The t- Statistic

The use of the standard deviation to determine an uncertainty limit is based on the assumption that we have a reasonably large sample, i.e. greater than 25. If that is not the case we run into the problem that the average we have determined is unlikely to be close enough to the true average, which is our required value. This is overcome by calculating the quantity 

In this expression, as before, 5 is the calculated standard deviation and n is the sample size. The quantity t is known as the r-value, and can be determined from statistical tables. Its precise value depends on the level of accuracy required together with a quantity known as the number of degrees of freedom, which is equal to - 1. Relevant values of t are given in Appendix 5. 

If we perform this calculation for the first data set given in Chapter 9, we have a confidence interval of 

Applying this method to the data used in the previous chapter, which also has t = 2.78, gives a confidence interval of 

The series of dyes represented by the formula in Fig. 10.1 exhibit different colours depending on the nature of the substituent X. These can be quantified in terms of the wavelength at which most light is absorbed, as shown in the following table. 

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In reference to the tensile-strength table, consider the summary statistics X and. s 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 oft values summarized in Table 3-5. 

In order to compare populations based on their respective samples, it is necessaiy to have some basis of comparison. This basis is predicated on the distribution of the t statistic. In effecd, the t statistic characterizes the way in which two sample means from two separate populations will tend to vaiy by chance alone when the population means and variances are equal. Consider the following ... 

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... 

Frequently, limit values are given as a numerical value without any exceeding level. Then, according to the t-statistics, a significant exceeding has to be stated if x — Ax > xCl> as illustrated in Fig. 8.3a. On the other hand,... 

The t-statistic follows what is known as the Student s t-distribution, after the statistician William Sealy Gosset (1876-1937) who published under the pseudonym StudenT. The shape of the t-distribution is similar to that of the normal distribution, but forms a family of curves distinguished by a parameter known as the degrees of freedom. The 5% critical point in the t-distribution always exceeds the normal value of 1.96, but is nevertheless close to 2.0 for all but quite small values of degrees of freedom. 

A particular use of the t-statistic is calculating confidence intervals (Cl). When we calculate the mean of a sample we do not expect that it will be exactly equal to the mean of the population from which the sample was drawn. Nonetheless, we can expect that it will be reasonably close to the population mean. A confidence interval provides an estimate as to how close. The 95% confidence interval is a random interval such that, in 95% of hypothetical replications of the sampling process, the confidence intervals obtained will include the true value of p The confidence interval for p is of the form x multiples of s.e.m. The multiple used is tl-a/2 (n-1), which is the 100(l-o/2) percentage point of the t-distribution with n-1 degrees of freedom. Thus, the 95% Cl (o=0.05) is given by ... 

The t statistic was used to eliminate the unknown parameter a2. Thus, Equation... 

To develop the T statistics and be able to test the above hypothesis, it is required to compute different sum of squares and cross products for %LC and also for sampling times [15,16,18]. The aggregated sum of squares of the sampling times is defined as... 

When the t statistic version of the equations (11) and (12) is used, the arithmetic values for the 95% confidence limits are smaller. The corresponding... 

There are two different ways of carrying out this test. The first one involves taking a single sample and analysing it by both methods a number of times. The usual procedure is to undertake a number of analyses (preferably not less than 6) for the chosen sample with both methods and calculate the value of the t-statistic. This is then compared with the tabular value for the appropriate degrees of freedom at the selected confidence level. If the calculated value is less than the tabulated t value then the mean values, and hence the methods, are accounted equivalent. This method has the advantage that the number of replicates undertaken for each method does not have to be equal. However, it is not always recognised that for this test to be valid the precision of the two methods should be equal. The method used to compare the precisions of methods is the F-ratio test and is carried out as part of the procedure. 

Results of the computations are shown below. The Hausman statistic is 25.1 and the t statistic for the Wu test is -5.3. Both are larger than the table critical values by far, so the hypothesis that least squares is consistent is rejected in both cases. 

The log-likelihood function at the maximum likelihood estimates is -28.993171. For the model with only a constant term, the value is -31.19884. The t statistic for testing the hypothesis that (3 equals zero is 5.16577/2.51307 = 2.056. This is a bit larger than the critical value of 1.96, though our use of the asymptotic distribution for a sample of 10 observations might be a bit optimistic. The chi squared value for the likelihood ratio test is 4.411, which is larger than the 95% critical value of 3.84, so the hypothesis that 3 equals zero is rejected on the basis of these two tests. 

The final statistical values that are reported for Equation 7.3 are the standard errors of the regression coefficients. These allow us to assess the significance of the individual terms by computing a statistic, called the t statistic, by dividing the regression coefficient by its standard error ... 

Using the definitions of the normal size distributions, the t-statistic can be formulated as follows [20] ... 

When two samples are veiy similar, t approaches zero when they are different, t approaches infinity. The value of f is used to calculate the P value using Student s f-test tables, given in the appendix of this book. The P value is tte probability that the two distribution means are the same that is, Aj = Ag. When the P value is greater than a critical accepted value (typically 5% [21] or the experimental error due to both sampling and size determination if it is lai ger) then the null hypothesis (Ho Aj = A2) is accepted (i.e., the two populations are considered to be the same). Ceramic powder size distributions are often represented by log-normal distributions and not by normal distributions. For this reason the t statistic must be augmented for use with lognormal distributions. Equation (2.59) can be modified for this purpose to... 

The regression coefficients (bi) and their standard errors, the confidence interval for Pi, and values of the t-statistic to test the null hypothesis Hq = Pi = 0 and their associated probabilities... 

At this point we need to realise that we have too few readings for the standard deviation to be a satisfactory indication of the uncertainty, and that we need to use the t-statistic to generate this. The appropriate value of t for a 95% confidence level with 10 — 1 or 9 degrees of freedom is 2.26. The confidence limit is therefore... 

It is common practice to compare lowest limits of detection or response to various analytes. A number of schemes have been introduced in order to provide such a threshold figure of merit, but pierhaps the most reliable is that of Burrell which employs an extension of the sensitivity concept. If measurement of is restricted to a series of n replicate analyses at a very low concentration, that is, where the precision becomes low, then a practical confidence interval can be imposed which will permit objective evaluation of a conservative detection limit, based on the t statistic appropriate to the determined for n observations. Skogerboe and Grant have demonstrated the application of d in the form, where (1 — a) is the confidence interval required. 

If you divide the coefficient by its standard deviation, the result is the t-statistic for that coefficient. The f-statistic is used to test hypotheses about the value of the coefficient. In a multiple regression model, the value of fln/< n shows the relative importance of each term in the model. 

Given that seven samples were taken and a 95% confidence interval is of interest, the approximation for the t statistic presented in Eq. [1-28] can be used ... 

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