One-tailed t-test

The mean measured activity per unit surface area are shown for airways and bifurcations separately in Table II. These data are for those segments which contained only airway lengths bifurcations. The results are given as the number of particles which deposit per cm2 for 10 particles which enter the trachea. This assumes that the particle and activity distributions are equivalent. For the 0.2 and 0.15 ym particles the surface density at the bifurcations is greater than that along the airway lengths at p <. 01 when the paired data are compared by a one tailed t-test. 

To determine if the levels of the specific phenolics were significantly different from the unfined wine, the Dunnett s one-tailed t-test was performed using Statistical Analysis Software (SAS) (SAS Institute, Cary, NC) Values were considered significantly different if p<0.05. 

The example we jnst looked at is called one-sample t-test. It compares the mean of analytical resnlts with a stated valne. This is a typical analytical question. The problem may be two-tailed as in onr example, where it doesn t matter, if the analytical valne is biased to the one or the other direction. Or the question conld be one-tailed, e.g. if we want to know whether the copper content analysed in an alloy is below the specificatioa... 

A CRM for vitamins in honey does not exist at present and it is unlikely that one could prepare a RM due to the known instability of analytes. Therefore, to evaluate the bias, one must analyse the material by independent methods (Leon-Ruiz et al. 2011) or, more commonly, evaluate the recovery of known amounts of analytes spiked to honey. The first method is based on the fact that there is a negligible statistical probability that two independent methods may provide data affected by same bias, whereas in the spiking/recovery technique, recovery values statistically indistinguishable from 100% e.g. for a two-tailed t-test) indicate the absence of a bias. Table 13.5 shows recovery values of between 85 and 108.7%. Although a number of these recoveries different from 100% probably indicate some bias, it has been previously observed (AOAC 1998) that for low concentration levels recoveries can normally differ from 100%. AOAC guidelines (AOAC 1998) consider recovery levels between 80%... 

The Michaelis-Menten model was fitted to the experimental data using standard nonlinear regression techniques to obtain estimates of and K (Fig. 4.1). Best-fit values of and K of corresponding standard errors of the estimates plus the number of values used in the calculation of the standard error, and of the goodness-of-fit statistic are reported in Table 4.3. These results suggest that succinate is a competitive inhibitor of fumarase. This prediction is based on the observed apparent increase in Ks in the absence of changes in Vmax (see Table 4.1). At this point, however, the experimenter cannot state with any certainty whether the observed apparent increase in Ks is a tme effect of the inhibitor or merely an act of chance. A proper statistical analysis has to be carried out. For the comparison of two values, a two-tailed t-test is appropriate. When more than two values are compared, a one-way analysis of variance (ANOVA),... 

For the one-sample t-test the null hypothesis is that the mean, p, of X for the population from which the sample was selected is equal to po The test is performed by choosing a significance level, ot, and calculating using the t-distribution the critical values on both tails needed to give a critical region whose volume is equal to a. This is the two-sided test version of the one-sample t-test. The use of slightly different null hypotheses of p > po or p < Po leads to one-sided test versions of the one-sample t-test. 

All statistical computations were performed using GraphPad Prism version 4.0a for Mac OS X (GraphPad Software, San Diego, California, USA). Values of experimental groups are shown as mean SEM unless otherwise stated. One-way ANOVA with Tukey s post-test analysis was used to determine statistical significance. Where appropriate, either two-tailed t-tests or Mann Whitney U tests were performed. A probability of P < 0.05 was considered to be statistically significant. 

Both one-tailed and two-tailed t-tests can be used, depending on circumstances, but two-tailed are often preferred Table 3). The application of all three t-test equations is demonstrated by the following examples. 

We used a two-tailed test. Upon rereading the problem, we realize that this was pure FeO whose iron content was 77.60% so that p = 77.60 and the confidence interval does not include the known value. Since the FeO was a standard, a one-tailed test should have been used since only random values would be expected to exceed 77.60%. Now the Student t value of 2.13 (for —to05) should have been used, and now the confidence interval becomes 77.11 0.23. A systematic error is presumed to exist. 

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. 

The abbreviated table on the next page, which gives critical values of z for both one-tailed and two-tailed tests at various levels of significance, will be found useful for purposes of reference. Critical values of z for other levels of significance are found by the use of Table 2.26b. For a small number of samples we replace z, obtained from above or from Table 2.26b, by t from Table 2.27, and we replace cr by ... 

The t-values in this table are for a two-tailed test. For a one-tailed test, the a values for each column are half of the stated value, column for a one-tailed test is for the 95% confidence level, a = 0.05. For example, the first... 

Ot = significance level, usually set at. 10,. 05, or. 01 t = tabled t value corresponding to the significance level Ot. For a two-tailed test, each corresponding tail would have an area of Ot/2, and for a one-tailed test, one tail area would be equal to Ot. If O" is known, then z would be used rather than the t. t = (x- il )/ s/Vn) = sample value of the test statistic. 

The critical values or value of t would be defined by the tabled value of t with (n — I) df corresponding to a tail area of Ot. For a two-tailed test, each tail area would be Ot/2, and for a one-tailed test there would be an upper-tail or a lower-tail area of Ot corresponding to forms 2 and 3 respectively. 

If we don t have one stated value, but two independent sets of data (e.g. two analytical results from different laboratories or methods) we have to use the two-sample t-test, because we have to consider the dispersion of both data sets. In the same way as above we have to look carefully, what our question is it may be two-tailed (are the results significantly different ) or one-tailed (is the result from method A significantly lower than that from method B )... 

By June 11 all of the field load Insects had died or left the trees. Leaf damage to the test trees when measured on June 3 was relatively light. Leaves exhibiting noticeable damage averaged 27.6 i 2.1% (S.E.) for the control trees and 49.0 i 4.7% for test trees (p < 0.01, one-tailed paired t test). Estimated leaf area loss averaged 2.5 + 0.2% for controls and 11.3 i 2.1% for test trees (p < 0.005, one-tailed paired t test). Damage to control trees was due to unidentified insects other than tent caterpillars. 

The 95% critical value from the t distribution for a one tailed test is -1.833. Therefore, we would not reject the hypothesis at a significance level of 95%. 

The primary outcome analysis was a series of one-tailed paired t-tests to compare each subject s placebo measurement to his or her treatment measurement at... 

There is no a priori reason to doubt that the Central Limit Theorem, and consequently the normal distribution concept, applies to trace element distribution, including Sb and Ba on hands in a human population, because these concentrations are affected by such random variables as location, diet, metabolism, and so on. However, since enough data were at hand (some 120 samples per element), it was of interest to test the normal distribution experimentally by examination of the t-Distribution. The probability density plots of 0.2 and 3 ng increments for Sb and Ba, respectively, had similar appearances. The actual distribution test was carried out for Sb only because of better data due to the more convenient half life of 122Sb. After normalization, a "one tail" test was carried out. 

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