Table of t-Values

We get the critical value from the table of t-values. We have a two-tailed question, we want to have 95% confidence level, and with 10 measurements we have 9 degrees of freedom. Therefore we get from the table a critical value of 2.262. 

The f-test is used to test a null hypothesis. The computed f-statistic (calculated using one of the relationships given below) is compared with the value foimd in a table of t values corresponding to the appropriate number of degrees of freedom and at the desired confidence level. If the computed f-statistic exceeds the value from the table, the null hypothesis is rejected. 

If you consult a table of t values in a statistics text (a portion of such a table is shown in Figure 11-15), you will find, for 95% probability and 3 degrees of freedom, that t = 3.18. Since the calculated f-statistic is less than f-critical from the table, the intercept value of 0.0011 does not differ significantly from zero. 

Table 12.15 shows the mean of four estimates of the affinity of an antagonist measured with radioligand binding and also in a functional assay. Equation 12.36 yields a value for t of 2.29. For np + n2 - 2 degrees of freedom, this value of t is lower than the t for confidence at the 95% level (2.447 see Appendix table of t values). This indicates that the estimate of antagonist potency by these two different assay methods does not differ at the 95% confidence level. It should be noted that the preceding calculation for pooled standard deviation assumes that the standard deviation for both populations is equal. If this is not the case, then the degrees of freedom are calculated by... 

This value is then compared to a table of t values. If the calculated t is greater than the tabled value, the treatment effect is considered to be statistically significant the hypothesis is accepted with a p = 0.05 chance of being wrong. 

An excerpt from a table of t values (Steele and Torrie, 1960) is given here ... 

Tables of t values can be consulted to give the probability that there is a difference between the two data sets. In this example, if the t value is greater than 5.96 we can say that there is a greater than 99.9% probability that there is a difference between the two data sets. Because the t value is 10.1 we can say that there is greater than 99.9 % probability that the data for the blue airplanes is different from the white airplane data. Another way to express this is to say that there is a significant difference.

In this equation, r) the absolute hardness, is one-half the difference between /, the ionization potential, and A, the electron affinity. The softness, a, is the reciprocal of T]. Values of t) for some molecules and ions are given in Table 8.4. Note that the proton, which is involved in all Brdnsted acid-base reactions, is the hardest acid listed, with t — c (it has no ionization potential). The above equation cannot be applied to anions, because electron affinities cannot be measured for them. Instead, the assumption is made that t) for an anion X is the same as that for the radical Other methods are also needed to apply the treatment to polyatomic... 

Thus, the distance /2a may be regarded as a measure of the width of the distribution A k) and is called the half width. The half width may be defined using 1/2 or some other fraction instead of 1/e. The reason for using 1/e is that the value of k at that point is easily obtained without consulting a table of numerical values. These various possible definitions give different numerical values for the half width, but all these values are of the same order of magnitude. Since the value of I (x, r) falls from its maximum value of (2jr) to 1/e of that value when x — v t equals v/lja, the distance flja may be considered the half width of the wave packet. 

In reference to the tensile-strength table, consider the summary statistics x and x by days. For each day, the t statistic could be computed. If this were repeated over an extensive simulation and the resultant t quantities plotted in a frequency distribution, they would match the corresponding distribution of t values summarized in Table 3-6. 

As introduced in sections 3.1.3 and 4.2.3, the Arrhenius equation is the normal means of representing the effect of T on rate of reaction, through the dependence of the rate constant k on T. This equation contains two parameters, A and EA, which are usually stipulated to be independent of T. Values of A and EA can be established from a minimum of two measurements of A at two temperatures. However, more than two results are required to establish the validity of the equation, and the values of A and EA are then obtained by parameter estimation from several results. The linear form of equation 3.1-7 may be used for this purpose, either graphically or (better) by linear regression. Alternatively, the exponential form of equation 3.1-8 may be used in conjunction with nonlinear regression (Section 3.5). Seme values are given in Table 4.2. 

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)... 

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). 

Finally, 15 sets of T values for both aromatic and aliphatic systems have been correlated with Equa tlons 30 - 32 with generally good results The data sets studied are set forth in Table IV, the statistics obtained are reported in Table V. These results supply further support for the imf model of transport para meters It seems highly likely then that Equations 30 -32 are generally applicable and that T values are in-dead aomPQfllta parameters. 

The first eolunm shows the degrees of freedom, whieh is obtained from the number of measurements. Tables with t-values might be found in all statistics textbooks and relevant standards. 

This problem has Shelly traveling one less than twice as far as Shirley. You could make a table of possible values for the distances they traveled. You d probably be interested only in whole-number values, which won t solve the problem if the answer is a fraction, but you may get fairly close to the answer. Table 4-1 has some possible numbers or distances, starting with Shirley going 1 mile and ending to keep the total distance from getting larger than 17. 

From the t table, the t value of 3.182 is found in the row for sample size of 4 and in the column for 95% confidence level. The precision of a single value is therefore... 

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. 

In order to compare the results of an analysis technique with the true solution it is essential that the MWD be known before the analysis is started. Consequently, computer simulated data generated by assuming a molecular weight distribution and subsequently using Equation 3 to generate g(x) for a series of t values, was used for analysis. Five different MWDs were used, four of which were generated using GEX functions as models (two GEX functions for bimodal peaks), and Table I shows the GEX parameters used for each of these MWDs. 

An extensive list of T-values for aromatic substituents appears in Table 1.4. Pi values for side chains of amino acids in peptides have been well characterized and are easily available (130-132). Aliphatic fragments values were developed a few years later. For a more extensive list of substituent value constants, refer to the extensive compilation by Hansch et al. (133). Initially, the T-system was applied only to substitution on aromatic rings and when the hydrogen being replaced was of innocuous character. It was apparent from the... 

We would then look np in the table of t for 9 degrees of freedom its value for the 5% level (95% chance of being right is equivalent to a 5% chance of being wrong). Here t is 2.26. Then the limits dzL on either side of the sample mean, where a = the standard deviation of the population and n is the number of individuals in the sample, are... 

Having found the value of t, we enter the table of t with degrees of freedom (n, + —2). If our value of t exceeds that for the 5% level we may take it... 

Looking up this value of t in a more complete table of t than the one in the Appendix, we find it corresponds to a level of significance of approximately 25%. That is to say, in the absence of any difference between the two foundries, if we repeatedly sampled from them we would get as large an apparent difference between them as we did here 1 out of every 4 times. The present result, therefore, is only a very slight indication of the possible superiority of Foundry B. 

We also have t = its value in the table of t corresponding to (n + nj —2) — (5 + 2 — 2) = 5 degrees of freedom and to the significance level chosen. If we select the 5% level of significance (95% chance of being correct in the prediction that the true value of the difference in the means of the two samples lies between the limits stated) then for 5 degrees of freedom, t = 2.57. 

Generally, values for and a should be calculated for at least two extreme choices of T. Values so obtained must agree if the data obey Eq. (2.5). Equation (2.11) has been used in computing the values of a. and Tg listed in Table 2. 

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