Two tailed significance tests

Since there is no reason to believe that X must be either larger or smaller than p, the use of a two-tailed significance test is appropriate. The null and alternative hypotheses are... 

The variance for the sample of ten tablets is 4.3. A two-tailed significance test is used since the measurement process is considered out of statistical control if the sample s variance is either too good or too poor. The null hypothesis and alternative hypotheses are... 

What is a one-tailed significance test, and what is a two-tailed significance test ... 

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. 

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. 

The sorbent materials that performed best in the capacity and desorption efficiency tests were investigated further with respect to the stability of the sorbed analyte. Preliminary tests of analyte stability were conducted by a procedure similar to that in the desorption efficiency tests the procedure differed in that samples were stored 7 d prior to analysis rather than Id. To be acceptable, a sorbent material had to exhibit no statistically significant loss of analyte at the 0.05 significance level by a two-tailed t test. 

The results as summarized in Table V indicated that the stability was satisfactory for all three methods. To be acceptable, the average recovery of each analyte had to be at least 80%, and the difference between the average recovery on the first day and the average recovery after storage (for at least 7 d) had to be statistically insignificant at the 0.05 significance level by a two-tailed t test (9). 

Once the set of trial probabilities has been calculated, the probabilities that are less than or equal to the probability of the measured contingency table are summed. They can be summed in two ways. The first and easiest is to find the sum of all the Ps in the set. This sum gives the probabilities at both extremes, those that are more extreme in the direction of the measured table, and those that are more extreme in the other direction. This will give the two-tail / -value and the test is termed a two-tailed test. The two-tail consideration describes the probability that a measured contingency table as far away from the expected contingency table as was the measured consistency table would occur. If this probability is less than or equal to the two-tail significance level, a, chosen for the study, then the null hypothesis of no effect is rejected otherwise, the null hypothesis is accepted. 

The experiments described here are principally diagnostic in nature where cellular biomass was significantly enhanced in bottles after resource (iron or light) amendment, relative to control (or other) treatments, we infer that algal growth rates in the control (or other) treatments were limited by a deficiency in that resource. The statistical significance of differences between mean values of parameters measured in different treatments were assessed using a two-tailed r-test for comparisons between two treatments, or a one-way analysis of variance (ANOVA) for comparisons between three or more treatments, at a confidence level of 95% (P = 0.05). 

Some texts also present tables for a two-tailed F-test, but because we do not employ this in this book, we omit it. However, a two-tailed F statistic at 10% significance is the same as a one-tailed F statistic at 5 % significance, and so on. 

Based on the above two statistical significance test methods, it was found that there is no significant difference between ICP XRF methods for the analysis of Al, Ni, V, Ti Fe. However, it is also important in such cases to compare the standard deviations, i.e., the random errors of two sets of data. This comparison can take two forms, viz., whether ICP method is more precise than XRF method (one-tailed F-test) or ICP XRF differ in their preeision (two-tailed F-test). 

Data sets were analyzed using a two-population independent two-tailed t-test or an analysis of variance (anova). Figures show means s.e.m. In aU cases, P-values less than 0.05 were considered as significant. 

Statistical significance of differences in cell growth is tested using a one-sample Student s f-test (each condition compared to control) and a two-tailed f-test assuming unequal variances (different washing solutions compared to each other). 

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. 

Some values of ftei. at the 95% probability level are given in Table 2. The columns in the table correspond to the numbers of degrees of freedom for the numerator set of data, while the rows correspond to the number of degrees of freedom for the denominator set. Two versions of the table are available, depending on the exact purpose of the comparison to be made a one-tailed F-test will show whether the precision of one set of data is significantly better than the other, while a two-tailed F-test will show whether the two precisions are significantly different. 

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 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. 

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. 

Had the calculated value for t been less than 1.80 then there would have been no significance in the results and no apparent bias in the laboratory procedure, as the tables would have indicated a probability of greater than 1 in 10 of obtaining that value. It should be pointed out that these values refer to what is known as a double-sided, or two-tailed, distribution because it concerns probabilities of values both less and greater than the mean. In some calculations an analyst may only be interested in one of these two cases, and under these conditions the -test becomes single-tailed so that the probability from the tables is halved. 

---