Student’s /-statistics

As in univariate calibration, prediction intervals (Pis) can be constructed from the above estimated standard error of prediction, by means of a Student s /-statistic, as ... 

The data in this example represent an investigation of ammonia production in the presence of a particular catalyst. The measured yield in percent is shown in the far right column of Table 8.10. Suppose we wish to achieve a prediction error less than fi = 1.5% in an example, where the standard deviation (measurement error) is y.2 = 1.05 estimated with 11 measurements, i.e., with degrees of freedom v= 10. The critical value of Student s /-statistic is found to be t10A95 = 2.228. At IV = 16 experiments, we check to see if the desired level of accuracy is achieved and obtain Kj-a Jd = 2.228 0.438 x 1.05 =1.51 > 1.5. At N= 16 experiments, weobtain tvl aJd =1.48 < 1.5 therefore, we can stop at N = 17 and be assured that, 95% of the time, we will achieve a prediction error not worse than +1.5%, which is considerably smaller than the range of the variation in the response value. 

The scaling factor a and the adjusted degrees of freedom b are given in Box and Tiao (1973). For large ti, n2, the scaling factor a approaches 1 and the degrees of freedom b approach i + n2 — 2 so that the posterior distribution of rjkl — i)k2 is approximately the noncentral Student s /-statistic... 

Because we know (the standard deviation in the interpolated value Xq), we can use the Student s /-statistics to estimate to what extent Xo estimates the population mean This can be determined according to... 

Xu and x are upper and lower limits, f Is Student s statistics for (n-1) degrees of freedom and confidence probability (1-a)... 

Following this, Xc is calculated as 2) wo %x)/slope, with t being the one-tailed Student s statistic and wo-Syjx the standard error at zero concentration derived from the calibration the slope derives from the predicted vs. reference regression. On the other hand, Xj is wq Syjy)/... 

The t (Student s t) distribution is an unbounded distribution where the mean is zero and the variance is v/(v - 2), v being the scale parameter (also called degrees of freedom ). As v -> < , the variance —> 1 (standard normal distribution). A t table such as Table 1-19 is used to find values of the t statistic where... 

Statistical analysis. Values are given as the mean SEM. Data are represented as averages of independent experiments, performed in duplicate or triplicate. Statistical analyses were done using the Student s t-test. P < 0.05 was considered statistically significant. 

The Student s (W.S. Gossett) /-lest is useful for comparisons of the means and standard deviations of different analytical test methods. Descriptions of the theory and use of this statistic are readily available in standard statistical texts including those in the references [1-6]. Use of this test will indicate whether the differences between a set of measurement and the true (known) value for those measurements is statistically meaningful. For Table 36-1 a comparison of METHOD B test results for each of the locations is compared to the known spiked analyte value for each sample. This statistical test indicates that METHOD B results are lower than the known analyte values for Sample No. 5 (Lab 1 and Lab 2), and Sample No. 6 (Lab 1). METHOD B reported value is higher for Sample No. 6 (Lab 2). Average results for this test indicate that METHOD B may result in analytical values trending lower than actual values. 

The test to determine whether the bias is significant incorporates the Student s /-test. The method for calculating the t-test statistic is shown in equation 38-10 using MathCad symbolic notation. Equations 38-8 and 38-9 are used to calculate the standard deviation of the differences between the sums of X and Y for both analytical methods A and B, whereas equation 38-10 is used to calculate the standard deviation of the mean. The /-table statistic for comparison of the test statistic is given in equations 38-11 and 38-12. The F-statistic and f-statistic tables can be found in standard statistical texts such as references [1-3]. The null hypothesis (H0) states that there is no systematic difference between the two methods, whereas the alternate hypothesis (Hf) states that there is a significant systematic difference between the methods. It can be seen from these results that the bias is significant between these two methods and that METHOD B has results biased by 0.084 above the results obtained by METHOD A. The estimated bias is given by the Mean Difference calculation. 

This set of articles presents the computational details and actual values for each of the statistical methods shown for collaborative tests. These methods include the use of precision and estimated accuracy comparisons, ANOVA tests, Student s t-testing, The Rank Test for Method Comparison, and the Efficient Comparison of Methods tests. From using these statistical tests the following conclusions can be derived ... 

The confidence limits for the slope and intercept may be calculated using the Student s t statistic, noting Equations 61-27 through 61-30 below. 

In this case study, 11 different extraction methods (Table 1) were applied to a 1000 m line of 15 B/C-horizon soil samples (<0.25 mm) across the Talbot VMS Cu-Zn deposit in northern Manitoba, Canada (Fig. 1), followed by ICP-MS analysis. Student s t test and minimum t probability statistics provide a tool to rank the exploration performance of each method (Stanley 2003 Stanley Noble 2008). By... 

Student s t test statistics and t probability were calculated to quantify the contrast between anomalous and background populations in each extraction method data set (Student 1908 Stanley Noble 2008). Sample sites were designated anomalous based on the projection of mineralization and a fault zone in the cover rocks. For most methods, Zn... 

Fig. 2. t probability as a function of Student s t statistics for Enzyme Leach (a best case ) and Bioleach (b worst case )... 

The wheat bran used in these studies was milled for us from a single lot of Waldron hard red spring wheat. Other foods and diet ingredients were purchased from local food suppliers. Data from HS-I was analyzed statistically by Student s paired t test, each subject acting as his own control. A three-way analysis of variance (ANOVA) was performed to test for significant differences betwen diet treatments, periods and individuals in HS-II and HS-III. 

The analysis of rank data, what is generally called nonparametric statistical analysis, is an exact parallel of the more traditional (and familiar) parametric methods. There are methods for the single comparison case (just as Student s t-test is used) and for the multiple comparison case (just as analysis of variance is used) with appropriate post hoc tests for exact identification of the significance with a set of groups. Four tests are presented for evaluating statistical significance in rank data the Wilcoxon Rank Sum Test, distribution-free multiple comparisons, Mann-Whitney U Test, and the Kruskall-Wallis nonparametric analysis of variance. For each of these tests, tables of distribution values for the evaluations of results can be found in any of a number of reference volumes (Gad, 1998). 

Since the comparison of rxy with the table values may be considered a somewhat weak test, it is perhaps more meaningful to compare the tr value with values in a t-distribution table for N-2 degrees of freedom (df), as is done for Student s I-tcst. This will give a more exact determination of the degree of statistical correlation between the two groups. 

Lactic acid fermentation was the topic of a paper by Vaccari et al.35 In this work, lactic acid, glucose, and biomass were determined over the course of the reaction. The measurements were made in real time, using a bypass pump and flow-through cell for the NIR measurements. Instead of using normal chemomet-ric statistics, the authors used correlation coefficients, mean of differences, standard deviation, student s t-test, and the student test parameter of significant difference to evaluate the results. Under these restrictions, the results appeared fairly good, with the biomass having the best set of statistics. 

When the statistieally sophisticated psychologists realized what I was doing, they had a field day pointing out my failings unjustified assumptions, violations of statistical theory and other mathematical crimes. They talked about ordinal scales versus ratio scales and scolded me for not using analysis of variance instead of Chi-square and Student s T tests of significance. 

Statistics. The levels of significance between two sets of unpaired results were determined according to Student s t test. 

The other piece of information (in addition to bg and required to establish a confidence interval for a parameter estimate was not available until 1908 when W. S. Gosset, an English chemist who used the pseudonym Student (1908), provided a solution to the statistical problem [J. Box (1981)]. The resulting values are known as critical values of Student s t and may be obtained from so-called /-tables (see Appendix B for values at the 95% level of confidence). 

Let TS be a test statistic which approximates a Student s t-dlstrlbutlon... 

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