Sampling variation

The unknown quantities of interest described in the previous section are examples of parameters. A parameter is a numerical property of a population. One may be interested in measures of central tendency or dispersion in populations. Two parameters of interest for our purposes are the mean and standard deviation. The population mean and standard deviation are represented by p and cr, respectively. The population mean, p, could represent the average treatment effect in the population of individuals with a particular condition. The standard deviation, cr, could represent the typical variability of treatment responses about the population mean. The corresponding properties of a sample, the sample mean and the sample standard deviation, are typically represented by x and s, which were introduced in Chapter 5. Recall that the term parameter was encountered in Section 6.5 when describing the two quantities that define the normal distribution. In statistical applications, the values of the parameters of the normal distribution cannot be known, but are estimated by sample statistics. In this sense, the use of the word parameter is consistent between the earlier context and the present one. We have adhered to convention by using the term parameter in these two slightly different contexts. 

An expression that defines how individual observations are used to derive a numerical estimate is called an estimator (much like a formula is used to calculate a number). The sample mean. 

Inferential statistics comprises two distinct, although closely related, procedures. In each case observations from a sample are used to  

Inferences about a population are made on the basis of a sample taken from that population. The process of inferential statistics requires  

We discuss the former method, confidence intervals, first, after a necessary introduction to the concept of sampling variation. The latter method (hypothesis testing) is discussed later. First, however, it is useful to introduce a few other ideas. 

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The model of simple competitive antagonism predicts that the slope of the Schild regression should be unity. However, experimental data is a sample from the complete population of infinite DR values for infinite concentrations of the antagonist. Therefore, random sample variation may produce a slope that is not unity. Under these circumstances, a statistical estimation of the 95% confidence limits of the slope (available in most... 

This yields an estimate for the bias (intercept) a and slope b needed to correct predictions yg from the new (child) instrument that are based on the old (parent) calibration model, b. The virtue of this approach is its simplicity one does not need to investigate in any detail how the two sets of spectra compare, only the two sets of predictions obtained from them are related. The assumption is that the same type of correction applies to all future prediction samples. Variations in conditions that may have a different effect on different samples cannot be corrected for in this manner. 

Sternson et al. [58] used a high performance liquid chromatographic method for the analysis of miconazole in plasma. Miconazole was extracted from alkalinized plasma with n-heptane-isamyl alcohol (98.5 1.5) and separated by high performance performance liquid chromatography on p-Bondapak Ci8 with ultraviolet detection at 254 nm. The mobile phase was methanol-tetrahydrofuran-acetate buffer (pH 5) (62.5 5 32.5) containing 5 mmol octanesulfonate per liter. The flow rate was 2 mL/min. Recovery was 100%. The relative standard deviation for injection-to-injection reproducibility was 0.4% and that for sample-to-sample variation was 5% at high miconazole concentrations (30 pg/mL) and 1% at low (1 pg/mL) concentrations. The limit of detection was 250 ng/mL. 

Sample Preparation Effects Many methods require the sample to be treated in some way before the analyte can be determined. Examples include drying, grinding or blending of the sample, and extraction or digestion of the sample. Variations in the conditions under which these activities are carried out (e.g. extraction temperature and time, solvent composition) may affect the final result. 

On the eighteenth day the analysis of variance and the standard deviation presented in Table III show that the between samples variation was dominant and remained so throughout the season. 

Sampling with a frequency below the range will ensure that the process autocorrelation in effect reduces the sampling variation, dependent upon the effect of possibly combining increments into composite samples (see further below). 

We have seen in the previous chapter that it is not possible to make a precise statement about the exact value of a population parameter, based on sample data, and that this is a consequence of the inherent sampling variation in the sampling process. The confidence interval provides us with a compromise rather than trying to pin down precisely the value of the mean p or the difference between two means — p2> for example, we give a range of values, within which we are fairly certain that the true value lies. 

Then, % NG = 2.523VN/0.1W, where V is the vol of Ti(III) corrected for a blank, N is the normality of the Ti(Ilt), and W is the weight of the NG sample. Variations of this redox titration may be found in Refs 3, 4 8 Determination of Purity of NG by the Dewarda Method. This method involves the saponification of NG to glycerol and nitrate ion. The latter is.then reduced with Dewarda s alloy, a mixture of Al, Zn and Gi. The resultant ammonia is thin distd out and titrated (Refs 2, 5, 10 St 17)... 

Figures 1 and 2 illustrate how the determination of false-positive and -negative inference errors varies with sample size. In Figure 1, the sample size is 50. To be 95 percent confident that an observed incidence difference between the experimental and control groups is really evidence of carcinogenicity rather than sampling variation, the difference in cancer rates between the two groups must be at least 23 percent.

If researchers were certain their results were not the result of sampling variation, they would be said to be 100 percent confident of their results. If errors from sampling variation were only 5 percent likely, researchers would be 100 percent minus 5 percent (or 95 percent) confident of their results. 

In this paper, we define an accuracy standard in terms of its two statistical parameters. However, in order to evaluate the accuracy of a particular method in terms of its statistical parameters, we have the problem that estimates of the method s statistical parameters are themselves subject to random sampling variations because the estimates must be calculated from only a finite number of replicate samples. 

The principle of PCA consists of finding the directions in space—known as principal components (PCs)—along which the data points are furthest apart. It requires linear combinations of the initial variables that contribute most to making the samples different from each other. PCs are computed iteratively, with the first PC carrying the most information, that is, the most explained variance, and the second PC carrying most of the residual information not taken into account by the previous PC, and so on. This process can go on until as many PCs have been computed as there are potential variables in the data table. At that point, all between-sample variation has been accounted for, and the PCs form a new set of axes having two... 

From these simple calculations the ANOVA table can be constructed together with the HORRAT ratio. An important difference from the conventional ANOVA table is that it allows for the sample variation to be accounted for. The general ANOVA table and calculated values for S samples are shown in Tables 28 and 29 but the example uses only two. However, if more than two samples are desired/available the procedure can be extended. 

In analyzing a lot with random sample variation, you find a sampling standard deviation of 5%. Assuming negligible error in the analytical procedure, how many samples must be analyzed to give 95% confidence that the error in the mean is within 4% of the true value Answer the same question for a confidence level of 90%. 

Fat content and temperature have been related to the density of creams. Phipps (1969) devised a nomograph covering up to 50% fat and temperatures from 40 to 80 °C. Homogenization slightly increases the density of whole milk but not of skim milk, and sterilization decreases the density of both milks (Short 1956). These changes are very small and the sample-to-sample variation is large thus, they are essentially negligible. 

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