T-tests

In the next several sections, the theoretical distributions and tests of significance will be examined beginning with Student s distribution or t test. If the data contained only random (or chance) errors, the cumulative estimates x and 5- would gradually approach the limits p and cr. The distribution of results would be normally distributed with mean p and standard deviation cr. Were the true mean of the infinite population known, it would also have some symmetrical type of distribution centered around p. However, it would be expected that the dispersion or spread of this dispersion about the mean would depend on the sample size. 

The standard deviation of the distribution of means equals cr/N. Since cr is not usually known, its approximation for a finite number of measurements is overcome by the Student t test. It is a measure of error between p and x. The Student t takes into account both the possible variation of the value of x from p on the basis of the expected variance and the reliability of using 5- in... 

The t test can be applied to differences between pairs of observations. Perhaps only a single pair can be performed at one time, or possibly one wishes to compare two methods using samples of differing analytical content. It is still necessary that the two methods possess the same inherent standard deviation. An average difference d calculated, and individual deviations from d are used to evaluate the variance of the differences. 

The t test is also used to judge whether a given lot of material conforms to a particular specification. If both plus and minus departures from the known value are to be guarded against, a two-tailed test is involved. If departures in only one direction are undesirable, then the 10% level values for t are appropriate for the 5% level in one direction. Similarly, the 2% level should be used to obtain the 1% level to test the departure from the known value in one direction only these constitute a one-tailed test. More on this subject will be in the next section. 

To begin with, we must determine whether the variances for the two analyses are significantly different. This is done using an T-test as outlined in Example 4.18. Since no significant difference was found, a pooled standard deviation with 10 degrees of freedom is calculated... 

To determine if the systematic errors of the analysts are significant, an T-test is performed using sx and sd... 

A modified form of the t-test for comparing several sets of data. 

Brittle fracture is probably the most insidious type of pressure-vessel failure. Without brittle fracture, a pressure vessel could be pressurized approximately to its ultimate strength before failure. With brittle behavior some vessels have failed well below their design pressures (which are about 25 percent of the theoretical bursting pressures). In order to reduce the possibility of brittle behavior. Division 2 and Sec. Ill require impac t tests. 

In Fig. 30-25, representation of the fault detection monitoring activity, there appears to be two distinct time periods of unit operation with a transition period between the two. The mean parameter value and corresponding sample standard deviation can be calculated for each time. These means can be tested by setting the null hypothesis that the means are the same and performing the appropriate t-test. Rejecting the null hypothesis indicates that there may have been a shift in operation of the unit. Diagnosis (troubleshooting) is the next step. 

Brinellharte, /. Brinell hardness, brlnellieren, t.t, test with the Brinell machine. Brinell-probe, /. Brinell test or sample, -zahl, /. Brinell number. 

These strengths tpply only when t test is conducted with both ends fixed. When in use, the strength of these ropes may be significantly reduced if one end is free to rotate. 

Paired t-tests Changes in constitutive calcitonin receptor responses with 100 nM AC512. 

As can be seen from the analysis in Table 11.3, the paired t-test indicates that the effect of AC512 on the constitutive activity is significant at the 99% level of confidence (p<0.01 and AC512 is an inverse agonist and does decrease the constitutive receptor activity of calcitonin receptors). 

While a t-test can be used to determine if the means of two samples can be considered to come from the same population, paired data sets are more powerful to determine difference. 

Thus, the calculated value of F (1.87) is less than the tabulated value therefore the methods have comparable precisions (standard deviations) and so the t-test can be used with confidence. 

Example 7. The -test using samples of differing composition (the paired t-test). 

Fig. 3.1. Young s moduli E of the polymers A, B, C, D in the rubbery state against absolute temperature T (test frequency 0.01 Hz). Entropic elasticity is indicated by the proportionality of E to T [11]...

All results were evaluated using the Student t-test, at the 5 % level of statistical significance. 

The data to be examined are correlated with eq. (24). Successful correlation with eq. (24) is a necessary but not sufficient condition for the existence of case (a). Strong evidence for case (a) is provided by a confidence level of Ij greater than or equal to 90.0. The confidence level of p is obtained by means of a student t test of. If is not significant, then this fact implies either the existence of cases (b), (c), or (d), or the use of an incorrect steric parameter. The data are now correlated with eq. (2). If the correlations with... 

Does the found mean Xmean correspond to expectations The expected value E(x) written as /r (Greek mu), is either a theoretical value, or an experimental average underpinned by so many measurements that one is very certain of its numerical value. The question can be answered by the t-test explained in Section 1.5.2. A rough assessment is obtained by checking to see whether and Xmean are separated by more than 2 Sx or not if the difference Ax is larger, x ean is probably not a good estimate for /t. 

The most widely used test is that for detecting a deviation of a test object from a standard by comparison of the means, the so-called t-test. Note that before a f-test is decided upon, the confidence level must be declared and a decision made about whether a one- or a two-sided test is to be performed. For details, see shortly. Three levels of complexity, a, b, and c, and subcases are distinguishable. (The necessary equations are assembled in Table 1.10 and are all included in program TTEST.)... 

The same is true if another situation is considered if in a batch process a sample is taken before and after the operation under scrutiny, say, impurity elimination by recrystallization, and both samples are subjected to the same test method, the results from, say, 10 batch processes can be analyzed pairwise. If the investigated operation has a strictly additive effect on the measured parameter, this will be seen in the t-test in all other cases both the difference Axmean and the standard deviation will be affected. 

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