Table of Critical Values

Table of critical values for the triangle test for differences... 

The discussions of DW are on p. 69 and 181-192 of Draper and Smith (third edition -the second edition contains a similar but somewhat less extensive discussion). They also include an algorithm and tables of critical values for deciding whether the correlation is statistically significant or not. You might also want to check out page 64 for the proof that the linear correlation between residuals and predicted values from the calibration is zero. 

Having established that the standard deviations of two sets of data agree at a reasonable confidence level it is possible to proceed to a comparison of the mean results derived from the two sets, using the t-test in one of its forms. As in the previous case, the factor is calculated from the experimental set of results and compared with the table of critical values (Table 2.3). If /jX ) exceeds the critical value for the appropriate number of degrees of freedom, the difference between the means is said to be significant. When there is an accepted value for the result based on extensive previous analysis t is computed from equation (2.9)... 

Note The ASTM (American Society for Testing Material) uses a different test for rejection of an outlier, called the reduced central value z,- = (x, — x)/s, which has its own table of critical values. 

Table 21.4—Abridged table of critical values of Q (Dixon s test).

Each of these confidence intervals (the calculated interval and the critical interval) can be expressed in terms of b0, sh, and some value of t (see Equation 6.5). Because the same values of b0 and sh(i are used for the construction of these intervals, the information about the relative widths of the intervals is contained in the two values of /. One of these, /crit, is simply obtained from the table of critical value of / - it is the value used to obtain the critical confidence interval shown in Figure 6.7 or 6.8. The other, /calc, is calculated from the minimum confidence interval about b0 that will include the value zero and is obtained from a rearrangement of Equation 6.5. 

This approach yields additional information over and above that gained from tables of critical values, in that two test statistics that are both significant at the p<0.05 level may have associated probabilities that are quite different, e.g., p=0.044 and p=0.012. While both p- values are less than 0.05, the former is not that much less, while the latter is considerably less. This is why modem journal publishing practices prefer authors to provide the actual p-value associated with a test statistic. 

The one-sample t test will be used to test the null hypothesis. As there are 10 observations and assuming the change scores (the random variable of interest) are normally distributed, the test statistic will follow a t distribution with 9 df. A table of critical values for the t distribution (Appendix 2) will inform us that the two-sided critical region is defined as t < -2.26 and t > 2.26 - that is, under the null hypothesis, the probability of observing a t value < -2.26 is 0.025 and the probability of observing a t value > 2.26 is 0.025. 

Note the value of 2.26 was obtained from tables of critical values of Student s t statistics at the 95% confidence interval for n — l. k This infers that in taking 10 samples, an error of 5.14 ppm was tolerated, and that the concentration of lead in the sample should be expressed as 93 5.14 ppm. 

Note the value of 1.96 was obtained from tables of critical values of Student s t statistics at the 95% confidence interval for n = oo. f... 

Q calculated in this manner is compared with a table of critical values of g as a function of the number of data (Table 22.4). If Gcaicuiated greater than Gcriticai the value in question can be rejected. 

G can be compared to tables of critical values for G at a = 0.05, critical, calculated using equation 3.2 below. If G > Gcvmca then the suspect point is rejected. Note that in the case of Grubbs s test, we compare with tabulated critical values simply because the calculation of the probability associated with the value of G is nontrivial. We do have a formula for the calculation of Gcriticai-... 

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