Confidence intervals on the mean

Figure 2.8 95% confidence intervals on the means shown in figure 2.1 ... 

Even when an attempt is made to control variables in the manner proposed above, there will still be local variability that cannot be encompassed if sampling is restricted to a few soil pits. Liski (1996) has suggested, for a boreal forest on a sandy substrate in Finland, that a minimum of 30 samples are required for a 10% confidence interval on the mean value obtained for the carbon inventory, and the number of samples required is likely to be higher than this for more heterogeneous tree-grass ecosystems. Carter et al. (1998) found that 15 random soil samples were required from 1° X 1° grid cells in Queensland (Australia) to define the average SOC content of a cell to within 10% of the true mean, while about 40 samples were required for an estimate to within 5% of the mean. 

FIGURE 7. Temperature dependence of the tensile fracture properties ofHiPerComp Composites. Error bars represent 95% confidence intervals on the means. 

Figure 9.17 (a, b, c) The release of Ti, A1 and V from the Ti alloy after 1, 2 and 4 weeks of immersion D.L. indicates the detection limit of each element. Error bars represent the 95% confidence interval on the means. (Ducheyne and Healy, 1988.)... 

Figure 9. Average elemental C concentrations by the reflectance method in the Los Angeles Basin 1958-1972, in y-gm ((Q) air monitoring station). Error bounds represent a 95% confidence interval on the long-term means.

UNODC has also started to conduct yield surveys in some countries, measuring the yield of test fields, and to develop methodologies to extrapolate the yields from proxy variables, such as the volume of poppy capsules or the number of plants per plot. This approach is used in South-East Asia as well as in Afghanistan. All of this is intended to improve yield estimates, aiming at information that is independent from farmers reports. The accuracy of the calculated yields depends on a number of factors, including the number of sites investigated. In the case of Afghanistan the confidence interval for the mean yield results in the 2006 survey was, for instance, +/- 3% of the mean value (tt= 0.1). 

Based on a sample of 65 observations from a normal distribution, you obtain a median of 34 and a standard deviation of 13.3. Form a confidence interval for the mean. (Hint Use the asymptotic distribution. See Example 4.15.) Compare your confidence interval to the one you would have obtained had the estimate of 34 been the sample mean instead of the sample median. 

Confidence intervals can be generated for any population parameter. Specifically, a two-sided confidence interval about the mean is an interval that contains the true unknown mean with a specified degree of confidence, 100(1 - a)%. The form of this equation, which depends on the sampling plan, is as follows. For sampling plan 1... 

Using a criterion based on stages 2 and 3 of the USP 25 dissolution test, a lower one-sided 90% confidence interval for the population mean is 93.84 -1.311(3.47)/V30 = 93.01. Since the lower bound on the confidence interval for the mean is greater than Q, these results would pass the criterion. 

In Example 2.3, we have calculated that 14 samples are needed to reach the decision with a 95 percent level of confidence. To be on the safe side, we collected and analyzed 20 samples. The collected samples have the concentrations of lead ranging from 5 to 210mg/kg the mean concentration is 86 mg/kg the standard deviation is 63 mg/kg and the standard error is 14mg/kg. From Appendix 1, Table 2 we determine that the t-value for 19 degrees of freedom (the number of samples less one) and a one-sided confidence interval for a — 0.05 is 1.729. Entering these data into Equation 10, Appendix 1, we calculate the 95 percent confidence interval of the mean 86 24 mg/kg. The upper limit of the confidence interval is 110 mg/kg and it exceeds the action level. Therefore, the null hypothesis Hq p > lOOmg/kg, formulated in Example 2.2 is true, as supported by the sample data. Based on this calculation we make a decision not to use the soil as backfill. 

Frequentist methods are fundamentally predicated upon statistical inference based on the Central Limit Theorem. For example, suppose that one wishes to estimate the mean emission factor for a specific pollutant emitted from a specific source category under specific conditions. Because of the cost of collecting measurements, it is not practical to measure each and every such emission source, which would result in a census of the actual population distribution of emissions. With limited resources, one instead would prefer to randomly select a representative sample of such sources. Suppose 10 sources were selected. The mean emission rate is calculated based upon these 10 sources, and a probability distribution model could be fit to the random sample of data. If this process is repeated many times, with a different set of 10 random samples each time, the results will vary. The variation in results for estimates of a given statistic, such as the mean, based upon random sampling is quantified using a sampling distribution. From sampling distributions, confidence intervals are obtained. Thus, the commonly used 95% confidence interval for the mean is a frequentist inference... 

In order to compensate for the uncertainty incurred by taking small samples of size n, the / probability distribution shown in Figure 3.2 is used in the calculation of confidence intervals, replacing the normal probability distribution based on z values shown in Figure 3.1. When n > 30, the /-distribution approaches the standard normal probability distribution. For small samples of size n, the confidence interval of the mean is inflated and can be estimated using Equation 3.9... 

A confidence interval for the mean is derived from sample data and allows us to establish a range within which we may assert that the population mean is likely to lie. In the case of 95 per cent CIs, such statements will be correct on 95 per cent of occasions. In the remaining 5 per cent of cases, particularly misleading samples will produce intervals that are either too high or too low and do not include the true population mean. 

The confidence interval on the slope can be calculated from the variance of the slope in the same manner as was used to determine the confidence range for the mean. Thus for a 95 percent confidence interval on the slope... 

The great similarity of Equations 37 and 38 highlights the common mechanism of action of two superficially different types of amides. In each of the above examples, compounds having log P values below the optimum of log P0 were studied. Hence, addition of a term in (log P)2 did not result in an improved correlation in either case. The confidence intervals on the intercepts of Equations 37 and 38 are tighter than those on eight sets of parabolic equations correlating hypnotic activity of barbiturates in a variety of animals (22). However, for six of the parabolic equations with moderately good confidence intervals, a mean intercept... 

A chemist obtained the following data for the alcohol content of a sample of blood % C2H5OH 0.084, 0.089, and 0.079. Calculate the 95% confidence interval for the mean assuming (a) the three results obtained are the only indication of the precision of the method and (b) from previous experience on hundreds of samples, we know that the standard deviation of the method s = 0.005% C2H5OH and is a good estimate of a. 

Systematic error in an analytical method must be determined and corrected for. We have seen that systematic error is assessed by making a measurement on a certified reference material (sometimes just referred to as a CRM). The mean of a number of determinations, x, can be used to decide if the systematic error is significant by using the equation for a confidence interval of the mean... 

Figure 3.8 Means and 95% confidence intervals on the analysis of a sample of glucose by a spectrophotometric method and an enzyme method.

The assessment of bioequivalence is based on 90% confidence intervals for the ratio of the population geometric means (test/reference) for the parameters under consideration. This method is equivalent to two one-sided tests with the null hypothesis of bio-inequivalence at the 5% significance level. Two products are declared bioequivalent if upper and lower limits of the confidence interval of the mean (median) of log-transformed AUC and Cmax each fall within the a priori bioequivalence intervals 0.80-1.25. It is then assumed that both rate (represented by Cmax) and extent (represented by AUC) of absorption are essentially similar. Cmax is less robust than AUC, as it is a single-point estimate. Moreover, Cmax is determined by the elimination as well as the absorption rate (Table 2.1). Because the variability (inter- and intra-animal) of Cmax is commonly greater than that of AUC, some authorities have allowed wider confidence intervals (e.g., 0.70-1.43) for log-transformed Cmax, provided this is specified and justified in the study protocol. 

Confidence intervals on the slope of the line b and the intercept a for a given significance levd may be computed from the equations given by Till (1974 — p.97). Thus confidence intervals on values ofy for a number of values of x may be used to draw a confidence band on the regression line. This confidence band will be wider at the ends of the fitted line because there are more points near the mean values. 

Exercise 2.18. The two 95% confidence intervals for the mean mass calculated from the masses of single beans have the same width, as shown in Fig. 2.11. On the other hand, the intervals for the number of beans in 1 kg obtained from the same data have very different widths. For the bean weighing 0.1188 g the 95% confidence interval ranges from 5266 to 20,964 beans. For the one weighing 0.2673 g the limits are 2955 and 5098, giving the impression that this determination is more precise than the other. Is this true Why ... 

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... 

If you use quantitative data, be sure to include confidence limits. If you are unsure of how good the data are, say so. The fact that the numbers are questionable does not mean that they must be disregarded. Even if you use failure data that seem very unreasonable, you can still include upper and lower limits that would help bound the problem. It is possible to have fairly good confidence that the number lies somewhere between lower and upper bounds, even if you do not know the exact number. Also remember to include confidence intervals on the answer, not just the input data. Consult a reliability engineering book before manipulating failure data. 

The comprehensive certificate that accompanies each SRM 2100 set lists the fracture toughness for the set as well as the uncertainty associated with the estimate. For each billet the mean fracture toughness and the scatter in results as measured by the three test methods were statistically indistinguishable. The data were therefore pooled for each billet. The certified average fracture toughness in Table 6 is the grand mean of the pooled NIST database. The uncertainty U (with the subscript 1) is a 95 % prediction uncertainty for a single future observation and is based on the results of the NIST observations from the same normally-distributed population. The uncertainty Um (where the subscript m denotes the mean) is a 95 % confidence interval for the mean of five future observations, also based on the results of the NIST independently and randomly selected observations. The expanded uncertain-... 

Figure 16.4 The density of combed objects depends on the number of objects allowed to deposit on the surface prior to combing. Thus an increase in incubation time leads to an increase in density (a), based on measurements of AFM images (bj. The deposition rate can be increased by lowering the energy barrier, which in this case is done by lowering pH (c) or raising ionic strength (dj. The kinetics model (see text) is fitted in (a), (b), and (c) error bars represent the 95% confidence interval in the mean. ...

A further examination of the confidence intervals on the parameters shows that the F-B decay model gave the largest intervals and the power law decay model the lowest with the exponential law model having confidence intervals somewhere in between. In some cases the F-B model gave overlaps of the ko values at one temperature level to that at a higher temperature level which would mean these two values are not significantly different from each other. This was not found with the other two models which may indicate a somewhat better confidence with the results of this study in the power law or the exponential decay. 

Figure 4. Effects of mist treatments on the freezing temperatures needed to kill 50% of shoots (LTjq) harvested from red spruce seedlings on (a) 21 September, (b) 19 October, (c) 30 November, 1987. Dotted lines represent 95% confidence intervals about the mean. (Source Fowler et al. 1989).

Confidence intervals express (in quantitative terms) the percentage of times that the true (yet unknown) values of the mean and standard deviation will lie within a range of specified values. These values are specified based on the statistics of a limited number of samples. The confidence intervals for the mean, p, are the following ... 

In the MC approach ttie analysis starts with a set of sample parameters, in this case the sample mean and standard deviation, and constructs synttietic data sets with the same number of data points and with a random distribution of errors that mimic the best understanding of the error distribution in the measured data set. These synthetic data sets are then used to construct a distribution of measured parameters such as mean and standard deviation. This distribution of parameters is then used to make arguments about confidence intervals on the measured parameters. 

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