# 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. IntroductionThe 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).

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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.

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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).

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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.

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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.

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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

Received: 03/02/2011 / Accepted: 12/04/2013

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. Tabla 1.-Forma abreviada de la tabla de valores criticos simulados 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 resaltados en negrillado. 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 resaltados en negrillado. 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 distribucion t de Student para diferentes grados de libertad utiles 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 estadisticas. 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 basada en diferentes valores criticos. 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 basada en diferentes valores criticos. 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] p(Si[O.sub.2]).sub.adj] 67 47.54 [(Ti[O.sub.2]).sub.adj] 67 3.229 [([Al.sub.2][O.sub.3]).sub.adj] 67 15.73 [([Fe.sub.2][O.sub.3]).sub.adj] 67 2.206 [(FeO).sub.adj] 67 8.84 [(MnO).sub.adj] 66 0.2001 [(MgO).sub.adj] 66 5.98 [(CaO).sub.adj] 67 9.84 [([Na.sub.2]O).sub.adj] 67 3.88 [([K.sub.2]O).sub.adj] 67 1.60 [([P.sub.2][O.sub.5]).sub.adj] 66 0.803 [(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] p(Si[O.sub.2]).sub.adj] 2.17 82 [(Ti[O.sub.2]).sub.adj] 0.429 82 [([Al.sub.2][O.sub.3]).sub.adj] 1.95 82 [([Fe.sub.2][O.sub.3]).sub.adj] 0.240 82 [(FeO).sub.adj] 1.45 82 [(MnO).sub.adj] 0.0222 82 [(MgO).sub.adj] 2.48 82 [(CaO).sub.adj] 1.77 82 [([Na.sub.2]O).sub.adj] 1.04 82 [([K.sub.2]O).sub.adj] 0.64 82 [([P.sub.2][O.sub.5]).sub.adj] 0.252 82 [(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] p(Si[O.sub.2]).sub.adj] 47.87 1.64 [(Ti[O.sub.2]).sub.adj] 3.09 0.51 [([Al.sub.2][O.sub.3]).sub.adj] 15.49 2.02 [([Fe.sub.2][O.sub.3]).sub.adj] 2.247 0.402 [(FeO).sub.adj] 8.76 1.01 [(MnO).sub.adj] 0.1694 0.0144 [(MgO).sub.adj] 7.14 3.03 [(CaO).sub.adj] 9.46 1.21 [([Na.sub.2]O).sub.adj] 3.48 0.68 [([K.sub.2]O).sub.adj] 1.67 0.50 [([P.sub.2][O.sub.5]).sub.adj] 0.633 0.201 [(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 p(Si[O.sub.2]).sub.adj] true true [(Ti[O.sub.2]).sub.adj] true true [([Al.sub.2][O.sub.3]).sub.adj] true true [([Fe.sub.2][O.sub.3]).sub.adj] true true [(FeO).sub.adj] true true [(MnO).sub.adj] false false [(MgO).sub.adj] true false [(CaO).sub.adj] true true [([Na.sub.2]O).sub.adj] false false [([K.sub.2]O).sub.adj] true true [([P.sub.2][O.sub.5]).sub.adj] false false [(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

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Title Annotation: | articulo en ingles |
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Author: | Verma, S.P.; Cruz-Huicochea, R. |

Publication: | Journal of Iberian Geology |

Date: | Jan 1, 2013 |

Words: | 20111 |

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