared distribution is used as the basis for evaluating the chance deviation of the estimated variance from the true variance. Rearranging Eq. (11) results in a confidence interval for the true variance in terms of the degrees of freedom, the estimated variance, and the chi-squared distribution as... 

There is no rigorous mathematical method for combining the individual precisions, but the variances of the individual measurements may be combined. The precision, at the 95 percent confidence level, is defined as approximately twice the true standard deviation. Since the latter is the square root of the true variance, this permits evaluation of the precision for the combined operation. 

If we wish to calculate the probable upper limit for the true variance (say 9), at say the S% level of significance, we cdculate... 

The degree of mixing defined as the variance of the mass fraction of sand in the mix needs to be determined. It has to be compared with the variance for a fully segregated system and the ideal variance of a random mix. First, the random sample variance [Eq. (21-60)] is calculated, and with it an upper limit for the true variance can then be laid down. The sand s average concentration p in the whole 3-ton mix is estimated by using the random sample average l ... 

Therefore, whenever the normal form of the quadrature variance is negative, this component of the field is squeezed or, in other words, the quantum noise in this component is reduced below the vacuum level. For classical fields, there is no unity coming from the boson commutation relation, and the normal form of the quadrature component represents true variance of the classical stochastic variable, which must be positive. 

During a polymerization reaction, the viscosity of a reactor effluent is measured periodically to determine the reaction progress. The measurement is repeated so that = 6. If the true variance exceeds 0.8 cSt (cSt = centistokes), the measurements are repeated to achieve better precision. For this sample, the mean viscosity was determined to be 106.4 cSt with variance 1.9 cSF. At a confidence level of 95%, does the variance exceed the desired variability ... 

Note With the computers in use today to record chromatograms, it is easy to determine the true variance of a peak. Nevertheless, the old manual approximations have established themselves and chromatography software programs go to great length to emulate the old manual procedures.)... 

The quantity cr is the standard deviation of the estimate p. The true variance of p depends on the manner in which observations are made. Hence,... 

In general, the standard error for any normal population mean is equal to o/y/Ti. and the variance is equal to Jn. where n is the number of values that are used to calculate the mean and o is the population variance. For this analysis the term Si is an estimate of the true variance among the k different means. If the null hypothesis is true.. V- can be used to evaluate the population variance. To evaluate the variance for individual population values, the calculated S must be multiplied by the number of test values (replicates) used for each of the means, designated as u,. to give n Sl. which is an estimate of The variance of the individual measurements for each of the k treatments is calculated b ... 

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