Critical t-values

It is evident that the calculated t-value for the constant value, a, is less than the critical t-value. From the statistical viewpoint this value, then, is negligible. The data can then be recalculated according to the first order model without a constant value. Table III shows the result of this recalculation. There are no changes relating to the conclusions made concerning the author determination. 

The critical effect, Ecriticai, depends on the (tabulated) critical t-value, tcriticaij and on (SE)e. The tcridcai depends on the number of degrees of freedom associated with (SE) and is usually considered at a significance level a = 0.05 (occasionally also a = 0.01). An effect is considered significant if xl > Emtical- In a robustness test, (SE)e can be estimated in different... 

FIGURE 8 Standardized Pareto chart, tx, standardized effect of factor X tcriticai> critical t-value. 

The t value is the number of standard deviations that the single value differs from the mean value. This t value is then compared to the critical t value obtained from a t-table, given a desired statistical confidence (i.e., 90%, 95%, or 99% confidence) and the number of degrees of freedom (typically iV-1), to assess whether the value is statistically different from the other values in the series. In chemometrics, the t test can be useful for evaluating outliers in data sets. 

The critical t-value is taken from the t-table for the desired confidence level... 

LPRINT " CRITICAL T-VALUE AT 95 X CONF. LEVEL. . " T 4082 LPRINT iLPRINT V ... 

If tbias is greater than the critical t value at a given confidence level i.e. 95%) and f - 1) degrees of freedom, then there is evidence that the bias is significant. Another way to check for bias has been described in Section 4.5.5... 

It is worth noting that the critical t value for an infinite number of test objects at the 95% confidence limit is 1.96 and here, with a sample size of n = 3, the value is 4.3, so clearly the larger the number of test objects, the smaller the t critical value becomes. For example, for a sample size of n = 6 (and therefore 5 degrees of freedom), the t critical value is 2.57. This is useful, as n = 6 is a very common number of test objects to run in an analytical test, and so remembering the critical value saves one from hunting statistical tables. If the calculated value for a data set is less than 2.57, the null hypothesis is retained, and if it is greater than 2.57 the null hypothesis is rejected. 

Consequently, evaluation and selection of polymers is possible by determining the asrv vs. T relation from the rj = f(y. T,M) relationship (see Chap. 15). Then the working constraints, i.e. the temperature range, can be adapted to the critical T-values (see Fig. 24.6). 

Results The uncertainties associated with the slopes are very different and rii = 2, so that the pooled variance is roughly estimated as (V + V2)/2, see case c in Table 1.10 this gives a pooled standard deviation of 0.020 a simple f-test is performed to determine whether the slopes can be distinguished. (0.831 - 0.673)/0.020 = 7.9 is definitely larger than the critical t-value for p - 0.05 and/= 3 (3.182). Only a test for Hp. t > tc makes sense, so a one-sided test must be used to estimate the probability of error, most likely of the order p - 0.001 or smaller. 

The r-statistic is (207.6 - 199.6)/3.78 = 2.12. Entering Table 5 with 60 degrees of freedom, we obtain a critical t-value for a one-sided test at the 0.025 level of significance equal to 2.00. As the alternative hypothesis is that the two QC specimens differ, without regard to which has the higher and which has the lower value, the test is a two-sided test. The critical table value at 0.025, one-sided, is the critical value to use for a two-sided test at 0.05. The calculated statistic exceeds this critical value, and we reject the hypothesis that the two sets of QC samples were prepared identically. If we have the stock solutions used to prepare the two QC specimens, we would probably analyze them to see whether they have identical concentrations. 

The variable that is most important to the relative filtration time is the nitric acid concentration. This is followed by the rate of hydrogen peroxide addition. The remaining four variables are of less importance, and, in fact, are not statistically significant. The critical t value for significance at the 35% confidence level is 2.013 and these four variables have t values lower than this. 

Table A t Distribution gives the critical t values. These values are used to compare estimated...

Table A has two entries The column heads specify the value of

Od V "1 + 2 The critical t-value is (according to Table 2.6) t(l - a/2 i-b 2 2) = t(l-0.05/2 12) = 2.18. This means that the calculated t-value is greater than the critical value, that is, the two-sided test is significant. The differences in the titanium determination in the two laboratories cannot be explained by random errors. 

Gemperline et al. discussed an improved methodology for wavelength distance measurements in an article comparing Mahalanobis distance and SIMCA [15]. Gemperline et al. have normalized the Cl value to account for sample size and to allow for qualitative decisions to be made according to a probability threshold by comparison to critical t values, rather than a Qx threshold. 

If the variances are not statistieally equal, there is not an exaet solution to the comparison of the two means (this is the so-called Fisher-Behrens problem), although the statistic derived from eqn (A2.4) is distributed approximately as a Student s t. The degrees of freedom must be calculated from the variances and the number of experimental points (using the Smith Satterthwaite s equation ) or, alternatively and more conveniently, applying eqn (A2.5) to estimate a critical t-value. ... 

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