Confidence coefficient

Factors with regression coefficients having confidence coefficients > 90% None ANOVA F-ratIo 13.65... 

The frequency interpretation of the interval estimates on the unknown amounts is given by the following ( 27 ) With at least 1- a confidence, based on the sampling characteristics of the observations on the standards, at least P proportion of the interval estimates made from a particular calibration will contain the true amounts. The Bonferroni inequality insures the 1-a confidence since the confidence interval about the regression line and the upper bound on cr are each performed using a 1- a/2 confidence coefficient. Hence, the frequency interpretation states that at least (1-a) proportion of the standard calibrations are such that at least P proportion of the intervals produced by the method cover the true unknown amounts. For the remaining a proportion of standard calibrations the proportion of intervals which cover the true unknown values may be less than P. 

Table II displays numerical results for values of ag = 1.0 and 1.5 and =. 25,. 50, 1.00, 1.50 and 2.00. Values of J and K are determined for confidence intervals with upper error bound set at 50 percent and 100 percent and confidence coefficients at 95 percent and 99 percent. In general, the table shows that it is very expensive to go from 95 percent to 99 percent confidence. It is also significantly more expensive to obtain an estimate with a 50 percent error than a 100 percent error. For example, if Og = Ol = 1.00, to be 95 percent certain that the error in the estimated level of airborne asbestos is less than 50 percent requires J = 47 and K = 1 at a cost of 24,440. If a 100 percent error in the estimate is acceptable, J is reduced to 16, K remains at 1 and the cost is 8,320.

For the first sample in the computer simulation the 99 per cent confidence interval is (78.64, 81.80). This is a wider interval than the 95 per cent interval the more confidence we require the more we have to hedge our bets. It is fairly standard to use 95 per cent confidence intervals and this links with the conventional use of 0.05 (or 5 per cent) for the cut-off for statistical significance. Under some circumstances we also use 90 per cent confidence intervals and we will mention one such situation later. In multiple testing it is also sometimes the case that we use confidence coefficients larger than 95 per cent, again we will discuss the circumstances where this might happen in a later chapter. 

There are invariably rules for how to obtain the multiplying constant for a specific confidence coefficient, but as a good approximation and providing the sample sizes are not too small, using the value 2 for the 95 per cent confidence interval and 2.6 for the 99 per cent confidence interval would get you very close. 

There is a connection with what we are seeing here and the calculation of the confidence interval in Chapter 3. Recall Table 3.1 within Section 3.1.3, Changing the multiplying constant . It turns out that p-values and confidence intervals are linked and we will explore this further in a later chapter. The confidence coefficients for d.f. = 38 are 2.02 for 95 per cent confidence and 2.71 for 99 per cent confidence. If we were to look at the tjg distribution we would see that 2.02 cuts off the outer 5 per cent probability while 2.71 cuts off the outer 1 per cent of probability. 

One element that makes the link work is the correspondence between the significance level (5 per cent) and the confidence coefficient (95 per cent). If we were to use 1 per cent as the cut-off for significance then the same link would apply but now with the 99 per cent confidence interval. 

The probability 1-a is called the confidence coefficient of the confidence interval. The interval defined in this way is referred to as a confidence interval, and the ends of the interval are called confidence limits. The quantity (1-a) is the confidence coefficient, we can write down ... 

By choosing l-a=0.99, we can expect the confidence interval to contain the population parameter in some 99 out of 100 cases. But, as will be shown later, the confidence interval corresponding to the coefficient l(a=0.99, is greater than the one in the case l-a=0.95. This increase in confidence interval is the bad outcome of the confidence coefficient increase. Which of the 1-a confidence coefficient values to choose in the actual case depends on what error risk is acceptable. 

This indicates that to know the measurement random error it is not sufficient to know its magnitude only (confidence interval of measurement error) but also the significance level that facilitates the confidence estimate of the obtained measurements. Using the error mean square as a measurement accuracy property is convenient because that value in a normal distribution is associated with a confidence or confidence coefficient of 0.68 probability. The doubled error mean square 2S has... 

Equation (2.27) is used to determine the confidence interval or its limit of arithmetic measurement mean, to the actual measurement value for the given confidence coefficient and the number of measurements. 

No. SRS number in the state register (number - year of production) Alloy Alloy system Certified mass fraction of hydrogen, cm3 / 100 g Error of SRSs, cm3 / 100 g (confidence coefficient is 0.95) Rod (wire) dia., mm... 

Table 22.2 Values of the Student s bilateral confidence coefficient t (calculation of Student s distribution)...

To calculate tolerance limits, two values must first be specified C the proportion of the population to be covered (the "coverage") and P the confidence coefficient. For given values of C and P, one or two sided tolerance limits for the population can be calculated. 

The mean values not only describe the sample analyzed but also estimate the true average situation in the population from which it was drawn. However, a simple point estimate (in this case the sample mean) of a population quantity is not always this satisfactory. It is usually desirable to have some confidence interval estimate of the population quantity. This confidence interval is the one within which we are fairly certain that the true population quantity will be included. Employing both the sample mean and the standard error, an interval may be constructed (e.fif., 21.27 0.20 for the juice from the vine-ripened tomatoes). If it is assumed that the population sampled was normal, the above interval is one of approximately 68% confidence for estimating the population mean. It has become customary to calculate intervals of 95% and 99% confidence (7 = 0.95 or 7 = 0.99, where 7 is known as the confidence coefficient). If a 1007% confidence interval is desired for estimating/i (the mean of the population), and the population is assumed to be of normal form, one calculates two limits, Li and L (Li < L ), specifying the interval by means of the following equation ... 

The recently coined phrase legally binding analytical results has been enthusiastically adopted by numerous authorities in response to problems raised in the administration of justice by such statistically sophisticated concepts as confidence coefficient, confidence interval, and the like. Reproducibility has become the primary criterion applied to analytical test results in a legal setting, not necessarily correctness. This may appear to be an unscientific development, but it probably must be tolerated, at least within reasonable limits. In any case, this problem has been a subject of intense debate in recent years, offering the promise of welcome changes in the foreseeable future ( the theory of legal substantiation ). 

Example The reaction rate (at constant reaction conditions, i.e., constant concentration, temperature, etc.) is measured 10 times (n= 10, Table 5). The confidence coefficient is set to 95%, and the f value is then 2.262 (see Table 4 above). Thus... 

Finally, the sample mean, X, constitutes a so-called point estimate of the mean, fi, of the population horn which the sample was selected at random. Instead of a point estimate, an interval estimate of p, may be required along with an indication of the confidence that can be associated with the interval estimate. Such an interval estimate is called a confidence interval, and the associated confidence is indicated by a confidence coefficient. The length of the confidence interval varies directly with the confidence coefficient for fixed values of n, the sample size the larger the value of n, the shorter the confidence interval. Thus, for fixed values of the confidence coefficient, the limits that contain a parameter with a probability of 95% (or some other stated percentage) are defined as the 95% (or that other percentage) confidence limits for the parameter the interval between the confidence limits is referred to as the aforementioned confidence interval. ... 

The longitudinal distribution of region B obeys the normal distribution. Through t-test, the value of expectation p = 114.2822, the variance a = 0.00091, the confidence coefficient above 95% is [114.2821 114.2823]. 

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