T statistics

Harvey, D. T. Statistical Evaluation of Acid/Base Indicators, /. Chem. Educ. 1991, 68, 329-331. 

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... 

In a data set it may be desirable to ask the question Is any one value significantly different from the others in the sample A t statistic (for n — 1 degrees of freedom where the sample size is n) can be calculated that takes into account the difference of the magnitude of that one value (xj and the mean of the sample (x ) ... 

Nilsson, T., Statistical analysis of individnal variation and sampling technique when determining the pigment content in beetroot, Swed. J. Agric. Res., 3, 201, 1973. 

S B standard deviation of n measurements, and tp t-statistic, for a p% confidence level. 

MacGregor, J. F.. and Kourti, T., Statistical process control of multivariate processes, Coni. Eng. Prac. 3, 404-414 (1995). 

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

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

Yamaoka K, Nakagawa T, Uno T. Statistical moments in pharmacokinetics. J Pharmacokinet Biopharm 1978 6 547-558. 

Confidence intervals nsing freqnentist and Bayesian approaches have been compared for the normal distribntion with mean p and standard deviation o (Aldenberg and Jaworska 2000). In particnlar, data on species sensitivity to a toxicant was fitted to a normal distribntion to form the species sensitivity distribution (SSD). Fraction affected (FA) and the hazardons concentration (HC), i.e., percentiles and their confidence intervals, were analyzed. Lower and npper confidence limits were developed from t statistics to form 90% 2-sided classical confidence intervals. Bayesian treatment of the uncertainty of p and a of a presupposed normal distribution followed the approach of Box and Tiao (1973, chapter 2, section 2.4). Noninformative prior distributions for the parameters p and o specify the initial state of knowledge. These were constant c and l/o, respectively. Bayes theorem transforms the prior into the posterior distribution by the multiplication of the classic likelihood fnnction of the data and the joint prior distribution of the parameters, in this case p and o (Fignre 5.4). 

The ANOVA table shown in Table 2.14 indicates that there was no significant lack-of-fit of the model. Parameter estimates and t-statistics for this model are shown in Table 2.15. 

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 ... 

CVt = 0.25/1.96 = 0.128. The number 0.128 is the largest true precision for a net error at +25% at the 95% confidence level. The number 1.96 is the appropriate t - statistic from the t distribution at the same confidence level. Since the coefficient of variation of pump error is assumed to be 5%, a method should have a CV analysis <0.102 to meet the CV accuracy standard. Tables IV and V7 shows that the infrared technique meets this requirement. 

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... 

Parameter Estimate t Statistic p Value Parameter Estimate t Statistic p Value... 

An interlaboratory bias study is a limited form of method performance study used to determine the bias of a standard method or the bias introduced by laboratories that use the standard method. Laboratories are chosen for their competence in performing the method, and the organization is the same as for a method performance study. The number of laboratories in the study is determined by the statistics required. If the bias (6) is calculated as the difference between accepted reference value and mean of n laboratories results, the significance can be tested using the standard deviation of the mean, sR/ /n. The Student s t statistic is calculated 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. 

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