Confidence limits of the mean

instead of the distribution as such, the calculated mean Xmean is to be qualified  

95% confidence interval Cl(Xmean) 2 -t-SxIs/n centered on Xmean (1.12b) 

It is apparent that the confidence interval for the mean rapidly converges toward very small values for increasing n, because both (/) and 1/Vn become smaller. 

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Fig. 11-26 Decade-averaged data of Northern hemisphere tree ring records from 1750-1979 and 7th-degree polynomial fit of the data. The vertical extension of blocks represents 95% confidence limits of the mean. The open circles give the change of —0.65% in atmospheric CO2 observed from 1956 to 1978 by Keeling et al. (1979). (Adapted from Peng et al, 1983.)...

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).

Short-Cut Methods Based on the Range. For small samples ( from 3 to about 10) the range, R = largest) — (smallest), can be used to obtain rough uncertainty estimates. The range, when multiplied by the appropriate value of K2 from Table 4, gives an approximate but useful estimate of the standard deviation. The factor Tin Table 4 is equivalent to 1X2 yfN, where ris the critical value from the P= 95 percent column in Table 3 for the appropriate degree of freedom. Hence, to get a quick estimate of the 95 percent confidence limits of the mean of a few observations, simply calculate JR. In summary,... 

A test of the zero hypothesis shows that the slope of sample B.13 lies within the 98% confidence limits of the mean. Thus, it is not possible to find any significant diflFerence between the eleven lanthanide patterns, and the average distribution given in Table III has to be accepted as the true average lanthanide pattern for these eleven samples. 

When a small number of replicate analyses are performed their mean value, x is used to estimate the population mean, systematic errors (i.e., with unbiased measurements), a would be the true value of the analyte concentration. But because random errors occur x will not be exactly equal to a. We thus wish to define an interval within which a lies with a given degree of probability, and thanks to the central limit theorem we can apply normal distribution properties to the sampling distribution of the mean to do this. Remembering that the standard error of x is ajy/n (see above), the normal distribution shows that 95% of the values of x will lie within 1.96cr/v of the mean. That is, the 95% confidence limits of the mean are given by... 

For large samples, the confidence limits of the mean are given by... 

Calculate the 95% and 99% confidence limits of the mean for the nitrate ion concentration measurements in Table 2.1. 

The mean and standard deviation of these values are 100.5 mM and 3.27 mM respectively. There are six measurements and therefore 5 degrees of freedom. From Table A.2 the value of for calculating the 95% confidence limits is 2.57 and from equation (2.9) the 95% confidence limits of the mean are given by ... 

Calculate the 95% confidence limits of the mean. Is the spiked value of 50 ng mb within the 95% confidence limits ... 

If the 95% confidence limits of the means are calculated, the ratio of 8.5 1 may range from a minimum of 6.9 1 to a maximum of 10.6 1. Similarly, the ratio of 10.4 1 may range from 8.4 1 to 13.0 1. Therefore, within the 95% confidence limits, no two ratios are statistically different from each other... 

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