Sample values

Preferably the transferring lab provides a sample which has already been analyzed, with the certainty of the results being known (41). This can be either a reference sample or a sample spiked to simulate the analyte. An alternative approach is to compare the test results with those made using a technique of known accuracy. Measurements of the sample are made at the extremes of the method as well as the midpoint. The cause of any observed bias, the statistical difference between the known sample value and the measured value, should be determined and eliminated (42). When properly transferred, the method allows for statistical comparison of the results between the labs to confirm the success of the transfer. 

Assume that the table represents typical production-hne performance. The numbers themselves have been generated on a computer and represent random obseiwations from a population with I = 3.5 and a population standard deviation <7 = 2.45. The sample values reflect the way in which tensile strength can vary by chance alone. In practice, a production supervisor unschooled in statistics but interested in high tensile performance would be despondent on the eighth day and exuberant on the twentieth day. If the supeiwisor were more concerned with uniformity, the lowest and highest points would have been on the eleventh and seventeenth days. 

A scientific basis for the evaluation and interpretation of data is contained in the accompanying table descriptions. These tables characterize the way in which sample values will vary by chance alone in the context of individual obsei vations, averages, and variances. 

Example What midrange of tensile values will include 95 percent of the sample values Since P[—1.96 < < 1.96] =. 95, the corresponding values of a. are... 

Example What midrange of daily tensile averages will include 95 percent of the sample values ... 

Distribution of Averages The normal cui ve rehes on a knowledge of O, or in special cases, when it is unknown,. s can be used with the normal cui ve as an approximation when n > 30. For example, with /I > 30 the inteiwals x s and x 2.s will include roughly 68 and 95 percent of the sample values respectively when the distribution is normal. 

Note that on the average 90 percent of all such sample values would be expected to fall within the interval 2.132. 

In terms of the tensile-strength table previously given, the respective chi-square sample values for the daily, weekly, and monthly figures couldbe computed. The corresponding df woiJdbe 4, 24, and 99 respec tively. These numbers would represent sample values from the respec tive distributions which are summarized in Table 3-6. 

The F distribution, similar to the chi square, is sensitive to the basic assumption that sample values were selected randomly from a normal distribution. 

Example What values of t define the midarea of 95 percent for weekly samples of size 25, and what is the sample value of t for the second week ... 

Ot = significance level, usually set at. 10,. 05, or. 01 t = tabled t value corresponding to the significance level Ot. For a two-tailed test, each corresponding tail would have an area of Ot/2, and for a one-tailed test, one tail area would be equal to Ot. If O" is known, then z would be used rather than the t. t = (x- il )/ s/Vn) = sample value of the test statistic. 

If the null hypothesis is assumed to be true, say, in the case of a two-sided test, form 1, then the distribution of the test statistic t is known. Given a random sample, one can predict how far its sample value of t might be expected to deviate from zero (the midvalue of t) by chance alone. If the sample value oft does, in fact, deviate too far from zero, then this is defined to be sufficient evidence to refute the assumption of the null hypothesis. It is consequently rejected, and the converse or alternative hypothesis is accepted. 

Consider the hypothesis Ii = [Lo- If, iri fact, the hypothesis is correct, i.e., Ii = [Lo (under the condition Of = o ), then the sampling distribution of x — x is predictable through the t distribution. The obseiwed sample values then can be compared with the corresponding t distribution. If the sample values are reasonably close (as reflectedthrough the Ot level), that is, X andxg are not Too different from each other on the basis of the t distribution, the null hypothesis would be accepted. Conversely, if they deviate from each other too much and the deviation is therefore not ascribable to chance, the conjecture would be questioned and the null hypothesis rejected. 

Since the sample value of t falls within the acceptance region, accept Hq for lack of contrary evidence i.e., there is insufficient evidence to demonstrate that thickness differs between the two selected locations. 

The decision rule for each of the three forms would be to reject the null hypothesis if the sample value oft fell in that area of the t distribution defined by Ot, which is called the critical region. Other wise, the alternative hypothesis would be accepted for lack of contrary evidence. 

I. Under the null hypothesis, it is assumed that the respective two samples have come from populations with equal proportions pi = po. Under this hypothesis, the sampling distribution of the corresponding Z statistic is known. On the basis of the observed data, if the resultant sample value of Z represents an unusual outcome, that is, if it falls within the critical region, this would cast doubt on the assumption of equal proportions. Therefore, it will have been demonstrated statistically that the population proportions are in fact not equal. The various hypotheses can be stated ... 

Conclusion. Since the sample value exceeds the critical value, it would be concluded that the process is not in control. 

In the above hst, one assumes that sample values are independent (i.e., not correlated). 

Thus, the z = 0 surface (and equivalently the z = 1 surface) is generated from the original oxidized and reduced simulations. Additional simulations were performed to sample values of the polarization coordinate away from the minima, using parameters scaled according to Eq. (19). 

Shannon s sampling theorem states that A funetion f t) that has a bandwidth is uniquely determined by a diserete set of sample values provided that the sampling frequeney is greater than 2uj, . The sampling frequeney 2tJb is ealled the Nyquist frequeney. 

Procedures can also be used to analyze straight lines with respect to slope and position, compare sample values to standard population means, compare methods, and detect differences in small samples. 

By means of numerical convolution one can obtain Xg t) directly from sampled values of G t) and Xj(t) at regular intervals of time t. Similarly, numerical deconvolution yields Xj(t) from sampled values of G(t) and Xg(t). The numerical method of convolution and deconvolution has been worked out in detail by Rescigno and Segre [1]. These procedures are discussed more generally in Chapter 40 on signal processing in the context of the Fourier transform. 

Using this notation, X corresponds to any time series of data with xt being a sampled value, and Z represents the processed forms of the data (i.e., a pattern). The z, are the pattern features, wy is the appropriate label or interpretation, is the feature extraction or data analysis transformation, and l is the mapping or interpretation that must be developed. 

The forward prediction x,(m + 1) is computed by averaging the n sample values in the future of the (m + l)st one. Since future data are required, this computation is not possible in real time and hence is noncausal. 

In the second case the potential can be modulated between two values (a reference and a sample potential) while the spectral frequencies are slowly scanned, or else the spectral data can first be collected at a reference potential, after which this is stepped to the sampling value where a second spectrum is obtained. The change in reflectivity AR/R is then computed,... 

In effect, this represents the root of a statistical average of the squares. The divisor quantity (n — 1) will be referred to as the degrees of freedom. The sample value of the standard deviation for the data given is. 3686. 

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