Confidence intervals variance

FIG. 21-45 The size of the unilateral confidence interval (95 percent) as a function of the number n of samples taken, measured in multiples of [cf Eq. (21-62)]. Example If 31 samples are taken, the upper limit of the variances confidence interval assumes a value of 1.6 times that of the experimental sample... 

An overview of statistical methods covers mean values, standard deviation, variance, confidence intervals, Student s t distribution, error propagation, parameter estimation, objective functions, and maximum likelihood. 

While literature has discussed to a large extent the pros and cons of various probabilistic or non-probabilistic paradigms (Helton Oberkampf, 2004), it is assumed in this paper that probabilistic imcer-tainty treatment is acceptable for the cases considered. Decision criteria will generally involve the consideration of some peculiar quantities of interest on the distribution of Z (e.g. variance, confidence intervals, quantiles). 

The quality of the regression model is assessed in view of numerical and graphical information, which includes the model variance, confidence intervals on the parameter estimates, the linear correlation coefficient, residual and normal probability plots. The model variance is defined 5 = [(y-y) (y-y)]/v, where yand y are the measured and calculated vectors of the dependent variable respectively, v is the number of degrees of freedom (v= N- k +1)) and k is the number of independent variables included in the model. The linear correlation coefficient is defined by = [(y - (y - p)] /[(y -yf y- y)], where y is the mean of y. The variance and... 

Now, the main question is how to associate a variance (confidence interval) with the prediction, because statistical techniques do not consider the SAM option. The answer is by no means simple and approximations need to be considered. A reasonable pragmatic starting point is to adapt the equations we have seen for interpolation [eqns (2.20) to (2.26)] to the new situation. 

The probabilistic nature of a confidence interval provides an opportunity to ask and answer questions comparing a sample s mean or variance to either the accepted values for its population or similar values obtained for other samples. For example, confidence intervals can be used to answer questions such as Does a newly developed method for the analysis of cholesterol in blood give results that are significantly different from those obtained when using a standard method or Is there a significant variation in the chemical composition of rainwater collected at different sites downwind from a coalburning utility plant In this section we introduce a general approach to the statistical analysis of data. Specific statistical methods of analysis are covered in Section 4F. 

In the previous section we considered the amount of sample needed to minimize the sampling variance. Another important consideration is the number of samples required to achieve a desired maximum sampling error. If samples drawn from the target population are normally distributed, then the following equation describes the confidence interval for the sampling error... 

Confidence Interval for the Difference in Two Population Means The confidence intei val for a mean can be extended to include the difference between two population means. This intei val is based on the assumption that the respective populations have the same variance <7 ... 

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... 

Equation (2-95) gives the variance of y at any Xj. With this equation confidence intervals can be estimated, using Student s t distribution, for the entire range of Xj. In particular, when all Xj = 0, y = Oq. nd we find... 

The two-sided confidence intervals for the coefficients b and b, w hen and are random variables having t distributions with (n - 2) degrees of freedom and error variances of... 

While it is useful to know X(y ), knowing the CL(A ) or, alternatively, whether X is within the preordained limits, given a certain confidence level, is a prerequisite to interpretation, see Figure 2.11. The variance and confidence intervals are calculated according to Eq. (2.18). 

The variance about the mean, and hence, the confidence limits on the predicted values, is calculated from all previous values. The variance, at any time, is the variance at the most recent time plus the variance at the current time. But these are equal because the best estimate of the current time is the most recent time. Thus, the predicted value of period t+2 will have a confidence interval proportional to twice the variance about the mean and, in general, the confidence interval will increase with the square root of the time into the future. 

Secondly, knowledge of the estimation variance E [P(2c)-P (2c)] falls short of providing the confidence Interval attached to the estimate p (3c). Assuming a normal distribution of error In the presence of an Initially heavily skewed distribution of data with strong spatial correlation Is not a viable answer. In the absence of a distribution of error, the estimation or "krlglng variance o (3c) provides but a relative assessment of error the error at location x Is likely to be greater than that at location 2 " if o (2c)>o (2c ). Iso-varlance maps such as that of Figure 1 tend to only mimic data-posltlon maps with bull s-eyes around data locations. 

The estimate p (x) retained need not be at the center of the confidence interval ]qw(2L) (x) Since the confidence intervals are obtained directly, there is no need to calculate the estimation variance, nor to hypothesize any model for the error distribution. 

It is noted that the partial derivatives in the above variance expressions depend on time t and therefore the variances should be computed simultaneously with the state variables and sensitivity coefficients. Finally, the confidence intervals of the cumulative production of each well and of the total reservoir are calculated by integration with respect to time (Kalogerakis and Tomos, 1995). 

As can be seen from Table 1, the estimated coefficients b[0] are not equal to zero for different samples, whereas the estimated coefficients b[l] are close to 1 within confidence interval. That means that coefficients b[0] estimated for different points of the territory are generalized relative characteristics of elements abundance at the chosen sampling points. Statistical analysis has confirmed that hypotheses Hi and H2 are true with 95% confidence level for the data obtained by any of the analytical groups involved. This conclusion allowed us to verify hypothesis H3 considering that the estimated average variances of the correlation equation (1) are homogeneous for all snow samples in each analytical group. Hypothesis H3... 

The estimation of means, variances, and covariances of random variables from the sample data is called point estimation, because one value for each parameter is obtained. By contrast, interval estimation establishes confidence intervals from sampling. 

An application of the confidence interval concept central to most statistical assessment is the t2-test for small normal samples. Let us consider a normally distributed variable X with mean p and variance a2. It will be demonstrated below that for m observations with sample mean x and variance s2, the variable U defined as... 

The same principle applies to the confidence interval of variances. We find that keeping the same sample from the same normal distribution, the variable v such as... 

Both assumptions are mainly needed for constructing confidence intervals and tests for the regression parameters, as well as for prediction intervals for new observations in x. The assumption of normal distribution additionally helps avoid skewness and outliers, mean 0 guarantees a linear relationship. The constant variance, also called homoscedasticity, is also needed for inference (confidence intervals and tests). This assumption would be violated if the variance of y (which is equal to the residual variance a2, see below) is dependent on the value of x, a situation called heteroscedasticity, see Figure 4.8. 

The denominator n 2 is used here because two parameters are necessary for a fitted straight line, and this makes s2 an unbiased estimator for a2. The estimated residual variance is necessary for constructing confidence intervals and tests. Here the above model assumptions are required, and confidence intervals for intercept, b0, and slope, b, can be derived as follows ... 

We will describe an accurate statistical method that includes a full assessment of error in the overall calibration process, that is, (I) the confidence interval around the graph, (2) an error band around unknown responses, and finally (3) the estimated amount intervals. To properly use the method, data will be adjusted by using general data transformations to achieve constant variance and linearity. It utilizes a six-step process to calculate amounts or concentration values of unknown samples and their estimated intervals from chromatographic response values using calibration graphs that are constructed by regression. 

A second reason is that the use of local variance versus global variance can result in markedly different bands. The separate calculations of variance at levels throughout the range of standards produces a wider confidence interval at lower values as seen in Kurtz method. If a common variance is used as the variance estimate then a lower confidence interval is calculated at each point as is probably the case in Wegscheider s method. 

The variances of the parameter estimates can be used to set confidence intervals that would include the true value of the parameter a certain percentage of the time. In general, the confidence interval for a parameter p, based on is given by... 

A related confidence interval is used for estimating a single mean of several new values of response at a given point in factor space. It can be shown that the estimated variance of predicting the mean of m new values of response at a given point in factor space, s y,o, is... 

Figure 2. Decreased scar size in cell transplanted animals. The resnlts of the myocardial scar measurements are shown as means and corresponding 95% confidence intervals. Two tailed p values were calculated for equal variances and considered significant when p<0.05.

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