One-tailed

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 ... 

Let us digress a moment and consider when a two-tailed test is needed, and what a one-tailed test implies. We assume that the measurements can be described by the curve shown in Fig. 2.10. If so, then 95% of the time a sample from the specified population will fall within the indicated range and 5% of the time it will fall outside 2.5% of the time it is outside on the high side of the range, and 2.5% of the time it is below the low side of the range. Our assumption implies that if p does not equal the hypothesized value, the probability of its being above the hypothesized value is equal to the probability of its being below the hypothesized value. 

A one-tailed test is required since the alternative hypothesis states that the population parameter is equal to or less than the hypothesized value. 

As applied in Example 12, the F test was one-tailed. The F test may also be applied as a two-tailed test in which the alternative to the null hypothesis is erj A cr. This doubles the probability that the null hypothesis is invalid and has the effect of changing the confidence level, in the above example, from 95% to 90%. 

Examples of (a) two-tailed, (b) and (c) one-tailed, significance tests. The shaded areas in each curve represent the values for which the null hypothesis is rejected. 

Since significance tests are based on probabilities, their interpretation is naturally subject to error. As we have already seen, significance tests are carried out at a significance level, a, that defines the probability of rejecting a null hypothesis that is true. For example, when a significance test is conducted at a = 0.05, there is a 5% probability that the null hypothesis will be incorrectly rejected. This is known as a type 1 error, and its risk is always equivalent to a. Type 1 errors in two-tailed and one-tailed significance tests are represented by the shaded areas under the probability distribution curves in Figure 4.10. 

The value of fexp is compared with a critical value, f(a, v), as determined by the chosen significance level, a, the degrees of freedom for the sample, V, and whether the significance test is one-tailed or two-tailed. 

The value of fexp is then compared with a critical value, f(a, v), which is determined by the chosen significance level, a, the degrees of freedom for the sample, V, and whether the significance test is one-tailed or two-tailed. For paired data, the degrees of freedom is - 1. If fexp is greater than f(a, v), then the null hypothesis is rejected and the alternative hypothesis is accepted. If fexp is less than or equal to f(a, v), then the null hypothesis is retained, and a significant difference has not been demonstrated at the stated significance level. This is known as the paired f-test. 

If systematic errors due to the analysts are significantly larger than random errors, then St should be larger than sd. This can be tested statistically using a one-tailed F-test... 

This value of fexp is compared with the critical value for f(a, v), where the significance level is the same as that used in the ANOVA calculation, and the degrees of freedom is the same as that for the within-sample variance. Because we are interested in whether the larger of the two means is significantly greater than the other mean, the value of f(a, v) is that for a one-tail significance test. 

Individual comparisons using Fisher s least significant difference test are based on the following null hypothesis and one-tailed alternative hypothesis... 

Using equation 14.25, we can calculate values of fexp for each possible comparison. These values can then be compared with the one-tailed critical value of 1.73 for f(0.05, 18), as found in Appendix IB. For example, fexp when comparing the results for analysts A and B is... 

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. 

V is located along the vertical margin and the probability is given on the horizontal margin. (For a one-tailed test, given the probability for the left tail, the i value must be preceded by a negative sign.)... 

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. 

Fig. 5a,b. Tolane compounda a melts at 39 °C and is not mesogenic . b differs only in one tail substituent melts to a nematic phase at 61 °C which persists until 89 °C... 

Surfactants employed for w/o-ME formation, listed in Table 1, are more lipophilic than those employed in aqueous systems, e.g., for micelles or oil-in-water emulsions, having a hydrophilic-lipophilic balance (HLB) value of around 8-11 [4-40]. The most commonly employed surfactant for w/o-ME formation is Aerosol-OT, or AOT [sodium bis(2-ethylhexyl) sulfosuccinate], containing an anionic sulfonate headgroup and two hydrocarbon tails. Common cationic surfactants, such as cetyl trimethyl ammonium bromide (CTAB) and trioctylmethyl ammonium bromide (TOMAC), have also fulfilled this purpose however, cosurfactants (e.g., fatty alcohols, such as 1-butanol or 1-octanol) must be added for a monophasic w/o-ME (Winsor IV) system to occur. Nonionic and mixed ionic-nonionic surfactant systems have received a great deal of attention recently because they are more biocompatible and they promote less inactivation of biomolecules compared to ionic surfactants. Surfactants with two or more hydrophobic tail groups of different lengths frequently form w/o-MEs more readily than one-tailed surfactants without the requirement of cosurfactant, perhaps because of their wedge-shaped molecular structure [17,41]. 

CHIDIST is the Excel function for the one-tailed probability of the chi-squared distribution. 

Bacteriophage T7 Bacteriophage T7 and its close relative T3 are relatively small DNA viruses that infect Escherichia coli. (Some strains of Shigella and Pasteurella are also hosts for phage T7.) The virus particle has an icosahedral head and a very small tail. The virus particle is fairly complex, with S different proteins in the head and 3-6 different proteins in the tail. One tail protein, the tail fiber protein, is the means by which the virus particle attaches to the bacterial cell surface. Only female cells of Escherichia coli can be infected with T7 male cells can be infected but the multiplication process is terminated during the latent period. 

FIG. 3-65 Acceptance region for two-tailed test. For a one-tailed test, area = a on one side only. 

NOTE If Z(N) is greater than the absolute value of the z-statistic (Normal Curve one-tailed) we reject the null hypothesis and state that there is no significant difference in rl and r2 at the selected significance level. 

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. 

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