True mean

In the next several sections, the theoretical distributions and tests of significance will be examined beginning with Student s distribution or t test. If the data contained only random (or chance) errors, the cumulative estimates x and 5- would gradually approach the limits p and cr. The distribution of results would be normally distributed with mean p and standard deviation cr. Were the true mean of the infinite population known, it would also have some symmetrical type of distribution centered around p. However, it would be expected that the dispersion or spread of this dispersion about the mean would depend on the sample size. 

A binomial distribution has well-defined measures of central tendency and spread. The true mean value, for example, is given as... 

The shape of a normal distribution is determined by two parameters, the first of which is the population s central, or true mean value, p, given as... 

In the previous section we noted that the result of an analysis is best expressed as a confidence interval. For example, a 95% confidence interval for the mean of five results gives the range in which we expect to find the mean for 95% of all samples of equal size, drawn from the same population. Alternatively, and in the absence of determinate errors, the 95% confidence interval indicates the range of values in which we expect to find the population s true mean. 

The most commonly encountered probability distribution is the normal, or Gaussian, distribution. A normal distribution is characterized by a true mean, p, and variance, O, which are estimated using X and s. Since the area between any two limits of a normal distribution is well defined, the construction and evaluation of significance tests are straightforward. 

Consider the problem of assessing the accuracy of a series of measurements. If measurements are for independent, identically distributed observations, then the errors are independent and uncorrelated. Then y, the experimentally determined mean, varies about E y), the true mean, with variance C /n, where n is the number of observations in y. Thus, if one measures something several times today, and each day, and the measurements have the same distribution, then the variance of the means decreases with the number of samples in each day s measurement, n. Of course, other fac tors (weather, weekends) may make the observations on different days not distributed identically. 

The inference from the statistical calculations is that the true mean value of the carbon monoxide from the idling automobile has a 66.7% chance of being between 1.664% and 1.870%. The best single number for the carbon monoxide emission would be 1.767% (the mean value). 

The mean of several readings (x) will make a more reliable estimate of the true mean (yu) than is given by one observation. The greater the number of measurements (n), the closer will the sample average approach the true mean. The standard error of the mean sx is given by ... 

When a small number of observations is made, the value of the standard deviation s, does not by itself give a measure of how close the sample mean x might be to the true mean. It is, however, possible to calculate a confidence interval to estimate the range within which the true mean may be found. The limits of this confidence interval, known as the confidence limits, are given by the expression ... 

Example 4. f-Test when the true mean is known. 

Sxx is the sum of squares of the residuals, r, that are obtained when the average value Xmean is subtracted from each observation a ,. Xmean is the best estimate for the true mean fj.. When discussing theoretical concepts or when the standard deviation is precisely known, a small Greek sigma, a, is used in all other cases, the estimate Sx appears instead. 

What is the largest true mean that complies for u = 4 (double sampling, two determinations for each sample) ... 

Figure 4.34. The confidence limits of the mean of 2 to 10 repeat determinations are given for three forms of risk management. In panel A the difference between the true mean (103.8, circle ) and the limit L is such that for n = 4 the upper confidence limit (CLu, thick line) is exactly on the upper specification limit (105) the compound risk that at least one of the repeat measurements yi >105 rises from 23 n = 2) to 72% (n = 10). In panel B the mean is far enough from the SLj/ so that the CLu (circle) coincides with it over the whole range of n. In panel C the mean is chosen so that the risk of at least one repeat measurement being above the SLu is never higher than 0.05 (circle, corresponds to the dashed lines in panels A and B).

Table 4.30. The True Mean fji has to be at Least this Far from the Nearest SL...

Figure 4.35. The range available for the true mean /r as a function of the number of repeat measurements and the CV. The case discussed in the text is indicated by thick lines and circles. The SL are assumed to be 95 and 105%. For a CV = 2% the OOS risk is above 5% for n > 8, and for CV = 2.5%, n is restricted to 2. For SL = 90. .. 110%, the figure must be split in the middle and the upper part shifted by +5%, the lower part by -5%.

Value given is the sample size required to estimate the average emission rate with 95% confidence that the estimate will be within 20% of the true mean. 

Though these conditions will not be strictly satisfied in practical heat exchangers, the Ft values obtained from the curves will give an estimate of the true mean temperature difference that is sufficiently accurate for most designs. Mueller (1973) discusses these... 

Secondary t-p. "THE GOLDEN FLEECE, or, THE FLOWER OF TREASURES In which is succintly and methodically handled, the Stone of the philosophers, his excellent effect admirable Virtues and The better to attain to the Original true means of Perfection. Inriched with Figures representing the Colours to rise as they suooesstvely appear in the Practise of this Blessed Work. By that great Philosopher SOLOMON TRISMOSIN Master to Paracelsus"... 

The confidence interval gives the range of values within which the true mean is likely to lie, at a stated level of confidence. It is calculated by multiplying the standard deviation of the mean by the appropriate value of t(v>... 

Once the reliability of a replicate set of measurements has been established the mean of the set may be computed as a measure of the true mean. Unless an infinite number of measurements is made this true mean will always remain unknown. However, the t-factor may be used to calculate a confidence interval about the experimental mean, within which there is a known (90%) confidence of finding the true mean. The limits of this confidence interval are given by ... 

With what confidence can the mean of a set of experimental results be quoted Calculate the confidence interval (equation as a measure of the true mean (2.7))... 

If the standard deviation for a method is known, how many results must be Use the confidence interval method obtained to provide a reasonable estimate of the true mean (equation (2.7))... 

Figure 2.4(a) shows normal error curves (B and S) with true means pB and ps for blank and sample measurements respectively. It is assumed that for measurements made close to the limit of detection, the standard... 

It can be inferred that the true meaning of the love is a rose metaphor is that... 

The following data (Table 2) represent the weight percent measurements of an undesirable product in a process stream. Find a 90% confidence interval for its true mean. Assume normality and a = 0.25. 

Note that, by construction, all notional particles are identically distributed. Thus, in the absence of deterministic errors caused by using

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