T one-tailed

P(T < t) one-tail t Critical one-tail P(T < t) two-tail t Critical two-tail... 

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. 

Note. The tabulation is for one tail only, i.e. for positive values of t. For t the column headings for a must be doubled. 

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. 

Extra-pair copulations (N=15) In pair copulations (N=49) t-value P (one-tailed)... 

The table on this slide is an example for sneh t-valnes. In the first two rows are the eonfidenee levels for one-tailed (IT) and two-tailed (2T) questions. 

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... 

If we have a one-tailed question we have to use other t-values. In the table shown in this shde we have to select the level of confidence from the first row for one-tailed and from the second row for two-tailed questions. 

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 )... 

The electronic spectrum (36) of the pol5uner is dominated by a very broad ultraviolet band, with shoulders at 280 and 470 m/t, which tails into the visible region and is responsible for the deep brown color of the polymer. Very weak crystal field excitations are found at 640 and 880 m. From the latter transition one can estimate that for high-spin Fe +, Dq = 1100 cm i. This value is typical of Fe3+ in octahedral coordination with oxygen ligands, but the X-ray evidence (see below) indicates that the coordination is tetrahedral, so that Dq seems anomalously high. However, the coordination symmetry is actually lower than tetrahedral, since both hydroxide and oxide ligands are involved. 

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 normal distribution is commonly encountered in the cumulative form, that is, as the fraction of particles larger (oversized) or smaller (undersized) than a particular tt value. Since the total area under the normal curve equals unity, the area under one tail of the curve from t, to oo gives the fraction of the population having t values greater than the integration limit t . 

Tables are also available (e.g., see Beyer 1987) for the area under the normal curve between 7 (7 = 0) and one value of tt (i.e., they apply to one tail only). This area is known as the error function and is often symbolized as erf (t) ...

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