# Alternative approach for precise and accurate Student's t critical values and application in geosciences/ Metodo alterno para valores criticos de t de Student precisos y exactos con aplicacion en las geociencias.

1. Introduction

The Student's t significance test is among the most widely used statistical methods for comparing the means of two statistical samples (e.g., univariate data arrays in geosciences) as documented in several books (e.g., on geography by Ebdon, 1988; petroleum research and geosciences by Jensen et al., 1997; analytical chemistry or chemometrics by Otto, 1999 and Miller and Miller, 2005; biology and geology by Blaesild and Granfeldt, 2003; physical sciences by Bevington and Robinson, 2003; geochemistry or geochemometrics by Verma, 2005; criminology and justice by Walker and Maddan, 2005; all sciences by Kenji, 2006).

For application of the t test, critical values or percentage points are required for the pertinent degrees of freedom and the chosen confidence or significance level, generally being 95% or 0.05 (Miller and Miller, 2005) or 99% or 0.01 (Verma, 1997, 2005, 2009). Because statistical tests, including the Student t test, are applied at a pre-established confidence level, the accuracy and precision of critical values is of utmost importance, especially when the test probability estimates for two samples are close to the chosen level for the "decision" of hypotheses, as illustrated in the present work.

Similarly, the significance of linear correlation of two variables or vectors can be tested through the transformation of Pearson's linear correlation coefficient r to a Student's t value using the equation t = (r/[square root of 1-[[r.sup.2]]) x [square root of (n - 2)] where n is the number of sample pairs being regressed (Fisher, 1970; Miller and Miller, 2005). The absolute value of t calculated from the above equation is compared with the corresponding critical value for t (for (n - 2) degrees of freedom) at a chosen confidence level from the same Student's t tables. Although the parameter r can be directly used for testing the statistical significance of linear correlations, the limited nature of the tabulated critical values of r (e.g., Bevington and Robinson, 2003; Verma, 2005), for example, their scarceness for v > 100, should make the application of t test more appropriate and versatile.

The so far tabulated t values include confidence levels varying from 60% to 99.9% (two-sided or two-tailed) or from 80% to 99.95% (one-sided or one-tailed), for degrees of freedom (v) of 1(1)30(5)50(10)100 and 200, 500, 1000, and [infinity]; these are reported to two or three decimal places (e.g., Ebdon, 1988; Miller and Miller, 2005; Verma, 2005). The expression "1(1)30(5)50(10)100" means that the critical values are tabulated for all v from 1 to 30, 35, 40, 45, 50, 60, 70, 80, 90, and 100 (the numbers in parentheses refer to the step size for the initial and final values outside the brackets).

With the availability of modern analytical techniques, it is now possible to generate analytical data with greater precision than was possible in the past. A freely available software R (R Development Core Team, 2009) can be used to generate precise critical values to more decimal places than two or three currently available in tables of standard textbooks. Nevertheless, all currently available t values, including those in the software R, have been traditionally calculated from the consideration of Student's t distribution. According to the sampling theory, Student's t value represents the critical difference at a given confidence level between two small or finite samples drawn from a normal (Gaussian) distribution.

In the present paper, we used this alternative Monte Carlo type approach to simulate new precise and accurate critical values for the t test. Because such values could not be generated in a reasonable time for all sample sizes, we resorted to obtaining best-fit polynomial equations based on double and triple natural logarithm-transformations for interpolation or extrapolation of critical values as well as for probability estimates. Our results are fully consistent with the traditional approach, but our approach is more explicit especially for probability calculations. We also discuss the application of Student's t test to three case studies that highlight the importance of precise and accurate t values. Additional examples provide detailed account of arriving at central tendency and dispersion parameters for chemical variables in the geochemical reference materials granites G-2 and G-1 from U.S.A., as well as comparison of geochemical compositions of basic rocks from the Canary and Azores Islands.

2. Methodology

2.1. Monte Carlo simulation of Student's t critical values

This procedure has been recently used for generating precise and accurate critical values of 33 discordancy test variants (Verma and Quiroz-Ruiz, 2006a, 2006b, 2008, 2011; Verma et al., 2008), which have been useful for overall efficiency evaluation of these tests (Gonzalez-Ramirez et al., 2009; Verma et al., 2009) as well as for application of discordancy tests to experimental data. A different application of our Monte Carlo procedure deals with the evaluation of nuclear reactor performance (Espinosa-Paredes et al., 2010) and for evaluation of error propagation in ternary diagrams (Verma, 2012).

Our modified Monte Carlo type simulation procedure can be summarised in the following six steps:

1) Generating random numbers uniformly distributed in the space (0, 1), i.e., samples from a uniform U(0, 1) distribution: The Marsenne Twister algorithm of Matsumoto and Nishimura (1998) was employed, because this widely used generator has a very long ([2.sup.19937]-1) period, which is a highly desirable property for such applications (Law and Kelton, 2000). Thus, a total of 20 different and independent streams were generated, each one consisting of at least 100,000,000 or more random numbers (IID U(0, 1)). In this way, more than 2000,000,000 random numbers of 64 bits were generated.

2) Testing of the random numbers if they resemble independent and identically distributed IID U(0, 1) random variates: Each stream was tested for randomness using two- and three-dimensional plot method (Marsaglia, 1968; Law and Kelton, 2000). The simulated data uniformly filled the (0, 1) space as required by this randomness test in both two- and three-dimensions. Another test for randomness was also applied, which checks how many individual numbers are actually repeated in a given stream of random numbers, and if such repeat-numbers are few, the simulated random numbers can be safely used for further applications. On the average, only around one number out of 100,000 numbers in individual streams of IID U(0, 1) was repeated. Between two streams, the repeat-numbers were, on the average, around three in 200,000 combined numbers, amounting to about 150 in the combined total of 10,000,000 numbers for two streams. Thus, because the repeat-numbers were so few, all 20 streams were considered appropriate for further work.

3) Converting the random numbers to continuous random variates for a normal distribution N(0, 1): The polar method (Marsaglia and Bray, 1964) was employed instead of the somewhat slower trigonometric method (Box and Muller, 1958). Further, any other faster scheme such as the algorithm proposed by Kinderman and Ramage (1976) was not explored, because the polar method was fast enough for our purpose; furthermore, this method uses two independent streams of random numbers for generating one stream of normal random variates, which we considered an asset for our work. Two parallel streams of random numbers ([R.sub.1] and [R.sub.2]) were used for generating one set or stream of IID N(0, 1) normal random variates. Thus, from 20 different streams of IID U(0, 1) and by dividing them into two sets of 10 streams, 100 sets or streams of N(0, 1) were obtained, each one of the size ~100,000,000 or more. These N(0, 1) streams were found to be useful for simulations of critical values. The simulated data were graphically examined for normality (Verma, 2005). Practically, no repeat-numbers were found in tests with 100,000 numbers in these sets of random normal variates. Therefore, the data were considered of high quality to represent a normal distribution, and could be safely used for further applications.

4) Establishing the best simulation sizes: In order to determine the best simulation sizes, the results of mean critical values for 60% to 99.9% (two-sided confidence limits) and their respective standard error estimates were simulated for degrees of freedom (v) of 1, 2, 5, 10, 20, and 30, for 13 different simulation sizes between 10,000 and 100,000,000, and using only 10 independent streams of IID N(0, 1) normal random variates. Representative results for v = 20 are summarised in Figure 1, in which the mean critical values are shown by open circles and the standard errors by vertical error bars. Figure 1 shows that the critical values tend to stabilize as the standard errors sharply decrease with the simulation size increasing from 10,000 to 100,000,000. Therefore, for all final reports the simulation sizes were set at 100,000,000 for all degrees of freedom.

5) Computing test statistics from 100 different streams of random normal variates: For each v of Student s t critical value, several different combinations can be used to compute the critical values, because v = ([n.sub.x] + [n.sub.y] - 2) where [n.sub.x] and [n.sub.y] are, respectively, the sample sizes for the two statistical samples under consideration. The test statistic is given by the following equation (Verma, 2005):

t = [absolute value of [bar.x]-[bar.y]]/s([square root of (1/[n.sub.x]+1/[n.sub.y]))

where [absolute value of [bar.x] - [bar.y]] is the absolute difference between the two mean values, and s is the combined standard deviation of the two samples or data arrays. The parameter s was calculated as follows:

s = [square root of [[n.sub.x].summation over (i=1)][([x.sub.i]- [bar.x]).sup.2]+[[n.sub.y].summation over (i=1)][([y.sub.i]-[bar.y]).sup.2]/[(.sub.nx]+[n.sub.y]-2)

[FIGURE 1 OMITTED]

As an example, for v = 10 [n.sub.x] can vary from 1 to 6, with the corresponding [n.sub.y] varying from 11 to 6, thus obtaining 6 combinations which, when multiplied by the total number (100) of N(0, 1) streams, can provide 600 possible results of this t-statistic (v = 10). For smaller values of n, there will be less number of such combinations and vice versa. As another example, we can quote v = 20 [n.sub.x] can vary from 1 to 11, with the corresponding [n.sub.y] varying from 21 to 11, thus obtaining 11 combinations which, when multiplied by the total number (100) of N(0, 1) streams used in the present simulations, can provide 1100 possible results of this t-statistic (v = 20). Each set of calculations was carried out 100,000,000 times (as determined in this study; Figure 1).

6) Inferring critical values and evaluating their reliability: Critical values (percentage points) were computed for each of the possible sets of 100,000,000 simulated test statistic values for sample sizes of 1(1)30(5)100(10)200(50) 400(100)1000(200)2000(1000)6000. For example, for v = 10, 600 such sets were used. Each set of 100,000,000 t-statistic results were arranged from low to high values and critical values or percentage points were extracted for a total of 11 confidence levels (both two-sided and one-sided) from 50% to 99.9%. These were: confidence levels (two-sided) = 50%, 60%, 70%, 80%, 90%, 95%, 98%, 99%, 99.5%, 99.8%, and 99.9%, i.e., with significance levels [alpha] = 0.50, 0.40, 0.30, 0.20, 0.10, 0.05, 0.02, 0.01, 0.005, 0.002, and 0.001, as well as correspondingly one-sided of 75% to 99.95% with significance levels [alpha] of 0.25 to 0.0005. The final overall mean (central tendency) as well as standard deviation and standard error of the mean (dispersion) parameters for Student's t were computed from these sets of values. The standard error and the corresponding mean values were rounded following the flexible rules put forth by Bevington and Robinson (2003) and Verma (2005).

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

2.2. Polynomial fits for Student's t critical values

When a tabulated critical value is missing for a given v and [alpha], as is the case of the present work, interpolation or extrapolation of the available critical values is required. There is no clear indication on how this was done in the past except that, in an attempt to generate the best interpolated critical values, natural logarithm-transformation of v was proposed (Verma, 2009) as a means for obtaining highly precise interpolations using Statistica[C] software. We used highly precise critical values generated in this work to test 28 different regression models for obtaining both the best-fit interpolation and extrapolation equations. These models consisted of simple (i.e., without natural logarithm-transformation of n) polynomial regressions (quadratic to 8th order) to single natural-logarithm of n (ln(v), double natural-logarithm of n (ln(ln(v))) and triple natural-logarithm of n (ln(ln(ln(v)))) transformed quadratic to 8th order regressions. The best-fit equations were obtained from the combined criteria of four different fitting quality parameters: (i) the multiple-correlation coefficient ([R.sup.2]; Fig. 2); (ii) averaged sum of the squared residuals SSR/N where N is the total number of n used in the regression model (Fig. 3); (iii) averaged sum of the squared residuals of interpolation [(SSR/N).sub.int] where N is the total number n not used in obtaining the regression model that lie within the range of all n values of the regression model (Fig. 4); and (iii) averaged sum of the squared residuals of extrapolation [(SSR/N).sub.ext] where N is the total number of v that lie outside the range of all n values of the regression model (Fig.5).

2.3. Polynomial fits for Student's t probability calculations

The simulated critical values for Student's t were used to propose an explicit method based on best fit equations for the computation of probability or confidence level estimates of Student's t test corresponding to any two sets of statistical samples. Such probabilities can be calculated through commercial or freely available software packages (e.g., see Miller and Miller, 2005; Efstathiou, 2006), but the exact procedure is largely unknown. We present new "best-fit" equations based on these highly precise and accurate critical values for computing probabilities or confidence levels from them. Thus, the performance of these equations can be compared with the existing commercial software packages. The advantage is that our method is explicit and calculates the confidence level (%) that would correspond to the calculated t value for any two statistical samples.

3. Results

3.1. Critical Values

The results of our simulated critical values are presented in abridged form in Table 1. The respective standard errors are summarised in Table 2. All 76 critical values and their standard errors (Tables ES1 and ES2, respectively) are available on request to any of the authors (spv@ier. unam.mx or recrh@live.com). The actually simulated 67 critical values include the following values of v that were used for evaluating the different regression models: 1(1)30(5)100(10)160(20)200(50)400(100)1000(200)2000 In this nomenclature, the numbers in parentheses refer to the step-size or increment and the numbers before and after these parentheses are the initial and final v, respectively, for each increment. The trends of these new critical values are graphically shown in Figure 6, which highlights their non-linear nature in this bivariate plot. Similarly, new critical values were also simulated for arbitrarily chosen v = 105, 220, 380, 860, and 1100, used for testing the proposed equations for interpolation (see one such equation as inset in Figure 6) as well as for v = 3000, 4000, 5000, and 6000, used for testing of extrapolation equations. Thus, for each of the 11 confidence levels, a total of 76 critical values along with their respective standard errors were simulated (Tables ES1 and ES2).

These new critical values are highly precise because the average standard errors for two-sided confidence levels of 50%, 60%, 70%, 80%, 90%, 95%, 98%, 99%, 99.5%, 99.8%, and 99.9% are, respectively, as follows: 0.0000057, 0.0000066, 0.0000079, 0.0000096, 0.0000144, 0.0000231, 0.000054, 0.000115, 0.000278, 0.00105, and 0.00252. These errors, when expressed as relative standard errors in percent are, respectively, as follows: 0.00083%, 0.00076%, 0.00073%, 0.00069%, 0.00072%, 0.00080%, 0.0010%, 0.0012%, 0.0015%, 0.0022%, and 0.0028%. We also consider that these new critical values are accurate to the similar extent as the precision estimates, because our Monte Carlo procedure of generating random normal variates has been shown to be highly accurate (Verma, 2005; Verma and Quiroz-Ruiz, 2006a, 2006b, 2008; Verma et al., 2008). Finally, note that the most frequently used confidence levels of 95% (Miller and Miller, 2005) and 99% (Verma, 1997, 2005) are highlighted in boldface (Table ES1).

Our critical values for Student's t are consistent with those estimated by software R following standard methods (Fig. 7). Whereas the use of software R requires some programming work, our values are readily available in tabulated form as well as in electronic files. Both confidence levels of 95% and 99% most used in science and engineering applications as well as the extreme confidence levels of 50% and 99.9% are shown in Figure 7. For 50%, 95% and 99%, critical values of the present work agree with those calculated by software R within about 0.002%. For the extremely high confidence level of 99.9%, these differences reach higher values but are mostly within 0.005%. Our new critical values (Table 1 or ES1) are individually characterised by their standard error estimates (Table 2 or ES2). Nevertheless, we can conclude that the alternative approach of Monte Carlo simulation gives t critical values consistent with those obtained from the Student's t distribution (software R). In other words, we have empirically confirmed through high precision Monte Carlo simulation that the small-size sampling from normal distribution is represented by the Student's t distribution.

[FIGURE 4 OMITTED]

3.2. Critical value equations

The results of 28 regression models are summarised in Figures 2-5. Four best models from each of the four criteria (Figs. 2-5) are presented in Table 3 for 99% confidence levels (both two-sided and one-sided), whereas complete information for all confidence levels is given in Table ES3 (available from any of the authors), in which the best interpolation and extrapolation models are highlighted. None of the simpler polynomial regressions (a total of 7 models) was satisfactory (see models identified as q to p8 in Figures 2-5). New methodology of natural logarithm- (ln-) transformation of v provided better models (see the remaining 21 models in Figures 2-5), although single logarithm-transformation (ln(v) was not satisfactory (see models Iq to 18). Generally, the double(ln(ln(v))) and triple-logarithm (ln(ln(ln(v)))) transformations with the 4th to 8th order polynomials were the best models (see ll4 to ll8 and lll4 to lll8 in Table 3 or ES3 and Figures 2-5).

[FIGURE 5 OMITTED]

The best equations for all confidence levels are presented for 99% confidence level in Table 4 (complete information is provided in Table ES4 available from any of the authors). As expected, the interpolation equations provide better estimates (lower errors) than the extrapolation equations. Nevertheless, both sets of equations can be used for computing the critical values for all those v not included in Table ES1.

Recently, Verma (2012) has graphically shown how the log-transformation of the x-axis (degrees of freedom) provides "smoothing" of these curves, enabling thus a better fit to the data in the log-transformed space. Double or triple log-transformation can make these curves smoother than the simple log-transformation. Such transformations were successfully used by Verma and Quiroz-Ruiz (2008) for polynomial fits of critical values of discordancy tests. We suggest that log-transformations provide an efficient means for obtaining "best-fit" equations in other applications, in which polynomial fits without transformation fail to perform satisfactorily. We emphasise that this should be an important application of our procedure in many scientific and engineering fields.

The version of Student's t test for unequal variances (Ebdon, 1988; Jensen et al., 1997; Otto, 1999; Miller and Miller, 2005; Verma, 2005; Kenji, 2006) would also be objectively and best applied if we could estimate precise critical value for non-integer v. For such applications, the calculated v nearly always results in a non-integer number, and most text books (e.g., Ebdon, 1988; Verma, 2005) suggest truncating the v value to the integer number. We propose that it would be better to maintain the non-integer v and estimate the corresponding critical value. This would be certainly possible from the use of the critical value equations (Table ES4). The freely available software R also does this job, but it needs certain amount of programming work and the procedure is not as explicit as in this work.

3.3. Student's t confidence level calculations for two statistical samples

As an innovation, we report an explicit method to estimate the "critical" confidence level of Student's t test corresponding to any two statistical samples. Although such probability estimates can also be obtained from conventional software, the method of these calculations is not stated. Using log-transformation of critical values (Table ES1), we fitted "best" equations to our Student's t critical values; these equations for a few degrees of freedom are summarised in Table 5 (equations for all simulated degrees of freedom are listed in Table ES5 available from any of the authors). Confidence levels of Student's t test can be easily calculated by substituting the calculated t value for [t.sub.calc] in the appropriate equation proposed for given degrees of freedom for two statistical samples.

The applications to geochemical reference materials granites G-1 and G-2 from U.S.A. and basic rocks from Canary and Azores Islands presented in the next section will further clarify our proposed method.

4. Applications

It is clear that if the calculated t value ([t.sub.calc]) for a set of two statistical samples is widely different from the critical value at a given confidence or significance level and the required degrees of freedom, the statistical interpretation of Student's t test will not depend on the critical value tables, but if [t.sub.calc] is close to the tabulated critical value, the precision and accuracy of the critical value will largely determine the final interpretation and decision in favour of the null and alternate hypotheses ([H.sub.0] and [H.sub.1], respectively), i.e., either [H.sub.0] will be accepted and correspondingly [H.sub.1] will be rejected, or [H.sub.1] will be accepted and correspondingly [H.sub.0] will be rejected. We illustrate the importance of our precise critical values through a series of carefully chosen examples. For testing these hypotheses ([H.sub.0] and [H.sub.1] the strict 99% confidence level (two-sided) will be used. Of course, for such applications other confidence levels, such as 95%, could likewise be used, if desired.

The above interpretation cannot be directly compared with any commercial or freely available software, because the latter only provide the probability estimates (or significance level) corresponding to the [t.sub.calc] of the two statistical samples under evaluation. Therefore, we also computed such probability estimates from best-fitted equations (Table 5 or ES5) and compared them with independent estimates obtained from different software--Statistica[C], R, and Excel.

Further, for the applications presented here, only crude chemical compositions are evaluated. Nevertheless, the applications can be easily extended to log-transformed data (Aitchison, 1986; Egozcue et al., 2003; Agrawal and Verma, 2007) in order to comply with the coherent statistical treatment of compositional data.

4.1. Geochemistry

Sr-isotopic composition ([sup.87]Sr/[sup.86]Sr) of rocks provides constraints on geological processes (Faure, 1986, 2001). Therefore, the data quality plays an essential role in quantifying the relative importance of these processes. In this context, an example of [sup.87]Sr/[sup.86]Sr measured in the geochemical reference material JA-1 (andesite from Hakone volcano) from the Geological Society of Japan (GSJ; Internet address http://riodb02.ibase.aist.go.jp/geostand/) will be used. Let us assume that two independent trials or experiments involving two laboratories (LabA and LabB) were carried out. We further assume that each of these laboratories obtained 11 different measurements on this sample in each trial. For this purpose, we simulated, using our Monte Carlo procedure, fairly realistic data (Table 6) on this sample in the light of the actual measurements compiled by the GSJ. We would like to evaluate for each trial the null hypothesis ([H.sub.0]) that the two statistical samples of [sup.87]Sr/[sup.86]Sr (measurements from LabA and LabB) were drawn from the same population, i.e., there is no significant difference between them, and the alternate hypothesis ([H.sub.1]) that the two statistical samples of [sup.87]Sr/[sup.86]Sr were not drawn from the same population, i.e., there is a significant difference between them. The partial calculations as well as the calculated t and several tabulated critical values are summarised in Table 6. Because, in both trials, the calculated t values are less than the tabulated critical value (Miller and Miller, 2005), the null hypothesis [H.sub.0] will be true (or accepted) and, as a consequence, the alternate hypothesis [H.sub.1] will be false (or rejected), i.e., there is no significant difference between these sets of data from the two laboratories. Note that the t test is applied at the strict 99% confidence level. If the tabulated critical values by Verma (2005) were used, the interpretation will be just the opposite, i.e., for both trials, there is significant difference between these sets of data from the two laboratories. However, if the statistical inference were drawn from the new precise critical values for Student's t test, for Trial 1 [H.sub.0] will be accepted and [H.sub.1] rejected, whereas for Trial 2, [H.sub.0] will be rejected and [H.sub.1] accepted (Table 6). We therefore conclude that it is safer to use more precise and accurate critical values to draw statistical inferences.

In an analogous manner, we now compare the performance of our work with Statistica[C], R, and Excel results (Table 6). For Trail 1, the probability estimates are all >0.01, except for R (=0.01). Because the hypotheses [H.sub.0] and [H.sub.1] are being evaluated at 99% confidence level or, equivalently, at 0.01 significance level, the probability corresponding to the set of samples represented by Trail 1 should be >0.01 for [H.sub.0] to be accepted. Therefore, for Trail 1, Statistica[C], Excel and our work suggest that [H.sub.0] is accepted and correspondingly, [H.sub.1] is rejected. For Trail 2, on the other hand, because the probability estimates from all packages and this work are <0.01 (Table 6), the interpretation would be that [H.sub.1] is accepted and correspondingly, [H.sub.0] is rejected.

4.2. Chemistry

Our second example concerns simulated data for paracetamol concentration in tablets (Miller and Miller, 2005). We envision the experiment either by using two analytical methods (Trial 1) or by two analysts using the same method (Trial 2); the results are summarised in Table 7. Similar to the geochemical application, the inference at 99% confidence level will depend on the critical values used for evaluating these experiments. For Trial 1 (results of two analytical methods; Table 7), using any of the literature critical values (Miller and Miller, 2005; Verna, 2005, 2009), [H.sub.0] will be accepted and [H.sub.1] will be rejected, whereas in the light of the new interpolated critical value obtained from the best-fit equation (Table 4 or ES4), [H.sub.0] will be rejected and [H.sub.1] will be accepted. On the other hand, for Trial 2 (results of two analysts; Table 7), [H.sub.0] will be accepted and [H.sub.1] will be rejected according to the literature critical values (Miller and Miller, 2005; Verma, 2005), but [H.sub.0] will be rejected and [H.sub.1] will be accepted from Verma (2009) as well as from the present work. The presently available precise critical values and best-fit equations could therefore be advantageously used in all future applications in chemistry.

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

In terms of probability estimates from this work and their comparison with Statistica[C], R and Excel (Table 7), all results are consistent to infer that [H.sub.1] is accepted and correspondingly, [H.sub.0] is rejected, because all probabilities are <0.01.

4.3. Medicine

Our third example deals with monitoring the change in glucose levels of a group of patients with schizophrenia or schizoaffective disorder over several weeks of treatment with an antipsychotic medication (Lindenmayer et al., 2003). We present just one set of simulated data in Table 8. The application of Student's t test at 99% confidence level shows that [H.sub.0] is accepted and [H.sub.1] is rejected when the literature critical values are used, but the opposite is the case when the presently simulated precise critical value from the best-fit equation (Table 4 or ES4) is used. Thus, the medical treatment will be considered effective only from the present critical values. The same conclusion of successful medical treatment ([H.sub.0] rejected and [H.sub.1] accepted) is also consistently reached from the probability estimates (Table 8).

4.4. Geochemical reference material granite G-1 from U.S.A.

In the field of geochemistry, the most important use of Student's t test could be related to the evaluation of data quality of traditionally available reference materials (RM; see Verma, 2012). Geochemical data for major- and trace-elements in RM are generally obtained from the application of different analytical methods in laboratories worldwide (e.g., Gladney et al, 1991, 1992; Imai et al, 1995; Verma, 1997, 1998; Velasco-Tapia et al, 2001; Marroquin-Guerra et al, 2009; Pandarinath, 2009a).

The geochemical data for granite G-1 were compiled from a published report by Gladney et al. (1991). These authors applied the so called "two standard deviation method" to eliminate discordant observations and also reported their "recommended" or "consensus" values for different elements. This particular granite sample is one of the first international geochemical reference materials proposed long ago by the United States Geological Survey (e.g., see Flanagan, 1967). The compiled data were first processed by applying all thirty-three discordancy tests at 95% confidence level through DODESSYS software (Verma and Diaz-Gonzalez, 2012). This confidence level was chosen to make comparable the multiple test method of Verma (1997) used here with the "two standard deviation method" practiced by Gladney et al. (1991). The statistical results from the outlier-free data of G-1 (for the same chemical elements as for G-2) are summarised in Table 9. Also included for comparison are the consensus values from Gladney et al. (1991). Elements with at least five valid observations were reported.

With the aim of objectively comparing these two sets of results, one-sided t test, in combination with Fisher F test (Verma, 2005; Cruz-Huicochea and Verma, 2013) was applied to these data (Table 9) at the same 95% confidence level as done for DODESSYS. The null hypothesis ([H.sub.0]) was that this work provided the statistically similar standard deviation (from F test) and mean values (t test) as the literature values, whereas the alternate hypothesis ([H.sub.1]) was that this work provided lower or higher standard deviation and lower or higher mean, respectively, from the F and t tests. The following elements showed significantly lower standard deviation for this work as compared to the literature values: Si, Fe, K, Nd, Sm, Ho, Tm, Yb, Co, Li, and Sr. None of the elements showed statistically significantly higher standard deviation for this work than that reported by Gladney et al. (1991). This implies that the multiple test method of Verma (1997) practiced here performs better (provides less dispersion) than the two standard deviation method used by Gladney et al. (1991). The application of t test showed that none of the mean values obtained in the present work was significantly lower or higher than the literature values at 95% confidence level.

4.5. Geochemical reference material granite G-2 from U.S.A.

In order to establish central tendency (mean) and dispersion (standard deviation, standard error of the mean, or confidence interval) parameters, the data from different analytical methods should first be evaluated from significance tests (Verma, 1998). However, the application of F and t tests requires that the user assures that the individual groups of data have been drawn from normal populations (see Jensen et al., 1997). Objective ways to achieve this goal can be found in Barnett and Lewis (1994), Verma et al. (2009), or Gonzalez-Ramirez et al. (2009).

For geochemical data from two different analytical methods and at the chose confidence level, depending on the results of whether the null hypothesis [H.sub.0] (both sets of statistical samples drawn from the same normal population) is accepted and correspondingly, the alternate hypothesis [H.sub.1] is rejected, or otherwise, i.e., [H.sub.0] is rejected and correspondingly, [H.sub.1] is accepted. Thus, these data from two different methods can or cannot be combined for arriving at pooled statistical parameters (Verma, 1998). In other words, in the first result of significance tests ([H.sub.0] accepted) the data from the two methods can be combined to calculate the central tendency and dispersion parameters. In the second result ([H.sub.1] accepted), the identity for one or more analytical methods different from the remaining groups will have to be maintained whereas the similar method groups could be combined. The statistical parameters will then be calculated individually for them.

A computer program was written in Java that is much more efficient than the available software for statistical processing of multi-variate geochemical data, especially those arising from inter-laboratory trials. An updated version of this program (UDASYS--Univariate Data Analysis SYStem; Verma et al., 2013) with all discordancy and significance tests and capable of efficiently processing extensive experimental databases, is now available from the authors.

We used the current preliminary version of this program to process an unpublished compilation (by S.P. Verma and R. Gonzalez-Ramirez) of geochemical data for granite G-2 (a reference material from the United States Geological Survey, U.S.A.). Prior to the application of t test at 99% confidence level, F test was applied at 99% confidence level to all data sets in this file to determine the type of t-statistics applicable for each pair of groups or statistical samples. Although application of ANOVA could be a better procedure to statistically handle these extensive geochemical data (e.g., Jensen et al., 1997), we highlight the application of t test to all possible groups of data (or pairs of statistical samples). For the application of t test, the implicit assumption is that both samples be drawn from normal population(s), which was tested and its validity assured through software DODESSYS (Verma and Diaz-Gonzalez, 2012) by applying all single-outlier type tests (Verma et al, 2009) at 99% confidence level. The method grouping was the same as that proposed by Velasco-Tapia et al. (2001).

As a result of the application of t-test to all pairs of data for G-2, no statistically significant differences were observed at 99% confidence level for six major-elements (Si, Ti, Al, Mn, [H.sub.2][O.sup.+], and [H.sub.2][O.sup.-], all expressed as %m/m), nine rare earth elements (Pr, Nd, Sm, Gd, Dy, Ho, Er, Yb, and Lu), eighteen more common trace-elements (B, Be, Cr, Cu, Ga, Hf, Li, Nb, Pb, Rb, Sb, Sr, Ta, U, V, Y, Zn, and Zr) and other ten trace-elements (Ag, As, Bi, Br, C, Hg, Mo, S, Se, and W). Confidence levels were also individually calculated for all pairs of method groups. One pair of method groups showed differences in their mean values for two major-elements (Fe and Mg), five rare earth elements (La, Ce, Eu, Tb, and Tm), and five trace-elements (Cs, Sc, Cd, Cl, and F), whereas two or more groups of data showed differences for the remaining four major-elements (Ca, Na, K, and P) and six trace-elements (Ba, Co, Ni, Th, Ge, and Tl). The data obtained from the method groups showing significant differences from the remaining methods were left out before applying discordancy tests to the combined data and computing statistical parameters.

The applications of discordancy and significance tests as explained above enabled us to compute statistical parameters for geochemical data of G-2 (Table 10). These data include both central tendency and dispersion parameters for all ten major-elements from Si to P, water content ([H.sub.2][O.sup.+] and [H.sup.2][O.sup.-]), fourteen rare earth elements from La to Lu, twenty-four commonly measured trace-elements from B to Zr and nineteen other trace-elements from Ag to W (Table 10). These values for sixty-nine chemical parameters, including the lower and upper 99% confidence limits of the mean, will be useful for calibrating analytical instruments and evaluating data quality of individual laboratories. The application of t test has been for arriving at reliable statistical estimates (Table 10) for granite G-2.

4.6. Comparison of geochemical reference materials G-1 and G-2 from U.S.A.

The granite standard G-2, collected a few km away from the site of G-1, was proposed to replace the already exhausted supply of G-1. We considered it interesting to evaluate the hypothesis that there are no significant differences between the chemical composition of the two standards (Table 11) by applying Student t test (two-sided) at 99% confidence level (see Verma, 2005 for more details). We also evaluated if the new standard G-2 showed higher or lower concentrations than the older standard G-1 by applying Student t test (one-sided) at 99% confidence level. However, the complete chemical data for each standard were newly processed for discordant outliers by applying all single as well as multiple outlier discordancy tests (Verma and Diaz-Gonzalez, 2012) and t test was applied to such discordant outlier free data sets for each element. The results are summarised in Table 11.

The following elements showed significant differences between G-1 and G-2 at 99% confidence level (see [H.sub.0] "false" in the column "Two-sided' in Table 11; this word could as well be "rejected" instead of false): all major elements (Si, Ti, Al, Fe, Mn, Mg, Ca, Na, K, and P); [H.sub.2][O.sup.+] and [H.sub.2][O.sup.-]; rare earth elements La, Sm, Eu, Gd, and Lu; and most other commonly measured trace elements (B, Ba, Be, Co, Cr, Cs, Cu, Ga, Hf, Li, Nb, Ni, Pb, Rb, Sb, Sr, Ta, Th, U, V, Zn, and Zr); and several less frequently measured trace elements (As, Au, Bi, F, Hg, In, Mo, Sn, Tl, and W). Consequently, the alternate hypothesis ([H.sub.1]) of the existence of a statistically significant difference would be true or accepted for these elements. On the contrary, the following elements did not show significant differences between G-1 and G-2 (see [H.sub.0] "true" in the column "Two-sided' in Table 11; this word could as well be "accepted" instead of true): rare earth elements Ce, Pr, Nd, Tb, Dy, Ho, Er, Tm, and Yb; and trace elements Y, Ag, Br, C, Cd, Cl, Ge, Ir, S, and Se.

Similarly, the same elements also showed significantly higher or lower concentration values (see [H.sub.0] "false" in the column "One-sided" in Table 11); Y is also added to this list. Consequently, as for the significant differences all elements listed except Y (see [H.sub.0] "true" in the column "One-sided" in Table 11), the one set of data are not higher or lower from the other set.

Therefore, we can safely conclude that the more recently prepared granite reference material G-2 has significantly different chemical composition than the earlier granite material G-1 although both were sampled from nearby localities in the same intrusive body.

4.7. Basic rocks from Canary and Azores Islands

Both groups of islands in the Atlantic Ocean probably originated by similar tectonic processes and, therefore, chemical compositions of similar magma types from these islands are likely to be similar. It may be interesting to explore the application of significance tests (F and t) to geochemical data from these islands.

Geochemical data for igneous rocks from two groups of Islands were compiled from the following sources: for Canary Islands, Abley et al. (1998), Krochert and Buchner (2009), Longpre et al. (2009), Praegel and Holm (2006), and Thirlwall et al. (2000); and for Azores, Beier et al. (2006), de Lima et al. (2011), Storey et al. (1989), and Turner et al. (1997). The major element data were first adjusted to 100% on an anhydrous basis after Feoxidation adjustment and CIPW norms were calculated through SINCLAS software (Verma et al., 2002, 2003). Both sets of data were processed for discordant outliers through DODESSYS (Verma and Diaz-Gonzalez, 2012) and then F and t tests were applied to them at 99% confidence level. The results are summarised in Table 12.

Most major elements and normative minerals, rare earth elements Eu-Lu, and several trace elements did not show significant differences between the Canary and Azores Islands (see "true" in the [H.sub.0] "Two-sided" column of Table 12). The elements or normative mineral parameters that showed significant differences at 99% confidence level (see "false" in the [H.sub.0] "Two-sided" column of Table 12) were as follows: major elements [(MnO).sub.adj], [([Na.sub.2]O).sub.adj], and ([P.sub.2][O.sub.5]) ; normative minerals [(Ne).sub.norm], [(Fs).sub.norm], [(Fo).sub.norm], [(Ol).sub.norm], and [(Ap).sub.norm]; Mg_value; rare earth elements La, Ce, Pr, Nd, and Sm; and trace elements Cr, Nb, Pb, Sr, Ta, Th, U, Y, Zn, and Zr.

Basic rocks from the Canary Islands showed significantly higher concentrations than the Azores Islands for the following elements (see "false" in the H0 "One-sided" column and the mean concentrations in Table 12): [(MnO).sub.norm] [([Na.sub.2]O).sub.adj][([P.sub.2][O.sub.5]).sub.adj], [(Ne).sub.norm], [(Fs).sub.norm] and [(Ap).sub.norm]; rare earth elements La, Ce, Pr, Nd, and Sm; and trace elements Nb, Sr, Ta, Th, U, Y, Zn, and Zr. On the other hand, a few elements in basic rocks of the Canary Islands showed lower concentrations than the Azores Islands: [(MgO).sub.adj], [(Fo).sub.norm], [(Ol).sub.norm], Mg_value, Cr, and Pb. Once established such similarities and differences on strictly statistical basis, geological and petrogenetic reasons can then be explored to explain them.

4.8. Other scientific and engineering fields

Although to limit the length of this paper we have formulated only a few examples, numerous such cases can be built from all other areas of science and engineering where quantitative data are interpreted for their statistical significance. Just to mention a few areas where these critical values will be useful, they are: agriculture, astronomy, biology, biomedicine, biotechnology, criminology and justice, environmental and pollution research, food science and technology, geochronology, meteorology, nuclear science, palaeontology, petroleum research, quality assurance and assessment programs, soil science, structural geology, water research, and zoology.

The correct procedure would be to fulfil the requirement that the statistical samples should have been drawn from normal populations without any statistical contamination, which should be ascertained through discordancy tests (Barnett and Lewis, 1994; Verma, 1997, 2005, 2012; Verma et al, 1998) before the application of Student's t test. The new software DODESSYS (Verma and Diaz-Gonzalez, 2012) should prove useful for this purpose.

Finally, numerous researchers who have applied F and t tests to their data (e.g., Diaz-Gonzalez et al., 2008; Armstrong-Altrin, 2009; Hernandez-Martinez and Verma, 2009; Gomez-Arias et al., 2009; Gonzalez-Marquez and Hansen, 2009; Madhavaraju and Lee, 2009; Marroquin-Guerra et al., 2009; Pandarinath, 2009a, 2009b; Alvarez del Castillo et al., 2010; Zeyrek et al., 2010; Torres-Alvarado et al., 2011; Wani and Mondal, 2011), will also benefit from these new critical values for their future work.

5. Conclusions

New highly precise and accurate critical values have been generated for Student's t test. Best-fit regression equations based on double or triple natural-logarithm transformations of degrees of freedom have also been proposed for computing critical values for other degrees of freedom not-tabulated in the present work, including fractional degrees of freedom. These critical values agree with those provided by software R. Although only a few examples highlight the importance of new critical values for inferring the validity of the null or alternate hypothesis, this work should be useful for many other scientific and engineering fields. Application of significance and discordancy tests to geochemical reference materials G-1 and G-2 from U.S.A. and to basic rocks from the Canary and Azores Islands enabled us to successfully characterize and objectively compare a large number of chemical parameters. Such a procedure is therefore recommended to be routinely used in all areas of geosciences.

http://dx.doi.org/10.5209/rev_JIGE.2013.v39.n1.41747

Acknowledgements

We are grateful to Alfredo Quiroz-Ruiz for help and guidance in the procurement and use of modern computing facilities and participation in the initial stage of this work. The Sistema Nacional de Investigadores (SNI-Conacyt) provided the necessary financial support for purchasing the special computing facilities and for assigning an assistantship to the second author (RCH); both these actions enabled the development of the research work leading to this paper on critical values and other reports currently under preparation as well as for the culmination of Bachelor level thesis of RCH in Computer Engineering. We also thank the two reviewers (Dr. Umran Serpen and Prof. Jeffery T Walker) for kindly reading our earlier manuscript; we were glad to see that both of them highly appreciated our work, suggesting publication in its present form. In any case, we have tried to clarify our presentation at places where we felt necessary. We are also grateful to the editors-in-chief Dr. Jose Lopez-Gomez and Dr. Javier Martin-Chivelet, two associate editors Dr. Maria Belen Munoz Garcia and Dr. Raul de la Horra, and two additional anonymous reviewers from the Universidad Complutense de Madrid for suggestions for improvement of this work.

References

Ablay, G.J., Carroll, M.R., Palmer, M.R., Marti, J., Sparks, R.S.J. (1998): Basanite-phonolite lineages of the Teide-Pico Viejo volcanic complex, Tenerife, Canary Islands. Journal of Petrology 39, 905-936. doi: 10.1093/petroj/39.5.905

Agrawal, S., Verma, S.P. (2007): Comment on "Tectonic classification of basalts with classification trees" by Pieter Vermeesch (2006). Geochimica et Cosmochimica Acta 71, 3388-3390.

Aitchison, J. (1986): The statistical analysis of compositional data. Chapman and Hall, London, 416 p.

Alvarez del Castillo, A., Santoyo, E., Garcia-Valladares, O., Sanchez-Upton, P (2010): Evaluacion estadistica de correlaciones de fraccion volumetrica de vapor para la modelacion numerica de flujo bifasico en pozos geotermicos. Revista Mexicana de Ingenieria Quimica 9, 285-311.

Armstrong-Altrin, J.S. (2009): Provenance of sands from Cazones, Acapulco, and Bahia Kino beaches, Mexico. Revista Mexicana de Ciencias Geologicas 26, 764-782.

Barnett, V, Lewis, T. (1994): Outliers in Statistical Data. Third edition, John Wiley, Chichester. 584p.

Beier, C., Haase, K.M., Hansteen, T.H. (2006): Magma evolution of the Sete Cidades volcano, Sao Miguel, Azores. Journal of Petrology 47, 1375-1411. doi: 10.1093/petrology/egl014

Bevington, P.R., Robinson, D.K. (2003): Data Reduction and Error Analysis for the Physical Sciences. Third edition, McGraw Hill, New York. 320p.

Blaesild, P, Granfeldt, J. (2003): Statistics with Applications in Biology and Geology. Chapman & Hall/CRC, Boca Raton. 555p.

Box, G.E.P., Muller, M.E. (1958): A note on the generation of random normal deviates. Annales of Mathematical Statistics 29, 610-611. doi:10.1214/aoms/1177706645

Cruz-Huicochea, R., Verma, S.P. (2013): New critical values for F and their use in the ANOVA and Fisher's F tests for evaluating geochemical reference material granite G-2 (U.S.A.) and igneous rocks from the Eastern Alkaline Province (Mexico). Journal of Iberian Geology 39,13-30. doi:10.5209/rev_JIGE.2013.v39.n1.41746.

de Lima, E.F., Machado, A., Nardi, L.V.S., Saldanha, D.L., Azevedo, J.M.M., Sommer, C.A., Waichel, B.L., Chemale Jr, F., de Almeida, D.P.M. (2011): Geochemical evidence concerning sources and petrologic evolution of Faial Island, Central Azores. International Geology Review 53, 1684-1708. doi.org/10.1080/00206814.2010.496248

Diaz-Gonzalez, L., Santoyo, E., Reyes-Reyes, J. (2008): Tres nuevos geotermometros mejorados de Na/K usando herramientas computacionales y geoquimiometricas: aplicacion a la prediccion de temperaturas de sistemas geotermicos. Revista Mexicana de Ciencias Geologicas 25, 465-482.

Ebdon, D. (1988): Statistics in Geography. Second edition, Basic Blackwell, Oxford. 232p.

Efstathiou, C.E. (2006): Estimation of type I error probability from experimental Dixon's "Q" parameter on testing for outliers within small size data sets. Talanta 69, 1068-1071. doi:10.1016/j.talanta.2005.12.031

Egozcue, J.J., Pawlowsky-Glahn, V., Mateu-Figueras, G., Barcelo-Vidal, C. (2003): Isometric logratio transformations for compositional data analysis. Mathematical Geology 35, 279-300.

Espinosa-Paredes, G., Verma, S.P, Vazquez-Rodriguez, A., Nunez-Carrera, A. (2010): Mass flow rate sensitivity and uncertainty analysis in natural circulation boiling water reactor core from Monte Carlo simulations. Nuclear Engineering and Design 240, 1050-1062. doi:org/10.1016/j.nucengdes.2010.01.012

Faure, G. (1986): Principles of Isotope Geology. Second edition, Wiley, New York. 653p.

Faure, G. (2001): Origin of Igneous Rocks. The Isotopic Evidence. Springer, Berlin. 496p.

Fisher, R.A. (1970): Statistical Methods for Research Workers. Fourteenth edition, Oliver and Boyd, Edinburgh. 362p.

Flanagan, F.J. (1967): U.S. Geological Survey silicate rock standards. Geochimica et Cosmochimica Acta 31, 289-308.

Gladney, E.S., Jones, E.A., Nickell, E.J., Roelandts, I. (1991): 1988 compilation of elemental concentration data for USGS DTS-1, G-1, PCC-1, and W-1. Geostandards Newsletter 15, 199-396. doi: 10.1111/j.1751-908X.1991.tb00114.x

Gladney, E.S., Jones, E.A., Nickell, E.J., Roelandts, I. (1992): 1988 compilation of elemental concentration data for USGS AGV-1, GSP1 and G-2. Geostandards Newsletter 16, 111-300. doi:10.1111/j.1751908X.1992.tb00492.x

Gomez-Arias, E., Andaverde, J., Santoyo, E., Urquiza, G. (2009): Determinacion de la viscosidad y su incertidumbre en fluidos de perforacion usados en la construccion de pozos geotermicos: aplicacion en el campo de Los Humeros, Puebla, Mexico. Revista Mexicana de Ciencias Geologicas 26, 516-529.

Gonzalez-Marquez, L.C., Hansen, A.M. (2009): Adsorcion y mineralizacion de atrazina y relacion con parametros de suelos del DR 063 Guasave, Sinaloa. Revista Mexicana de Ciencias Geologicas 26, 587-599.

Gonzalez-Ramirez, R., Diaz-Gonzalez, L., Verma, S.P. (2009): Eficiencia relativa de 15 pruebas de discordancia con 33 variantes aplicadas al procesamiento de datos geoquimicos. Revista Mexicana de Ciencias Geologicas 26, 501-515.

Hernandez-Martinez, J.L., Verma, S.P. (2009): Resena sobre las metodologias de campo, analiticas y estadisticas empleadas en la determinacion y manejo de datos de los elementos de tierras raras en el sistema suelo-planta. Revista de la Facultad de Ciencias Agrarias Universidad Nacional de Cuyo 41, 153-189.

Imai, N., Terashima, S., Itoh, S., Ando, A. (1995): 1994 compilation of analytical data for minor and trace elements in seventeen GSJ geochemical reference samples, "igneous rock series". Geostandards Newsletter 19, 135-213. doi: 10.1m/j.1751-908X.1995.tb00158.x

Jensen, J. L., Lake, L.W., Corbett, P.W.M., Goggin, D.J. (1997): Statistics for Petroleum Engineers and Geoscientists. Prentice-Hall, Upper Saddle River. 390p.

Kenji, G. K. (2006): 100 statistical tests. Third edition, SAGE publications, London. 242p.

Kinderman, A.J., Ramage, J.G. (1976): Computer generation of normal random variables. Journal of the American Statistical Association 71, 893-896.

Krochert, J., Buchner, E. (2009): Age distribution of cinder cones within the Bandas del Sur Formation, southern Tenerife, Canary Islands. Geological Magazine 146, 161-172. doi: org/10.1017/ S001675680800544X

Law, A.M., Kelton, W.D. (2000): Simulation Modeling and Analysis. Third edition, McGraw Hill, Boston. 400p.

Lindenmayer, J.-P., Czobor, P., Volavka, J., Citrome, L., Sheitman, B., McEvoy, J.P., Cooper, T.B., Chakos, M., Lieberman, J.A. (2003): Changes in glucose and cholesterol levels in patients with schizophrenia treated with typical or atypical antipsychotics. American Journal of Psychiatry 160, 290-296. doi:10.1176/appi.ajp.160.2.290

Longpre', M.-A., Troll, V.R., Walter, T.R., Hansteen, T.H. (2009): Volcanic and geochemical evolution of the Teno massif, Tenerife, Canary Islands: Some repercussions of giant landslides on ocean island magmatism. Geochemistry Geophysics Geosystems 10, Q12017. doi:10.1029/2009GC002892

Madhavaraju, J., Lee, YI. (2009): Geochemistry of the Dalmiapuram Formation of the Uttatur Group (Early Cretaceous), Cauvery basin, southeastern India: Implications on Provenance and Paleo-redox conditions. Revista Mexicana de Ciencias Geologicas 26, 380-394.

Marroquin-Guerra, S.G., Velasco-Tapia, F., Diaz-Gonzalez, L. (2009): Evaluacion estadistica de Materiales de Referencia Geoquimica del Centre de Recherches Petrographiques et Geochimiques (Francia) aplicando un esquema de deteccion y eliminacion de valores desviados. Revista Mexicana de Ciencias Geologicas 26, 530-542.

Marsaglia, G. (1968): Random numbers fall mainly in the planes. National Academy of Science Proceedings 61, 25-28.

Marsaglia, G., Bray, T.A. (1964): A convenient method for generating normal variables. SIAM Review 6, 260-264.

Matsumoto, M., Nishimura, T. (1998): Mersenne Twister: A 623-dimensionally equidistributed uniform pseudorandom number generator. ACM Transactions of Modelling and Computer Simulations 8, 3-30.

R Development Core Team (2009): R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. URL http://www.R/project.org

Miller, J. N., Miller, J. C. (2005): Statistics and Chemometrics for Analytical Chemistry. Fifth edition, Pearson-Prentice-Hall, Harlow. 268p.

Otto, M. (1999): Chemometrics. Statistics and computer application in Analytical Chemistry. Wiley-VCH, Weinheim. 314p.

Pandarinath, K. (2009a): Evaluation of geochemical sedimentary reference materials of the Geological Survey of Japan (GSJ) by an objective outlier rejection statistical method. Revista Mexicana de Ciencias Geologicas 26, 638-646.

Pandarinath, K. (2009b): Clay minerals in SW Indian continental shelf sediments cores as indicators of provenance and paleomonsoonal conditions: a statistical approach. International Geology Review 51, 145-165. doi:10.1080/00206810802622112

Pragel, N.-O., Holm, P.M. (2006): Lithospheric contributions to high-MgO basanites from the Cumbre Vieja volcano, La Palma, Canary Islands and evidence for temporal variation in plume influence. Journal of Volcanology and Geothermal Research 149, 213-239. doi: 10.1016/j.jvolgeores.2005.07.019

Storey, M., Wolff, J.A., Norry, M.J., Marriner, G.F. (1989): Origin of hybrid lavas from Agua de Pau volcano, Sao Miguel, Azores, in Saunders, A. D., and Norry, M. J., eds., Magmatism in the ocean basins, Oxford, Geological Society Special Publication No. 42, 161-180.

Thirlwall, M.F., Singer, B.S., Marriner, G.F. (2000): [sup.39]Ar-[sup.40]Ar ages and geochemistry of the basaltic shield stage of Tenerife, Canary Islands, Spain. Journal of Volcanology and Geothermal Research 103, 247-297. doi: 10.1016/S0377-0273(00)00227-4

Torres-Alvarado, I.S., Smith, A.D., Castillo-Roman, J. (2011): Sr, Nd and Pb isotopic and geochemical constraints for the origin of magmas in Popocatepetl volcano (central Mexico) and their relationship with the adjacent volcanic fields. International Geology Review 53, 84-115. doi:10.1080/00206810902906738

Turner, S., Hawkesworth, C., Rogers, N., King, P. (1997): U-Th isotope disequilibria and ocean island basalt generation in the Azores. Chemical Geology 139, 145-164. doi: org/10.1016/S0009-2541(97)00031-4

Velasco-Tapia, F., Guevara, M., Verma, S.P. (2001): Evaluation of concentration data in geochemical reference materials. Chemie der Erde 61, 69-91.

Verma, S.P. (1997): Sixteen statistical tests for outlier detection and rejection in evaluation of International Geochemical Reference Materials: example of microgabbro PM-S. Geostandards Newsletter. The Journal of Geostandards and Geoanalysis 21, 59-75. doi: 10.1111/j.1751-908X.1997.tb00532.x

Verma, S.P. (1998): Improved concentration data in two international geochemical reference materials (USGS basalt BIR-1 and GSJ peridotite JP-1) by outlier rejection. Geofisica Internacional 37, 215-250.

Verma, S.P. (2005): Estadistica Basica para el Manejo de Datos Experimentales: Aplicacion en la Geoquimica (Geoquimiometria). Universidad Nacional Autonoma de Mexico, Mexico, D. F. 186p.

Verma, S.P. (2009): Evaluation of polynomial regression models for the Student's t and Fisher F critical values, the best interpolation equations from double and triple natural logarithm transformation of degrees of freedom up to 1000, and their applications to quality control in science and engineering. Revista Mexicana de Ciencias Geologicas 26, 79-92.

Verma, S.P. (2012): Geochemometrics. Revista Mexicana de Ciencias Geologicas 29, 276-298.

Verma, S.P., Diaz-Gonzalez, L. (2012): Application of the discordant outlier detection and separation system in the geosciences: International Geology Review 54, 593-614. doi: 10.1080/00206814.2011.569402

Verma, S.P., Quiroz-Ruiz, A. (2006a): Critical values for six Dixon tests for outliers in normal samples up to sizes 100, and applications in science and engineering. Revista Mexicana de Ciencias Geologicas 23, 133-161.

Verma, S.P., Quiroz-Ruiz, A. (2006b): Critical values for 22 discordancy test variants for outliers in normal samples up to sizes 100, and applications in science and engineering. Revista Mexicana de Ciencias Geologicas 23, 302-319.

Verma, S.P., Quiroz-Ruiz, A. (2008): Critical values for 33 discordancy test variants for outliers in normal samples for very large sizes of 1,000 to 30,000. Revista Mexicana de Ciencias Geologicas 25, 369-381.

Verma, S.P., Quiroz-Ruiz, A. (2011): Corrigendum to Critical values for 22 discordancy test variants for outliers in normal samples up to sizes 100, and applications in science and engineering [Rev. Mex. Cienc. Geol., 23 (2006), 302-319]. Revista Mexicana de Ciencias Geologicas 28, 202.

Verma, S.P., Orduna-Galvan, L.J., Guevara, M. (1998): SIPVADE: A new computer programme with seventeen statistical tests for outlier detection in evaluation of international geochemical reference materials and its application to Whin Sill dolerite WS-E from England and Soil-5 from Peru. Geostandards Newsletter: The Journal of Geostandards and Geoanalysis 22, 209-234. doi: 10.1111/j.1751908X.1998.tb00695.x

Verma, S.P., Torres-Alvarado, I.S., Sotelo-Rodriguez, Z.T. (2002): SINCLAS: standard igneous norm and volcanic rock classification system. Computers & Geosciences 28, 711-715. doi: org/10.1016/S00983004(01)00087-5

Verma, S.P., Torres-Alvarado, I.S., Velasco-Tapia, F. (2003): A revised CIPW norm. Schweizerische Mineralogische und Petrographische Mitteilungen 83, 197-216.

Verma, S.P., Quiroz-Ruiz, A., Diaz-Gonzalez, L. (2008): Critical values for 33 discordancy test variants for outliers in normal samples up to sizes 1000, and applications in quality control in Earth Sciences. Revista Mexicana de Ciencias Geologicas 25, 82-96.

Verma, S.P., Diaz-Gonzalez, L., Gonzalez-Ramirez, R. (2009): Relative efficiency of single-outlier discordancy tests for processing geochemical data on reference materials. Geostandards and Geoanalytical Research 33, 29-49. doi: 10.1111/j.1751-908X.2009.00885.x

Verma, S.P., Cruz-Huicochea, R., Diaz-Gonzalez, L. (2013): Univariate data analysis system: deciphering mean compositions of island and continental arc magmas, and influence of the underlying crust. International Geology Review, in press. doi: 10.1080/00206814.2013.810363.

Walker, J.T., Maddan, S. (2005): Statistics in Criminology and Criminal Justice. Analysis and Interpretation. Second edition, Jones and Bartlett Publishers, Sudbury, Mass., USA. 427p.

Wani, H., Mondal, M.E.A. (2011): Evaluation of provenance, tectonic setting, and paleoredox conditions of the Mesoproterozoic-Neoproterozoic basins of the Bastar craton, Central Indian shield: Using petrography of sandstones and geochemistry of shales. Lithosphere 3, 143-154. doi: 10.1130/L74.1

Zeyrek, M., Ertekin, K., Kacmaz, S., Seyis, C., Inan, S. (2010): An ion chromatography method for the determination of major anions in geothermal water samples. Geostandards and Geoanalytical Research 34, 67-77. doi: 10.1111/j.1751-908X.2009.00020.x

S. P. Verma (1) *, R. Cruz-Huicochea (2)

(1) Departamento de Sistemas Energeticos, Instituto de Energias Renovables, Universidad Nacional Autonoma de Mexico, Priv. Xochicalco s/no., Col. Centro, Apartado Postal 34, Temixco 62580, Morelos, Mexico. spv@ier.unam.mx

(2) Posgrado en Ingenieria (Energia), Instituto de Energias Renovables, Universidad Nacional Autonoma de Mexico, Priv. Xochicalco s/no., Col. Centro, Apartado Postal 34, Temixco 62580, Morelos, Mexico. recrh@live.com

* corresponding author

```Table 1.-Abridged form of simulated critical value table of Student t
test. The abbreviations are as follows: [t.sup.cv.sub.50ts]-critical
value of t for two-sided (ts) 50% confidence level;
[t.sup.cv.sub.75ts]--critical value of t for one-sided (os) 75%
confidence level; and similar symbols are used for other columns. The
more frequently used confidence levels are marked in boldface.

para la prueba de t de Student. Las abreviaturas son las siguientes:
[t.sup.cv.sub.50ts]--el valor critico de t para dos colas (ts) nivel
de confianza 50%; [t.sup.cv.sub.75ts]--el valor critico de t para una
cola (os) nivel de confianza 75%; y simbolos similares se usaron para
las otras columnas. Los niveles de confianza mas usados han sido

Two-sided CL           50%                    60%
Two-sided SL           0.50                   0.40
One-sided CL           75%                    80%
One-sided SL           0.25                   0.20
v(df)          [t.sup.cv.sub.50ts]    [t.sup.cv.sub.60ts]
[t.sup.cv.sub.75os]    [t.sup.cv.sub.80os]
1.0000136              1.3763807
2                   0.8164885              1.0606517
3                   0.7648982              0.9784738
4                   0.7406960              0.9409621
5                   0.7266861              0.9195479
6                   0.71756197             0.9057104
7                   0.71113773             0.8960278
8                   0.70638975             0.88889369
9                   0.70271989             0.88340214
10                  0.69981044             0.8790541
15                  0.69119456             0.86623779
20                  0.6869600              0.8599715
25                  0.6844278              0.8562327
30                  0.6827641              0.8537724
40                  0.6806737              0.8506991
50                  0.6794206              0.8488612
60                  0.6786080              0.8476578
80                  0.67757172             0.84613753
100                 0.67694855             0.84522811
150                 0.67612862             0.84402537
200                 0.6757101              0.8434174
400                 0.6751006              0.8425213
600                 0.6748943              0.8422214
800                 0.6747807              0.8420569
1000                0.6747324              0.8419752

Two-sided CL           70%                    80%
Two-sided SL           0.30                   0.20
One-sided CL           85%                    90%
One-sided SL           0.15                   0.1
v(df)          [t.sup.cv.sub.70ts]    [t.sup.cv.sub.80ts]
[t.sup.cv.sub.85os]    [t.sup.cv.sub.90os]
1.9626083               3.077666
2                   1.3861911              1.8855913
3                   1.2497811              1.6377376
4                   1.1895678              1.5332070
5                   1.1557757              1.4758941
6                   1.1341590              1.4397674
7                   1.1191594              1.4149212
8                   1.1081493              1.3968232
9                   1.0997126              1.3830348
10                  1.0930532              1.3721800
15                  1.07352282             1.34059701
20                  1.0640185              1.3253411
25                  1.0583813              1.3163424
30                  1.0546676              1.3104108
40                  1.0500388              1.3030681
50                  1.0472899              1.2987037
60                  1.0454769              1.2958257
80                  1.0431958              1.2922269
100                 1.04183610             1.2900783
150                 1.04002824             1.28722112
200                 1.0391193              1.2857865
400                 1.0377814              1.2836758
600                 1.0373362              1.2829467
800                 1.0371012              1.2826179
1000                1.0369683              1.2824003

Two-sided CL           90%                     95%#
Two-sided SL           0.10                   0.05#
One-sided CL           95%#                   97.5%
One-sided SL          0.05#                   0.025
v(df)          [t.sup.cv.sub.90ts]     [t.sup.cv.sub.95ts]
[t.sup.cv.sub.95os]    [t.sup.cv.sub.97.5os]
6.313726                12.70648
2                   2.9199454                4.302583
3                   2.3533422               3.1824461
4                   2.1318520               2.7764709
5                   2.0150635               2.5706039
6                   1.9432031               2.4469490
7                   1.8945732               2.3646182
8                   1.8595443               2.3060045
9                   1.8331124               2.2621405
10                  1.8124617               2.2281519
15                  1.7530417               2.1314479
20                  1.7247240               2.0859472
25                  1.7081287               2.0595269
30                  1.6972655               2.0422851
40                  1.6838528               2.0210644
50                  1.6758933               2.0085488
60                  1.6706451               2.0002966
80                  1.6641254               1.9900591
100                 1.6602380               1.9839625
150                 1.6550845               1.9759082
200                 1.6525015               1.9718839
400                 1.6486858               1.9659428
600                 1.6473705               1.9638981
800                 1.6467646               1.9629293
1000                1.6463813               1.9623318

Two-sided CL            98%                      99%#
Two-sided SL            0.02                    0.01#
One-sided CL            99%#                    99.5%
One-sided SL           0.01#                    0.005
v(df)           [t.sup.cv.sub.98ts]      [t.sup.cv.sub.99ts]
[t.sup.cv.sub.99os]     [t.sup.cv.sub.99.5os]
31.82117                 63.6605
2                     6.964659                 9.925064
3                     4.540802                 5.841159
4                    3.7470196                 4.604111
5                    3.3649657                 4.032188
6                    3.1427120                3.7074540
7                    2.9979522                3.4995422
8                    2.8964851                3.3554063
9                    2.8214342                3.2498186
10                   2.7637882                3.1692831
15                   2.6024840                2.9467167
20                   2.5279932                2.8453151
25                   2.4850622                2.7874162
30                   2.4573009                2.7500471
40                   2.4232606                2.7044590
50                   2.4032702                2.6777969
60                   2.3901240                2.6602785
80                   2.3738513                2.6386770
100                  2.3642287                2.6258889
150                  2.3514615                2.6090102
200                  2.3451314                2.6006393
400                  2.3357197                2.5881811
600                  2.3325262                2.5839896
800                  2.3310267                2.5819814
1000                 2.3300840                2.5807617

Two-sided CL           99.5%                    99.8%
Two-sided SL           0.005                    0.002
One-sided CL           99.75%                   99.9%
One-sided SL           0.0025                   0.001
v(df)           [t.sup.cv.sub.99.5s]    [t.sup.cv.sub.99.8ts]
[t.sup.cv.sub.99.75os]   [t.sup.cv.sub.99.9os]
127.3270                 318.262
2                     14.08901                 22.32674
3                     7.453738                 10.21582
4                     5.597755                 7.173050
5                     4.773381                 5.893499
6                     4.316880                 5.207832
7                     4.029425                 4.785345
8                    3.8325482                 4.500858
9                    3.6896319                 4.296806
10                   3.5814565                 4.143787
15                   3.2860435                3.7328521
20                    3.153335                 3.551717
25                   3.0781518                 3.450202
30                    3.029821                 3.385212
40                   2.9711960                 3.306969
50                    2.93696                  3.261488
60                   2.9145618                3.2317721
80                   2.8869870                3.1952620
100                  2.8706395                3.1737332
150                  2.8491737                3.1454784
200                  2.8384848                3.1314241
400                   2.822711                 3.110746
600                   2.817389                 3.103850
800                   2.814853                 3.100460
1000                 2.8132926                3.0984256

Two-sided CL           99.9%
Two-sided SL           0.001
One-sided CL           99.95%
One-sided SL           0.0005
v(df)          [t.sup.cv.sub.99.9ts]
[t.sup.cv.sub.99.95os]
636.541
2                     31.59805
3                     12.92492
4                     8.610084
5                     6.869000
6                     5.958930
7                     5.408129
8                     5.041424
9                     4.780890
10                    4.587094
15                    4.072783
20                    3.849393
25                    3.725082
30                    3.646024
40                    3.551148
50                    3.496126
60                    3.460233
80                    3.416342
100                  3.3904678
150                  3.3565400
200                   3.339775
400                   3.314961
600                   3.306849
800                   3.302718
1000                  3.300340

Note: The more frequently used confidence levels are marked in
boldface indicated with #.

Table 2.-Standard error values for simulated critical values of the
Student t test. The abbreviations are as follows:
[t.sup.vc.sub.50ts]--critical value of t for two-sided (ts) 50%
confidence level; [t.sup.vc.sub.75os]--critical value of t for one-
sided (os) 75% confidence level; and similar symbols are used for
other columns. The more frequently used confidence levels are marked
in boldface.

Tabla 2.-Valores del error estandar para valores criticos simulados
para la prueba de t de Student. Las abreviaturas son las siguientes:
[t.sup.vc.sub.50ts]--el valor critico de t para dos colas (ts) nivel
de confianza 50%; [t.sup.vc.sub.50os]--el valor critico de t para una
cola (os) nivel de confianza 75%; y simbolos similares se usaron para
las otras columnas. Los niveles de confianza mas usados han sido

Two-sided CL           50%                    60%
Two-sided SL           0.50                   0.40
One-sided CL           75%                    80%
One-sided SL           0.25                   0.20
v(df)          [t.sup.cv.sub.50ts]    [t.sup.cv.sub.60ts]
[t.sup.cv.sub.75os]    [t.sup.cv.sub.80os]
1                   0.0000163              0.0000245
2                   0.0000079              0.0000095
3                   0.0000071              0.0000083
4                   0.0000050              0.0000060
5                   0.0000050              0.0000059
6                   0.00000442             0.0000049
7                   0.00000422             0.0000051
8                   0.00000379             0.00000438
9                   0.00000388             0.00000428
10                  0.00000407             0.0000045
15                  0.00000294             0.00000325
20                  0.0000057              0.0000068
25                  0.0000058              0.0000065
30                  0.0000055              0.0000062
40                  0.0000052              0.0000058
50                  0.0000050              0.0000056
60                  0.0000046              0.0000050
80                  0.00000388             0.00000437
100                 0.00000353             0.00000392
150                 0.00000292             0.0c000319
200                 0.0000056              0.0000062
400                 0.0000082              0.0000084
600                 0.0000100              0.0000117
800                 0.0000083              0.0000087
1000                0.0000047              0.0000055

Two-sided CL           70%                     80%
Two-sided SL           0.30                    0.20
One-sided CL           85%                     90%
One-sided SL           0.15                    0.1
v(df)          [t.sup.cv.sub.70ts]     [t.sup.cv.sub.80ts]
[t.sup.cv.sub.85os]     [t.sup.cv.sub.90os]
1                   0.0000373                0.000071
2                   0.0000129               0.0000184
3                   0.0000105               0.0000146
4                   0.0000076               0.0000096
5                   0.0000069               0.0000092
6                   0.0000059               0.0000075
7                   0.0000062               0.0000073
8                   0.0000051               0.0000060
9                   0.0000050               0.0000059
10                  0.0000054               0.0000064
15                  0.00000372              0.00000447
20                  0.0000079               0.0000094
25                  0.0000077               0.0000095
30                  0.0000072               0.0000094
40                  0.0000067               0.0000078
50                  0.0000063               0.0000074
60                  0.0000057               0.0000064
80                  0.0000048               0.0000054
100                 0.00000448              0.0000050
150                 0.00000354              0.0000419
200                 0.0000068               0.0000075
400                 0.0000100               0.0000110
600                 0.0000116               0.0000125
800                 0.0000102               0.0000122
1000                 0.000006               0.0000071

Two-sided CL            90%                      95%#
Two-sided SL            0.10                    0.05#
One-sided CL            95%#                    97.5%
One-sided SL           0.05#                    0.025
v(df)           [t.sup.cv.sub.90ts]      [t.sup.cv.sub.95ts]
[t.sup.cv.sub.95os]     [t.sup.cv.sub.97.5os]
1                    0.000207                 0.00056
2                    0.0000376                 0.000076
3                    0.0000243                0.0000424
4                    0.0000146                0.0000230
5                    0.0000139                0.0000208
6                    0.0000102                0.0000153
7                    0.0000102                0.0000142
8                    0.0000085                0.0000125
9                    0.0000082                0.0000118
10                   0.0000086                0.0000129
15                   0.0000060                0.0000078
20                   0.0000121                0.0000161
25                   0.0000118                0.0000172
30                   0.0000131                0.0000165
40                   0.0000098                0.0000148
50                   0.0000093                0.0000118
60                   0.0000088                0.0000111
80                   0.0000070                0.0000091
100                  0.0000065                0.0000089
150                  0.0000054                0.0000071
200                  0.0000092                0.0000120
400                  0.0000144                0.0000179
600                  0.0000176                0.0000231
800                  0.0000167                0.0000208
1000                 0.0000091                0.0000120

Two-sided CL            98%                      99%#
Two-sided SL            0.02                    0.01#
One-sided CL            99%#                    99.5%
One-sided SL           0.01#                    0.005
v(df)           [t.sup.cv.sub.98ts]      [t.sup.cv.sub.99ts]
[t.sup.cv.sub.99os]     [t.sup.cv.sub.99.5os]
1                     0.00222                   0.0061
2                     0.000183                 0.000340
3                     0.000086                 0.000143
4                    0.0000436                 0.000077
5                    0.0000351                 0.000058
6                    0.0000272                0.0000427
7                    0.0000249                0.0000373
8                    0.0000199                0.0000301
9                    0.0000191                0.0000261
10                   0.0000199                0.0000292
15                   0.0000125                0.0000172
20                   0.0000248                0.0000357
25                   0.0000230                0.0000290
30                   0.0000245                0.0000313
40                   0.0000199                0.0000281
50                   0.0000173                0.0000237
60                   0.0000161                0.0000205
80                   0.0000135                0.0000174
100                  0.0000119                0.0000161
150                  0.0000099                0.0000127
200                  0.0000183                0.0000224
400                  0.0000279                0.0000368
600                  0.0000313                0.0000406
800                  0.0000281                0.0000368
1000                 0.0000174                0.0000508

Two-sided CL           99.5%                    99.8%
Two-sided SL           0.005                    0.002
One-sided CL           99.75%                   99.9%
One-sided SL           0.0025                   0.001
v(df)           [t.sup.cv.sub.99.5s]    [t.sup.cv.sub.99.8ts]
[t.sup.cv.sub.99.75os]   [t.sup.cv.sub.99.9os]
1                      0.0173                   0.073
2                     0.00069                  0.00168
3                     0.000242                 0.00048
4                     0.000120                 0.000246
5                     0.000096                 0.000182
6                     0.000064                 0.000120
7                     0.000055                 0.000100
8                    0.0000438                 0.000073
9                    0.0000386                 0.000070
10                   0.0000409                 0.000066
15                   0.0000246                0.0000386
20                    0.000048                 0.000074
25                   0.0000411                 0.000065
30                    0.000046                 0.000069
40                   0.0000397                 0.000053
50                   0.0000314                 0.000046
60                   0.0000277                0.0000422
80                   0.0000227                0.0000345
100                  0.0000210                0.0000316
150                  0.0000179                0.0000263
200                  0.0000304                0.0000423
400                   0.000046                 0.000070
600                   0.000054                 0.000065
800                   0.000051                 0.000072
1000                 0.0000266                0.0000419

Two-sided CL           99.9%
Two-sided SL           0.001
One-sided CL           99.95%
One-sided SL           0.0005
v(df)          [t.sup.cv.sub.99.9ts]
[t.sup.cv.sub.99.95os]
1                      0.181
2                     0.00353
3                     0.00097
4                     0.000426
5                     0.000272
6                     0.000190
7                     0.000138
8                     0.000114
9                     0.000108
10                    0.000097
15                    0.000056
20                    0.000100
25                    0.000087
30                    0.000099
40                    0.000079
50                    0.000063
60                    0.000061
80                    0.000047
100                  0.0000427
150                  0.0000356
200                   0.000059
400                   0.000081
600                   0.000008
800                   0.000099
1000                  0.000054

Note: The more frequently used confidence levels are marked in
boldface indicated with #

Table 3.-Evaluation of best-fit critical value equations obtained
from 65 critical values of Student t distribution for degrees of
freedom (v) from 3 to 2000 and 99% confidence level.
[R.sup.2]--multiple-correlation coefficient; (*)--Next to the
best-fit (i.e., second best-fit) according to this particular
criterion; [**]--Third best-fit according to this particular
criterion; {***}--Fourth best-fit according to this particular
criterion. For explanation of ll and lll functions see Table 4.

Tabla 3.-Evaluacion de las ecuaciones obtenidas del mejor ajuste de
65 valores criticos de la distribucion t de Student para grados de
libertad (v) de 3 a 2000 y nivel de confianza de 99%.
[R.sup.2]--coeficiente de correlacion multiple; (*)--Proximo al
mejor-ajuste (i.e., el segundo mejor-ajuste) de acuerdo con este
criterio particular; [**]--El tercer mejor-ajuste de acuerdo con este
criterio particular; {***}--El cuarto mejor-ajuste de acuerdo con este
criterio particular. Para la explicacion de las funciones ll and lll
ver la Tabla 4.

Statistical decision criteria for
regression models and equations

Confidence            Fitting quality parameter
level                 (complete data in Fig. 2)

Best-fit
Two-    One-     [R.sup.2]          equation type
sided   sided                       (*)[**]
{***}

98%     99%      0.9999999949       ll8
(0.9999999561)     (lll8)
[0.9999919546]     [ll6,ll7]
{0.9999999395}     {ll5}

99%     99.80%   0.9999999949       ll8
(0.1999999732)     (lll8)
[0.99999 99408]    [ll6, ll7]
{0.9999999399}     {lll6, lll7}

Statistical decision criteria for
regression models and equations

Confidence           Mean squared residuals from
level                 65 fitted simulated data
(complete data in Fig. 3)

Best-fit
Two-    One-     SSR/N                  equation type
sided   sided    (N=65)                 (*)[**]{***}

98%     99%      6.5609x[l0.sup.-10]    ll8
(5.6238x[10.sup.-9])   (lll8)
[5.8139x[10.sup.-9]    [ll6, ll7]
{7.7461x[10.sup.-9}    {ll5}

99%     99.80%   1.3673x[10.sup.-9]     ll8
(7.1542x[10.sup.-9])   (lll8)
[1.5102x[10.sup.-8]    [ll6, ll7]
{1.6052x[10.sup.-8]}   {1l6, ll7}

Statistical decision criteria for
regression models and equations

Confidence            Mean square d residuals from 5 interpolated
level                                 simulated data
(complete data in Fig. 4)

Best-fit          Proposed
Two-    One-     [(SSR/N).sub.int]      equation type   interpolation
sided   sided    (N=5)                  (*)[**]{***}      equation

98%     99%      1.3402x[10.sup.-9]     ll8                  118
(5.1967x[10.sup.-9])   (ll5)
[6.9333x[10.sup.-9]]   [ll4]
{9.2727x[10.sup.-9]    {ll6, ll7}

99%     99.80%   2.1983x[10.sup.-9]     ll8                  118
(1.5653x[10.sup.-8])   (lll6, lll7)
[1.7794x[10.sup.-8]]   [ll6, ll7]
{1.8891x[10.sup.-8]}   {lll8}

Statistical decision criteria for regression
models and equations

Confidence              Mean squared residuals from 4 extrapolated
level                                simulated data
(complete data in Fig. 5)

Best-fit          Proposed
Two-    One-     [(SSR/N).sub.ext]      equation type   extrapolation
sided   sided    (N=4)                  (*)[**]           equation
{***}

98%     99%      2.0816x[10.sup.-8]     ll4                  ll4
(5.1879x[10.sup.-8])   (ll5)
[4.8977x[10.sup.-7]]   [ll6, ll7]
{6.7863x[10.sup.-7]}   {ll6, ll7}

99%     99.80%   1.0758x[10.sup.-6]     lll6, lll7
(2.0901x[10.sup.-6])   -ll8                lll6
[2.2253x[10.sup.-6]]   [ll6, ll7]
{2.5576x[10.sup.-6]}   {lll8}

Table 4.-The best-fit critical value equations of Student t
distribution for 99% confidence levels.

Tabla 4.-Evaluacion de las ecuaciones para el mejor ajuste de valores
criticos de la distribucion t de Student para nivel de confianza de
99%.

Confidence level             Best-fit interpolation ([sub.int]) and
xxtrapolation ([sub.ext]) oquntions
Two-    One-     Equation
sided   sided    type        Critical value oquntions
(Table 3)

98%     99%      ll8         [([t.sup.cv.sub.98ts]).sub.int] =
(4.94002103 [+ or -] 0.00012294) -
(4.5022554 [+ or -] 0.0018112) x
(ln(ln(v))) + (2.785764 [+ or -]
0.008079) x [(ln(ln(v))).sup.2]--
(0.485943 [+ or -] 0.016415) x
[(ln(ln(v))).sup.3] -(0.305189 [+ or -]
0.017282) x [(ln(ln(v))).sup.4] +
(0.216257 [+ or -] 0.009455) x
[(ln(ln(v))).sup.5] -(0.0506048 [+ or -
] 0.0022587) x [(ln(ln(v))).sup.6] +
(0.001324793 [+ or -] 0.000062583) x
[(ln(ln(v))).sup.8]

ll4         [([t.sup.cv.sub.98ts).sub.ext] =
(4.94237021 [+ or -] 0.00014897) -
(4.5412456 [+ or -] 0.0007948) x
(ln(ln(v))) + (2.9690047 [+ or -]
0.0013459) x [(ln(ln(v))).sup.2] -
(0.8658555 [+ or -] 0.0008820) x
[(ln(ln(v))).sup.3] + (0.09497881 [+ or
-] 0.00019659) x [(ln(ln(v))).sup.4]

99%     99.50%   ll8         [([t.sup.cv.sub.99ts]).sub.int] =
(6.48557253 [+ or -] 0.00017748) -
(7.3462223 [+ or -] 0.0026146) x
(ln(ln(v))) + (5.418770 [+ or -]
0.011663) x [(ln(ln(v))).sup.2] -
(1.721741 [+ or -] 0.023697) x
[(ln(ln(v))).sup.3] -(0.0930165 [+ or -
] 0.024949) x [(ln(ln(v))).sup.4] +
(0.274850 [+ or -] 0.013649) x
[(ln(ln(v))).sup.5] -(0.0779639 [+ or -
] 0.0032607) x [(ln(ln(v))).sup.6] +
(0.00221624 [+ or -] 0.00009035) x
[(ln(ln(v))).sup.8]

lll6        [([t.sup.cv.sub.99ts).sub.ext] =
(2.94230900 [+ or -] 0.00003147) -
(1.12134505 [+ or -] 0.00018085) x
(ln(ln(ln(v)))) + (0.78201429 [+ or -]
0.00031542) x [(ln(ln(ln(v)))).sup.2] +
(0.3639632 [+ or -] 0.0009774) x
[(ln(ln(ln(v)))).sup.3] -(0.20510287 [+
or -] 0.00048796) x
[(ln(ln(ln(v)))).sup.4] -(0.1945693 [+
or -] 0.0012343) x
[(ln(ln(ln(v)))).sup.5] -(0.04167330 [+
or -] 0.00042639) x
[(ln(ln(ln(v)))).sup.6]

Table 5.-Examples of the best-fit critical value equations of Student
t distribution for some degrees of freedom useful for Student t
probability estimates of two samples. * Best fit equation for Student
t confidence level calculations for any given degrees of freedom is
obtained by multiplying the "Expression" by 100. In other words, this
value gives the confidence level corresponding to the calculated t
value ([t.sub.calc]) for two statistical samples.

Tabla 5.-Ecuaciones para el mejor ajuste de valores criticos de la
para las estimaciones de probabilidad de t de Student de dos
muestras. * La ecuacion con el mejor ajuste para calcular el nivel de
confianza de t de Student para un grado de libertad se obtiene
mediante la multiplicacion de la expresion por 100. En otras
palabras, este valor proporciona el nivel de confianza que
corresponde al valor de t calculado ([t.sub.calc]) para dos muestras

Degrees of   Expression *
freedom

1            (0.4812181 [+ or -] 0.0035654)+(0.3895921 [+ or -]
0.0119223)-(ln([t.sub.calc]))-(0.0961044 [+ or -]
0.0149182)-[(ln([t.sub.calc])).sup.2]-(0.0052704 [+ or
-] 0.0091705)-[(ln([t.sub.calc])).sup.3]+(0.0077850 [+
or -] 0.0030085)-[(ln([t.sub.calc])).sup.4]-(0.0016238
[+ or -]
0.0005190)x[(ln([t.sub.calc])).sup.5]+(0.0001257 [+ or
-] 0.0000392)x[(ln([t.sub.calc])).sup.6]-(0.0000003 [+
or -] 0.0000001)x(ln([t.sub.calc]))8

2            (0.8870722 [+ or -] 0.0000194)+(0.1898799 [+ or -]
0.0002231)-(ln(ln([t.sub.calc])))-(0.0360825 [+ or -]
0.0003502)-[(ln(ln([t.sub.calc]))).sup.2]-(0.0749472 [+
or -]
0.0010577x[(ln(ln([t.sub.calc]))).sup.3]+(0.0081918 [+
or -] 0.0022984)-[(ln(ln([t.sub.calc]))).sup.4]+
(0.0280723 [+ or -]
0.0004311x[(ln(ln([t.sub.calc]))).sup.5]-(0.0029625 [+
or -] 0.0016119)-[(ln(ln([t.sub.calc]))).sup.6]-
(0.0035579 [+ or -] 0.0007959)-
[(ln(ln([t.sub.calc]))).sup.7]

3            (0.9273622 [+ or -] 0.0000131)+(0.1665987 [+ or -]
0.0000425)-(ln(ln([t.sub.calc])))-(0.0712199 [+ or -]
0.0004531)-[(ln(ln([t.sub.calc]))).sup.2]-(0.0762271 [+
or -]
0.0008672x[(ln(ln([t.sub.calc]))).sup.3]+(0.0265879 [+
or -] 0.0009273)-
[(ln(ln([t.sub.calc]))).sup.4]+(0.0445214 [+ or -]
0.0021473)-[(ln(ln([t.sub.calc]))).sup.5]-(0.0064676 [+
or -] 0.0002020j(ln(ln([t.sub.calc])))6-(0.0118609 [+
or -] 0.0009450)-(ln(ln([t.sub.calc])))7

4            (0.6282261 [+ or -] 0.0005813)+(0.4044682 [+ or -]
0.0065688)-(ln([t.sub.calc]))+(0.0992476 [+ or -]
0.0256664)-[(ln([t.sub.calc])).sup.2]-(0.2905062 [+ or
-] 0.0474512)-[(ln([t.sub.calc])).sup.3]+(0.0972464 [+
or -] 0.0460441)-[(ln([t.sub.calc])).sup.4]+(0.0215562
[+ or -] 0.0233383)-[(ln([t.sub.calc])).sup.5]-
(0.0138298 [+ or -] 0.0051782)-
(ln([t.sub.calc]))6+(0.000497 8 [+ or -]
0.0001241)x(ln([t.sub.calc]))8

5            (0.6374292 [+ or -] 0.0003591)+(0.4314525 [+ or -]
0.0046919)-(ln([t.sub.calc]))+(0.0279714 [+ or -]
0.0206944)-[(ln([t.sub.calc])).sup.2]-(0.1475031 [+ or
-] 0.0427944)-[(ln([t.sub.calc])).sup.3]-(0.0876243 [+
or -] 0.0462785)-[(ln([t.sub.calc])).sup.4]+(0.1399846
[+ or -] 0.0261045)-[(ln([t.sub.calc])).sup.5]-
(0.0450171 [+ or -]
0.0064427)x[(ln([t.sub.calc])).sup.6]+(0.0014487 [+ or
-] 0.0001910)-[(ln([t.sub.calc])).sup.8]

10           (0.6596781 [+ or -] 0.0003212)+(0.4507556 [+ or -]
0.0054577)-(ln([t.sub.calc]))+(0.0467337 [+ or -]
0.0272937)-[(ln(t.sub.calc)).sup.2]-(0.2133822 [+ or -
] 0.0594172)x[(ln([t.sub.calc])).sup.3]-(0.0596279 [+
or -] 0.0635323)x[(ln([t.sub.calc])).sup.4]+(0.1300124
[+ or -] 0.0328510)x(ln([t.sub.calc]))5-(0.0357628 [+
or -] 0.0065689)x[(ln([t.sub.calc])).sup.6]

20           (0.6701141 [+ or -] 0.0000467)+(0.4882536 [+ or -]
0.0011574)(ln([t.sub.calc]))-(0.1197997 [+ or -]
0.0079376)[(ln([t.sub.calc])).sup.2]+ (0.2532794 [+ or
-] 0.0240805)x[(ln([t.sub.calc])).sup.3]-(0.7727793 [+
or -] 0.0373642)[(ln([t.sub.calc])).sup.4]+(0.6579161
[+ or -] 0.0299526)[(ln([t.sub.calc]).sup.5]-
(0.1968662 [+ or -]
0.0104615)x[(ln([t.sub.calc])).sup.6]+(0.0066465 [+ or
-] 0.0006191)x[(ln([t.sub.calc])).sup.8]

30           (0.6742051 [+ or -] 0.0000581)+(0.4898511 [+ or -]
0.0016161)(ln([t.sub.calc]))-(0.1077921 [+ or -]
0.0117754)x[(ln([t.sub.calc])).sup.2]+ (0.2209211 [+ or
-] 0.0374719)x[(ln([t.sub.calc])).sup.3]-(0.7409656 [+
or -] 0.0607231)x[(ln([t.sub.calc])).sup.4]+(0.6301292
[+ or -] 0.0507433)x[(ln([t.sub.calc])).sup.5]-
(0.1821154 [+ or -]
0.0184581)x[(ln([t.sub.calc])).sup.6]+(0.0049689 [+ or
-] 0.0011836)x[(ln([t.sub.calc])).sup.8]

40           (0.6762561 [+ or -] 0.0000472)+(0.4908872 [+ or -]
0.0014026)x[(ln([t.sub.calc]))-(0.1031019 [+ or -]
0.0105392)x[(ln([t.sub.calc])).sup.2]+ (0.2075551 [+ or
-] 0.0343664)x[(ln([t.sub.calc)).sup.3]-(0.7271241 [+
or -] 0.0569294)(ln(tc,lk))4+(0.6150750 [+ or -]
0.0485792)x[(ln([t.sub.calc]).sup.5]-(0.17 30506 [+ or
-] 0.0180358x[(ln([t.sub.calc])).sup.6]+(0.0038491 [+
or -] 0.0012031x[(ln([t.sub.calc])).sup.8]

50           (0.6775041 [+ or -] 0.0000523)+(0.4909687 [+ or -]
0.0016203)x(ln([t.sub.calc]))-(0.0959895 [+ or -]
0.0124116)x[(ln([t.sub.calc])).sup.2]+ (0.1857113 [+ or
-] 0.0410752)x[(ln([t.sub.calc])).sup.3]-(0.6965743 [+
or -] 0.0689530)x[(ln([t.sub.calc])).sup.4]+(0.5873487
[+ or -] 0.0595873)x[(ln([t.sub.calc])).sup.5]-
(0.1606595 [+ or -]
0.0223969)x[(ln([t.sub.calc])).sup.6]+(0.0026971 [+ or
-] 0.0015318x[(ln([t.sub.calc])).sup.8]

100          (0.6799697 [+ or -] 0.0000394)+(0.4921354 [+ or -]
0.0013371)x(ln([t.sub.calc]))-(0.0887807 [+ or -]
0.0106548)x[(ln([t.sub.calc])).sup.2]+ (0.1626809 [+ or
-] 0.0363343)x[(ln([t.sub.calc])).sup.3]-(0.6655234 [+
or -] 0.0626486)x[(ln([t.sub.calc])).sup.4]+(0.5532806
[+ or -] 0.0555308)x[(ln([t.sub.calc])).sup.5]-
(0.1420660 [+ or -]
0.0213945x[(ln([t.sub.calc])).sup.6+(0.0005679 [+ or -
] 0.0015361)x[(ln([t.sub.calc])).sup.8]

1000         (0.6822162 [+ or -] 0.0000361)+(0.4925292 [+ or -]
0.0013468)x(ln([t.sub.calc]))-(0.0762137 [+ or -]
0.0111361)x[(ln([t.sub.calc])).sup.2]+ (0.1195291 [+ or
-] 0.0390311)x[(ln([t.sub.calc])).sup.3]-(0.5965984 [+
or -] 0.0689553)x[(ln([t.sub.calc])).sup.4]+(0.4836937
[+ or -] 0.0625404)x[(ln([t.sub.calc])).sup.5]-
(0.1091439 [+ or -]
0.0246388)x[(ln([t.sub.calc])).sup.6]-(0.0026889 [+ or
-] 0.0018483)x[(ln([t.sub.calc])).sup.8]

Table 6.-Simulated [sup.87]Sr-[sup.86]Sr values in geochemical
reference material JA-1 and application of Student's t test based on
different critical values.

Tabla 6.-Valores de la relacion [sup.87]Sr-[sup.86]Sr simulados en el
material de referencia geoquimica JA-1 y aplicacion de la prueba t de
Student basada en diferentes valores criticos.

[sup.87]Sr/[sup.86]Sr

Trial 1              Trial 2

LabA         LabB       LabA       LabB

0.703671   0.703613   0.703795   0.703625
0.703564   0.703389   0.703544   0.703594
0.703663   0.703547   0.703583   0.70348
0.703573   0.703543   0.703672   0.703594
0.703561   0.703537   0.703621   0.703471
0.703647   0.703506   0.703503   0.703628
0.703730   0.703496   0.703511   0.703466
0.703753   0.703557   0.703678   0.703452
0.703555   0.703602   0.703643   0.703423
0.703520   0.703433   0.703568   0.703488
0.703635   0.703640   0.703739   0.703509

Statistical parameter
Name                            Symbol         Values       Values
(Trial 1)    (Trial 2)
Number of measurements of       [n.sub.x]      11           11
[sup.87]Sr/[sup.86]
Sr from LabA
Number of measurements of       [n.sub.y]      11           11
[[sup.87]Sr/[sup.86]Sr
from LabB
Mean value of [sup.87]Sr/       [bar.x]        0.703625     0.703623
[sup.86]Sr from LabA
Mean value of [sup.87]Sr/       [bar.y]        0.703533     0.703521
[sup.86]Sr from LabB
Standard deviation of           [S.sub.x]      0.000076     0.000093
[sup.87]Sr/[sup.86]Sr
from LabA
Standard deviation of           [S.sub.y]      0.000075     0.000075
[sup.87]Sr/[sup.86]Sr
from LabB
Calculated Student's            [t.sub.calc]   2.84521033   2.84750347
t statistic for evaluating
[H.sub.0] and [H.sub.1]

Critical value of Student's     [([t.sup.cv.   2.85         2.85
t for 99% confidence            sub.99ts])
Miller (2005)                   .sub.MM]

Critical value of Student's     [([t.sup.cv.   2.845        2.845
t for 99% confidence            sub.99ts])
level (two-sided) and v         .sub.V]
of 20 from Verma (2005)

Critical value of Student's     [([t.sup.cv.   2.8453151    2.8453151
t for 99% confidence            sub.99ts])
level (two-sided) and v         .sub.tw]
of 20 from Table 1 or ES1
(this work)

[H.sub.0] : there is no statistically significant difference between
[sup.87]Sr/[sup.86]Sr from the two laboratories (LabA and LabB).

[H.sub.1] : there is statistically significant difference between
[sup.87]Sr/[sup.86]Sr from the two laboratories (LabA and LabB).

Decision (Trial 1): [t.sub.calc] < [([t.sup.cv.sub.99ts]).sub.MM] ;
[t.sub.calc] > [([t.sup.cv.sub.99ts]).sub.V]; or [t.sub.calc] <
[([t.sup.cv.sub.99ts]).sub.tw]

Decision (Trial 2): [t.sub.calc] < [([t.sup.cv.sub.99ts]).sub.MM] ;
[t.sub.calc] > [([t.sup.cv.sub.99ts]).sub.V]; or [t.sub.calc] <
[([t.sup.cv.sub.99ts]).sub.tw]

Calculated Student's t Probability for evaluating [H.sub.0] and
[H.sub.1]

Statistica                                     0.0100029    0.0099521
R (programming language)                       0.01000      0.009952
Excel                                          0.0100029    0.0099521
This work                                      0.0100005    0.0099497

Table 7.-Simulated paracetamol concentration in tablets by two
methods or by two analysts and application of Student's t test based
on different  critical values.

Tabla 7.-Concentacion simulada de paracetamol en tabletas por dos
metodos o por dos analistas y aplicacion de la prueba t de Student

Paracetamol (% m/m)

Trial 1                    Trial 2

MetA      MetB      AnalA     AnalB

84.32     83.97     84.15     83.78
84.05     83.45     83.98     84.19
83.85     84.13     83.97     84.1
83.95     83.91     83.76     83.79
84.13     83.79     83.96     83.92
83.79     84.03     84.03     83.85
83.94     83.86     84.24     83.89
83.76     83.73     83.93     83.85
84.08     83.9      83.86     84.04
84.13     83.74     83.96     84.01
83.98     84.05     83.94     83.87
84.01     83.75     83.89     83.87
84.13     84.03     83.88     83.82
83.89     83.94     84.28     83.76
84.17               84.13
84.15               84.06
84.08               84.37
84.12               84.26
83.93               84.35
84.05               84.39

Statistical parameter

Name                                       Symbol

Number of measurements of paracetamol      [n.sub.x]
from MetA or AnalA
Number of measurements of paracetamol      [n.sub.y]
from MetB or AnalB
Mean value of paracetamol from MetA or     [bar.x]
Mean value of paracetamol from MetB or     [bar.y]
Standard deviation of paracetamol from     [s.sub.x]
MetA or AnalA
Standard deviation of paracetamol from     [s.sub.y]
MetB or AnalB

Calculated Student's t statistic for       [t.sub.calc]
evaluating [H.sub.0] and [H.sub.1]

Linearly interpolated critical value of    [([t.sup.cv.sub.99ts])
Student's t for 99% confidence level       .sub.MM]
(two-sided) and v of 32 from Miller and
Miller (2005)

Linearly interpolated critical value of    [([t.sup.cv.sub.99ts])
Student's t for 99% confidence level       .sub.v1]
(two-sided) and v of 32 from Verma
(2005)

Best interpolated critical value of        [([t.sup.cv.sub.99ts])
Student's t for 99% confidence level       .sub.v2]
(two-sided) and v of 32 from Verma
(2009)

Best interpolated critical value of        [([t.sup.cv.sub.99ts])
Student's t for 99% confidence level       .sub.tw]
(two-sided) and v of 32 (this work)

Name                                       Values       Values
(Trial 1)    (Trial 2)
Number of measurements of paracetamol      20           20
from MetA or AnalA
Number of measurements of paracetamol      14           14
from MetB or AnalB
Mean value of paracetamol from MetA or     84.0255      84.0695
Mean value of paracetamol from MetB or     83.877143    83.91
Standard deviation of paracetamol from     0.13983      0.189055
MetA or AnalA
Standard deviation of paracetamol from     0.175781     0.128362
MetB or AnalB

Calculated Student's t statistic for       2.73893989   2.73954086
evaluating [H.sub.0] and [H.sub.1]

Linearly interpolated critical value of    2.743        2.743
Student's t for 99% confidence level
(two-sided) and v of 32 from Miller and
Miller (2005)

Linearly interpolated critical value of    2.7396       2.7396
Student's t for 99% confidence level
(two-sided) and v of 32 from Verma
(2005)

Best interpolated critical value of        2.73920      2.73920
Student's t for 99% confidence level
(two-sided) and v of 32 from Verma
(2009)

Best interpolated critical value of        2.7385076    2.7385076
Student's t for 99% confidence level
(two-sided) and v of 32 (this work)

[H.sub.0] : there is no statistically significant difference between
paracetamol from the two methods (MetA and MatB) or two analysts
(AnalA and AnalB).

[H.sub.1] : there is statistically significant difference between
paracetamol from the two methods (MetA and MatB) or two analysts
(AnalA and AnalB).

Decision {Trial 1): [t.sub.calc] < [([t.sup.cv.sub.99ts]).sub.MM] ;
[t.sub.calc] < [([t.sup.cv.sub.99ts]).sub.v1] ; [t.sub.calc] <
[([t.sup.cv.sub.99ts]).sub.v2] ; or [t.sub.calc] <
[([t.sup.cv.sub.99ts]).sub.tw]

Decision {Trial 2): [t.sub.calc] < [([t.sup.cv.sub.99ts]).sub.MM] ;
[t.sub.calc] < [([t.sup.cv.sub.99ts]).sub.v1] ; [t.sub.calc] <
[([t.sup.cv.sub.99ts]).sub.v2] ; or [t.sub.calc] <
[([t.sup.cv.sub.99ts]).sub.tw]

Calculated Student's t probability for evaluating [H.sub.0] and
[H.sub.1]

Statistica                                    0.0099888    0.0099740
R (programming language)                      0.009989     0.009974
Excel                                         0.0099888    0.0099740
This work                                     0.0099871    0.0099730

Table 8.-Simulated glucose levels of a group of patients before and
after medication and application of Student's t test based on
different critical values.

Tabla 8.-Niveles de glucosa simulados de un grupo de pacientes antes
y despues de la medicacion y y aplicacion de la prueba t de Student

Glucose level (mg/dl)

Baseline              14 weeks
Before medication After medication

87.2     73.7     62.4     112
97.5     66.9     185      146
86       122      79.5     147
102      133      81       146
85.2     85.2     109      92.8
104      86.4     144      142
112      78.8     81.9     99.8
93.7     99.6     140      110
109      92.3     131      104
81.7     99.6     61.7     143

135      89.2     99.8     130
85       96.5     69.3     133
61.9     80.2     108      88.6
83.3     104      81.6

Statistical parameter

Name                                 Symbol            Values

Number of measurements of glucose    [n.sub.x]         28
level (baseline)
Number of measurements of glucose    [n.sub.y]         27
level (14 weeks)
Mean value of glucose level          [bar.x]           93.9607143
(baseline)
Mean value of glucose level          [bar.y]           112.162963
(14 weeks)
Standard deviation of glucose        [S.sub.x]         17.39971
level (baseline)
Standard deviation of glucose        [S.sub.y]         31.39581
level (14 weeks)

Calculated Student's t statistic     [t.sub.calc]      2.67219998
for evaluating [H.sub.0] and
[H.sub.1]

Approximate critical value of        [([t.sup.cv.      [approximately
Student's t for 99% confidence       sub.99ts]).sub    equal to] 2.68
level (two-sided) and v of 53        .MM]
from Miller and Miller (2005)

Linearly interpolated critical       [([t.sup.cv.
value of Student's t for 99%         sub.99ts]).sub    2.6726
confidence level (two-sided) and     .v1]
v of 53 from Verma (2005)

Best interpolated critical value     [([t.sup.cv.      2.6726222
of Student's t for 99%               sub.99ts]).sub
confidence level (two-sided) and     .v2]
v of 53 from Verma (2009)

Best interpolated critical value     [([t.sup.cv.      2.6718070
of Student's t for 99%               sub.99ts]).sub
confidence level (two-sided) and     .tw]
v of 53 (this work)

[H.sub.0] : there is no statistically significant difference between
baseline and medication glucose levels.

[H.sub.1] : there is statistically significant difference between
baseline and medication glucose levels.

Decision : [t.sub.calc] < [([t.sup.cv.sub.99ts]).sub.MM] ;
[t.sub.calc] < [([t.sup.cv.sub.99ts]).sub.v1] ; [t.sub.calc] <
[([t.sup.cv.sub.99ts]).sub.v2] ; or [t.sub.calc] <
[([t.sup.cv.sub.99ts]).sub.tw]

Calculated Student's t probability for evaluating [H.sub.0] and
[H.sub.1]

Statistica                                             0.0099901
R (programming language)                               0.00999
Excel                                                  0.0099901
This work                                              0.0099873

Table 9.-Statistical parameters of element concentrations in
geochemical reference material granite G-1 from U.S.A.

Tabla 9.-Parametros estadisticos de concentraciones de elementos en
el material de referencia geoquimica, el granito G-1 de U.S.A.

Element                           This work

Number of        Mean         Standard
([micro]g/g)      observations    ([t.sub.tw])    deviation
([n.sub.tw])                   ([s.sub.tw])

Si (%)                 112          33.851         0.104
Ti (%)                 148           1.484         0.187
Al (%)                 133           7.535         0.125
Fe (%)                 125           1.348         0.076
Mn (%)                 155           0.213         0.046
Mg (%)                 135           0.2315        0.0416
Ca (%)                 139           0.984         0.056
Na (%)                 126           2.473         0.089
K (%)                  125           4.558         0.074
P (%)                  83            0.378         0.066
[H.sub.2]              60            0.337         0.069
[O.sup.+] (%)
[H.sub.2]              53            0.0460        0.0203
[O.sup.-] (%)

La                     44          106.8          23.1
Ce                     21          173.3          19.4
Pr                     12           17.42          2.96
Nd                     19           53.7           6.9
Sm                     18            7.87          0.55
Eu                     24            1.197         0.176
Gd                     15            5.11          0.95
Tb                     14            0.632         0.226
Dy                     14            2.358         0.414
Ho                      9            0.432         0.056
Er                     11            1.354         0.370
Tm                      8            0.1641        0.0294
Yb                     23            0.866         0.172
Lu                     15            0.1550        0.0305
Ba                     76         1158           134
Be                     29            3.06          0.79
Co                     42            2.295         0.367
Cr                     56           19.8           5.5
Cs                     15            1.713         0.318
Cu                     70           12.65          3.26
Ga                     44           18.19          2.55
Hf                     13            5.36          0.55
Li                     32           22.74          2.97
Nb                     19           22.57          4.30
Pb                     87           43.5           9.8
Rb                     47          217.8           9.6
Sb                     20            0.290         0.077
Sc                     28            3.09          0.53
Sr                     88          254.1          25.0
Ta                     10            1.503         0.391
Th                     34           50.5           6.5
U                      30            3.49          0.57
V                      49           17.61          3.71
Y                      32           12.37          2.05
Zn                     44           44.0           7.3
Zr                     70          201.2          28.5

Ag                      8            0.0436        0.0057
As                     12            0.730         0.140
Au                     12            0.00327       0.00095
Bi                      8            0.0710        0.0245
C                      14          194.7          31.0
Cd                      8            0.0489        0.0252
Cl                      9           56.9          17.8
F                      33          699           105
Ge                      9            1.133         0.101
Hg                     13            0.0983        0.0290
In                      9            0.02433       0.00269
Ir                      5            0.00258       0.00326
Mo                     26            6.35          1.03
S                      14          129            59
Sn                     28            3.51          1.16
Tl                     19            1.221         0.110
W                       8            0.4212        0.0372

Element                  Literature (Gladney et al, 1991)

Number of         Mean          Standard
([micro]g/g)      observations                      deviation
([n.sub.lit])   ([x.sub.lit])   ([s.sub.lit])

Si (%)                 122          33.83           0.14
Ti (%)                 144           1.500          0.170
Al (%)                 129           7.53           0.11
Fe (%)                 136           1.36           0.10
Mn (%)                 156           0.214          0.044
Mg (%)                 132           0.232          0.038
Ca (%)                 135           0.982          0.052
Na (%)                 124           2.47           0.08
K (%)                  138           4.54           0.11
P (%)                  82            0.380          0.060
[H.sub.2]              64            0.34           0.08
[O.sup.+] (%)
[H.sub.2]              58            0.046          0.023
[O.sup.-] (%)
La                     43          103             21
Ce                     25          173             24
Pr                     12           17              3
Nd                     22           57             11
Sm                     22            7.9            1.2
Eu                     26            1.2            0.21
Gd                     15            5.1            0.9
Tb                     13            0.600          0.190
Dy                     13            2.44           0.3
Ho                     13            0.430          0.140
Er                     11            1.35           0.37
Tm                     12            0.185          0.060
Yb                     26            0.96           0.31
Lu                     17            0.156          0.036
Ba                     81         1150            150
Be                     28            3.0            0.7
Co                     46            2.3            0.5
Cr                     61           20              6
Cs                     16            1.68           0.33
Cu                     67           12              3
Ga                     47           18.0            2.6
Hf                     13            5.4            0.6
Li                     35           23              4
Nb                     18           21              3
Pb                     86           45              9
Rb                     53          218             12
Sb                     21            0.30           0.08
Sc                     33            3.0            0.6
Sr                     96          253             35
Ta                     10            1.500          0.390
Th                     36           51              7
U                      31            3.450          0.520
V                      53           18              4
Y                      39           14              4
Zn                     46           45              8
Zr                     67          201             23

Ag                      8            0.044          0.006
As                     12            0.73           0.14
Au                     13            0.0032         0.001
Bi                     10            0.073          0.029
C                      15          200             40
Cd                      7            0.055          0.018
Cl                     10           53             20
F                      34          690            110
Ge                     10            1.09           0.17
Hg                     14            0.085          0.030
In                      9            0.024          0.003
Ir                      5            0.0025         0.0034
Mo                     31            6.8            1.7
S                      15          130             60
Sn                     29            3.6            1.2
Tl                     22            1.230          0.130
W                       9            0.430          0.044

Table 10.-Statistical parameters of element concentrations in
geochemical reference material granite G-2 from U.S.A.

Tabla 10.-Parametros estadisticos de concentraciones de elementos en
el material de referencia geoquimica, el granito G-2 de U.S.A

Element ([micro]g/g)   [n.sub.tw]   [x.sub.tw]    [S.sub.tw]

Si (%)                    127         32.251        0.223
Ti (%)                    158          0.2928       0.0244
Al (%)                    147          8.143        0.185
Fe (%)                    182          1.859        0.075
Mn (%)                    168          0.0256       0.0048
Mg (%)                    134          0.4614       0.0372
Ca (%)                    140          1.399        0.058
Na (%)                    137          3.027        0.091
K (%)                     162          3.716        0.089
P (%)                      82          0.0591       0.0055
[H.sub.2]                  30          0.522        0.099
[O.sup.+] (%)
[H.sub.2]                  30          0.1047       0.0418
[O.sup.-] (%)

La                        115         89.0          7.3
Ce                        103        159.3         11.5
Pr                         24         17.83         2.65
Nd                         85         54.8          6.1
Sm                         85          7.088        0.301
Eu                        100          1.386        0.110
Gd                         50          4.30         0.82
Tb                         79          0.488        0.093
Dy                         38          2.376        0.330
Ho                         24          0.432        0.107
Er                         17          1.082        0.238
Tm                         20          0.188        0.090
Yb                         92          0.777        0.139
Lu                         82          0.1111       0.0262

B                          15          2.216        0.371
Ba                        133       1845          152
Be                         32          2.446        0.428
Co                        101          4.65         0.63
Cr                        102          8.66         2.01
Cs                         61          1.330        0.146
Cu                        104         10.98         2.75
Ga                         59         22.77         4.21
Hf                         57          7.92         0.94
Li                         49         34.1          6.1
Nb                         45         12.23         4.26
Ni                         78          4.71         2.37
Pb                         92         30.30         4.29
Rb                        149        170.2          9.4
Sb                         21          0.720        0.220
Sc                         66          3.477        0.271
Sr                        152        471.3         32.7
Ta                         52          0.877        0.125
Th                         99         24.80         1.78
U                          67          2.019        0.155
V                          82         35.5          6.3
Y                          47         11.05         2.32
Zn                        100         85.8          8.1
Zr                         93        307.7         31.0

Ag                         16          0.0406       0.0137
As                         15          0.415        0.281
Au                         10          0.000995     0.000183
Bi                         9           0.0360       0.0048
Br                         7           0.238        0.139
C                          26        234           75
Cd                         17          0.0308       0.0147
Cl                         19         87.4         35.4
F                          34       1279           66
Ge                         13          1.012        0.224
Hg                         31          0.0523       0.0150
In                         6           0.02983      0.00183
Ir                         6           0.000045     0.000049
Mo                         16          0.96         0.60
S                          15         82.5         31.6
Se                         7           0.072        0.063
Sn                         11          1.93         0.90
Tl                         24          0.907        0.144
W                          8           0.133        0.065

Element ([micro]g/g)   99% Confidence limits of
the mean ([mu])

Lower limit   Upper limit

Si (%)                   32.200        32.303
Ti (%)                    0.2878        0.2979
Al (%)                    8.103         8.183
Fe (%)                    1.844         1.873
Mn (%)                    0.0247        0.0266
Mg (%)                    0.4530        0.4698
Ca (%)                    1.387         1.412
Na (%)                    3.007         3.048
K (%)                     3.697         3.734
P (%)                     0.0575        0.0607
[H.sub.2]                 0.473         0.572
[O.sup.+] (%)
[H.sub.2]                 0.0836        0.1257
[O.sup.-] (%)

La                       87.2          90.8
Ce                      156.3         162.3
Pr                       16.31         19.35
Nd                       53.0          56.5
Sm                        7.002         7.174
Eu                        1.357         1.415
Gd                        4.00          4.61
Tb                        0.460         0.516
Dy                        2.231         2.522
Ho                        0.371         0.494
Er                        0.914         1.250
Tm                        0.130         0.245
Yb                        0.739         0.815
Lu                        0.1034        0.1187

B                         1.931         2.501
Ba                     1810          1879
Be                        2.238         2.653
Co                        4.49          4.82
Cr                        8.13          9.18
Cs                        1.281         1.380
Cu                       10.27         11.69
Ga                       21.31         24.23
Hf                        7.59          8.25
Li                       31.7          36.4
Nb                       10.52         13.94
Ni                        4.00          5.42
Pb                       29.12         31.48
Rb                      168.2         172.2
Sb                        0.583         0.856
Sc                        3.389         3.566
Sr                      464.4         478.2
Ta                        0.831         0.923
Th                       24.33         25.27
U                         1.969         2.070
V                        33.6          37.3
Y                        10.14         11.96
Zn                       83.7          87.9
Zr                      299.2         316.1

Ag                        0.0305        0.0507
As                        0.199         0.632
Au                        0.000807      0.001183
Bi                        0.0307        0.0413
Br                        0.043         0.433
C                       193           275
Cd                        0.0204        0.0412
Cl                       63.9         110.8
F                      1249          1310
Ge                        0.822         1.201
Hg                        0.0448        0.0597
In                        0.02681       0.03285
Ir                       (0.0000)       0.000125
Mo                        0.52          1.40
S                        58.2         106.8
Se                       (0.0)          0.160
Sn                        1.06          2.79
Tl                        0.824         0.989
W                         0.052         0.213

Table 11.-Statistical parameters of element concentrations in G-1 and
G-2 and application of Student t test to evaluate similarities and
differences between them.

Tabla 11.-Parametros estadisticos de las concentraciones de elementos
en G-1 y G-2 asi como la aplicacion de la prueba t de Student para
evaluar similitudes y diferencias entre ellas.

Element                Granite reference material G-1

[n.sub.G1]   [x.sub.G1]   [s.sub.G1]

Si (%)              114         33.851       0.113
Ti (%)              150          0.1491      0.0196
Al (%)              133          7.535       0.125
Fe (%)              141          1.368       0.118
Mn (%)              158          0.0213      0.0049
Mg (%)              136          0.2305      0.0431
Ca (%)              139          0.984       0.056
Na (%)              129          2.472       0.098
K (%)               130          4.548       0.087
P (%)                90          0.0376      0.0089
[H.sub.2]            60          0.337       0.069
[O.sup.+] (%)
[H.sub.2]            57          0.0502      0.0248
[O.sup.-] (%)

La                   44        106.8        23.1
Ce                   27        168.3        27.8
Pr                   12         17.42        2.96
Nd                   19         53.7         6.9
Sm                   19          7.82        0.59
Eu                   25          1.181       0.190
Gd                   15          5.11        0.95
Tb                   14          0.632       0.226
Dy                   18          2.91        1.15
Ho                   13          0.434       0.138
Er                   11          1.354       0.370
Tm                   10          0.181       0.048
Yb                   23          0.866       0.172
Lu                   15          0.155       0.0305

B                    11          1.71        0.51
Ba                   76       1158         134
Be                   29          3.06        0.79
Co                   44          2.242       0.433
Cr                   61         19.5         6.2
Cs                   15          1.713       0.318
Cu                   70         12.65        3.26
Ga                   44         18.19        2.55
Hf                   13          5.36        0.55
Li                   32         22.74        2.97
Nb                   19         22.57        4.30
Ni                   33          2.88        2.17
Pb                   87         43.5         9.8
Rb                   48        218.5        10.5
Sb                   23          0.321       0.112
Sc                   31          3.14        0.60
Sr                   94        252.6        32.8
Ta                   11          1.64        0.58
Th                   35         50.8         6.7
U                    32          3.38        0.71
V                    49         17.61        3.71
Y                    32         12.37        2.05
Zn                   45         43.5         8.1
Zr                   71        199.8        30.7

Ag                   8           0.0436      0.0057
As                   12          0.730       0.140
Au                   12          0.00327     0.00095
Bi                   9           0.0677      0.0250
Br                   6           0.433       0.413
C                    16        208          48
Cd                   8           0.0489      0.0252
Cl                   9          56.9        17.8
F                    33        699         105
Ge                   9           1.133       0.101
Hg                   15          0.0900      0.0347
In                   9           0.02433     0.00269
Ir                   5           0.00258     0.00326
Mo                   27          6.47        1.18
S                    15        121          65
Se                   3           0.00703     0.00090
Sn                   28          3.51        1.16
Tl                   21          1.244       0.127
W                    8           0.4212      0.0372

Element               Granite reference material G-2

[n.sub.G2]   [x.sub.G2]    [s.sub.G2]

Si (%)              128         32.236        0.236
Ti (%)              161          0.2932       0.0260
Al (%)              149          8.143        0.198
Fe (%)              188          1.861        0.076
Mn (%)              167          0.0259       0.0047
Mg (%)              134          0.4614       0.0372
Ca (%)              148          1.402        0.062
Na (%)              141          3.027        0.096
K (%)               170          3.716        0.095
P (%)                84          0.0594       0.0058
[H.sub.2]            30          0.522        0.099
[O.sup.+] (%)
[H.sub.2]            30          0.1047       0.0418
[O.sup.-] (%)

La                  139         90.9          9.4
Ce                  121        160.1         13.1
Pr                   24         17.83         2.65
Nd                   85         55.2          6.2
Sm                   90          7.124        0.389
Eu                  100          1.399        0.105
Gd                   50          4.30         0.82
Tb                   77          0.485        0.083
Dy                   34          2.375        0.257
Ho                   23          0.404        0.072
Er                   17          1.082        0.238
Tm                   21          0.193        0.091
Yb                   93          0.804        0.152
Lu                   79          0.1117       0.0238

B                    15          2.216        0.371
Ba                  142       1838          185
Be                   32          2.446        0.428
Co                  103          4.77         0.62
Cr                  101          8.70         1.97
Cs                   65          1.344        0.164
Cu                  103         10.91         2.68
Ga                   59         22.77         4.21
Hf                   58          7.95         0.97
Li                   49         34.1          6.1
Nb                   46         12.32         4.25
Ni                   78          4.72         2.27
Pb                   93         30.18         4.43
Rb                  150        170.1          9.6
Sb                   22         69.4         24.7
Sc                   77          3.583        0.326
Sr                  154        471.3         32.7
Ta                   48          0.868        0.096
Th                  103         24.77         1.89
U                    69          2.035        0.177
V                    84         36.3          6.6
Y                    49         11.00         2.52
Zn                  102         85.8          8.9
Zr                   96        310.6         33.8

Ag                   16          0.0406       0.0137
As                   15          0.415        0.281
Au                   10          0.000995     0.000183
Bi                   10          0.0362       0.0045
Br                   7           0.238        0.139
C                    26        234           75
Cd                   21          0.0306       0.0168
Cl                   28         82.5         40.1
F                    39       1300          102
Ge                   15          1.103        0.319
Hg                   28          0.0511       0.0129
In                   9           0.0332       0.0058
Ir                   6           0.430        0.044
Mo                   17          1.12         0.87
S                    16         98.6         42.5
Se                   7           0.072        0.063
Sn                   20          1.77         0.68
Tl                   27          0.882        0.177
W                    8           0.132        0.065

Element              Student t test
([H.sub.0])

Two-sided   One-sided

Si (%)           false       false
Ti (%)           false       false
Al (%)           false       false
Fe (%)           false       false
Mn (%)           false       false
Mg (%)           false       false
Ca (%)           false       false
Na (%)           false       false
K (%)            false       false
P (%)            false       false
[H.sub.2]        false       false
[O.sup.+] (%)
[H.sub.2]        false       false
[O.sup.-] (%)

La               false       false
Ce               true        true
Pr               true        true
Nd               true        true
Sm               false       false
Eu               false       false
Gd               false       false
Tb               true        true
Dy               true        true
Ho               true        true
Er               true        true
Tm               true        true
Yb               true        true
Lu               false       false

B                false       false
Ba               false       false
Be               false       false
Co               false       false
Cr               false       false
Cs               false       false
Cu               false       false
Ga               false       false
Hf               false       false
Li               false       false
Nb               false       false
Ni               false       false
Pb               false       false
Rb               false       false
Sb               false       false
Sc               false       false
Sr               false       false
Ta               false       false
Th               false       false
U                false       false
V                false       false
Y                true        false
Zn               false       false
Zr               false       false

Ag               true        true
As               false       false
Au               false       false
Bi               false       false
Br               true        true
C                true        true
Cd               true        true
Cl               true        true
F                false       false
Ge               true        true
Hg               false       false
In               false       false
Ir               true        true
Mo               false       false
S                true        true
Se               true        true
Sn               false       false
Tl               false       false
W                false       false

Table 12.-Statistical parameters of element concentrations in basic
rocks from the Canary and Azores Islands and application of Student t
test to evaluate similarities and differences between them.

Tabla 12.-Parametros estadisticos de las concentraciones de elementos
en rocas basicas de las Islas Canarias y de Azores y la aplicacion de
la prueba t de Student para evaluar similitudes y diferencias entre
ellas.

Element                                        Canary Islands

[n.sub.Canary]   [x.sub.Canary]

[(Or).sub.norm]                          67           9.46
[(Ab).sub.norm]                          67          23.5
[(An).sub.norm]                          67          20.76
[(Ne).sub.norm]                          67           5.07
[(En).sub.norm]                          67          12.0
[(Fs).sub.norm]                          67           6.49
[(Di).sub.norm]                          67          18.5
[(Fo).sub.norm]                          65           6.33
[(Fa).sub.norm]                          67           4.50
[(Ol).sub.norm]                          67          11.12
[(Mt).sub.norm]                          67           3.198
[(Il).sub.norm]                          67           6.13
[(Ap).sub.norm]                          66           1.86
Mg_value                                 67          53.3

La                                       27          66.0
Ce                                       27         137.0
Pr                                       13          14.73
Nd                                       24          56.8
Sm                                       24          10.38
Eu                                       18           3.21
Gd                                       14           8.47
Tb                                       5            1.032
Dy                                       19           6.57
Ho                                       5            0.944
Er                                       15           2.804
Tm                                       4            0.305
Yb                                       23           2.61
Lu                                       19           0.300
Ba                                       67         506
Co                                       17          41.1
Cr                                       37           9.9
Cu                                       44          62
Ga                                       53          21.68
Hf                                       6            7.72
Nb                                       66          90.6
Pb                                       55           3.53
Rb                                       67          36.1
Sc                                       48          20.7
Sr                                       67         951
Ta                                       6            6.27
Th                                       65           7.12
U                                        17           2.71
V                                        63         238
Y                                        67          34.4
Zn                                       56         109.7
Zr                                       67         347

Element                            Canary Islands   Azores Islands

[s.sub.Canary]   [n.sub.Azores]

[(Or).sub.norm]                      3.78                 82
[(Ab).sub.norm]                      8.4                  82
[(An).sub.norm]                      3.50                 80
[(Ne).sub.norm]                      3.56                 82
[(En).sub.norm]                      5.6                  82
[(Fs).sub.norm]                      2.13                 82
[(Di).sub.norm]                      7.5                  82
[(Fo).sub.norm]                      2.78                 81
[(Fa).sub.norm]                      1.36                 82
[(Ol).sub.norm]                      4.40                 82
[(Mt).sub.norm]                      0.347                82
[(Il).sub.norm]                      0.82                 82
[(Ap).sub.norm]                      0.58                 82
Mg_value                             7.2                  82

La                                  25.4                  75
Ce                                  43.0                  73
Pr                                   3.80                 45
Nd                                  13.0                  73
Sm                                   2.67                 73
Eu                                   0.62                 73
Gd                                   1.73                 48
Tb                                   0.172                72
Dy                                   1.67                 45
Ho                                   0.152                45
Er                                   0.77                 45
Tm                                   0.053                51
Yb                                   1.15                 73
Lu                                   0.070                71
Ba                                 201                    80
Co                                  17.5                  64
Cr                                   8.8                  63
Cu                                  46                    26
Ga                                   2.48                 54
Hf                                   1.45                 70
Nb                                  28.1                  38
Pb                                   1.71                 49
Rb                                  16.2                  81
Sc                                  10.3                  64
Sr                                 270                    82
Ta                                   1.87                 70
Th                                   3.18                 70
U                                    1.61                 64
V                                   80                    76
Y                                    7.9                  76
Zn                                  11.3                  70
Zr                                 100                    70

Element                                      Azores Islands

[x.sub.Azores]   [s.sub.Azores]

[(Or).sub.norm]                      9.87             2.96
[(Ab).sub.norm]                     23.9              7.4
[(An).sub.norm]                     21.35             3.08
[(Ne).sub.norm]                      2.99             1.99
[(En).sub.norm]                     11.7              5.3
[(Fs).sub.norm]                      5.36             1.40
[(Di).sub.norm]                     17.1              6.4
[(Fo).sub.norm]                      8.34             3.62
[(Fa).sub.norm]                      4.96             1.14
[(Ol).sub.norm]                     13.5              4.7
[(Mt).sub.norm]                      3.26             0.58
[(Il).sub.norm]                      5.87             0.97
[(Ap).sub.norm]                      1.47             0.47
Mg_value                            56.9              9.7

La                                  41.8             11.3
Ce                                  87.3             23.1
Pr                                  10.54             2.77
Nd                                  42.3             11.1
Sm                                   8.74             2.17
Eu                                   2.86             0.66
Gd                                   7.83             1.74
Tb                                   1.162            0.229
Dy                                   6.21             1.40
Ho                                   1.124            0.257
Er                                   3.03             0.70
Tm                                   0.400            0.091
Yb                                   2.39             0.49
Lu                                   0.331            0.068
Ba                                 477              142
Co                                  37.1             12.1
Cr                                 291              210
Cu                                  49.7             20.3
Ga                                  21.64             2.58
Hf                                   6.50             1.56
Nb                                  58.0             16.7
Pb                                   4.91             2.69
Rb                                  36.1             11.9
Sc                                  22.2              5.9
Sr                                 674              134
Ta                                   3.67             0.98
Th                                   4.43             1.43
U                                    1.259            0.385
V                                  247.8             36.2
Y                                   31.0              6.6
Zn                                  76.7             30.8
Zr                                 289               80

Element                                 Student t test
([H.sub.0])

Two-sided   One-sided

[(Or).sub.norm]                    true        true
[(Ab).sub.norm]                    true        true
[(An).sub.norm]                    true        true
[(Ne).sub.norm]                    false       false
[(En).sub.norm]                    true        true
[(Fs).sub.norm]                    false       false
[(Di).sub.norm]                    true        true
[(Fo).sub.norm]                    false       false
[(Fa).sub.norm]                    true        true
[(Ol).sub.norm]                    false       false
[(Mt).sub.norm]                    true        true
[(Il).sub.norm]                    true        true
[(Ap).sub.norm]                    false       false
Mg_value                           false       false

La                                 false       false
Ce                                 false       false
Pr                                 false       false
Nd                                 false       false
Sm                                 false       false
Eu                                 true        true
Gd                                 true        true
Tb                                 true        true
Dy                                 true        true
Ho                                 true        true
Er                                 true        true
Tm                                 true        true
Yb                                 true        true
Lu                                 true        true
Ba                                 true        true
Co                                 true        true
Cr                                 false       false
Cu                                 true        true
Ga                                 true        true
Hf                                 true        true
Nb                                 false       false
Pb                                 false       false
Rb                                 true        true
Sc                                 true        true
Sr                                 false       false
Ta                                 false       false
Th                                 false       false
U                                  false       false
V                                  true        true
Y                                  false       false
Zn                                 false       false
Zr                                 false       false
```