One-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. 

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

The actual number of occurrences observed is unlikely to be exactly XM, but rather will be distributed around XM as described by the Poisson distribution. The problem facing the FRA is to determine whether the number of occurrences observed for a particular railroad in a particular year has deviated upward from the mean number expected for that railroad. Statisticians make that determination by using a one-tailed significance test. Statisticians look to see how far the observed value is from the mean. It is called a one-tailed test because the FRA is only interested in railroads whose safety performance is declining, that is to say that the observed number of occurrences is greater than the mean. 

Letting A represent the results in Table 4.1 and B represent the results in Table 4.8, we find that the variances are sa = 0.00259 and sp = 0.00138. A two-tailed significance test is used since there is no reason to suspect that the results for one analysis will be more precise than that of the other. The null and alternative hypotheses are... 

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... 

The one-tailed F test is used to test whether the between-sample variance is significantly greater than the within-sample variance. Applying the F test we obtain ... 

The F-test is normally used to compare two variances or errors and ask either whether one variance is significantly greater titan the other (one-tailed) or whether it differs significantly (two-tailed). In this book we use only the one-tailed F-test, mainly to see whether one error (e.g. lack-of-fit) is significantly greater than a second one (e.g. experimental or analytical). 

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. 

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). 

Both hypothesis statements reflect the same objective, but there are significant differences in the decision criterion utilized in each hypothesis. The null hypothesis of case I assumes that the well is producing 100 or more barrels per day unless statistical evidence proves otherwise, resulting in rejection of Hq. The null hypothesis of case II assumes that the well production is inferior unless production records indicate that daily output is more than 100 barrels per day, which will result in rejection of Hq. Both tests are valid and are called one-tailed hypothesis tests under a given type I error. Consider again the original data with a = 0.05. Table 3 summarizes the calculations for the significance tests associated with cases I and II. 

To determine whether chemical cues from three males alone are sufficient to repel conspecific males, we placed three males in one arm of the Y-maze and left the other arm empty. Subject males were then allowed to choose between the two arms. Subjects significantly preferred the arm containing three males to the arm containing nothing but plain water (Table 1, III one-tailed binomial test, P = 0.003). 

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. 

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

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. 

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 )... 

Statistical Analysis. Analysis of variance (ANOVA) of toxicity data was conducted using SAS/STAT software (version 8.2 SAS Institute, Cary, NC). All toxicity data were transformed (square root, log, or rank) before ANOVA. Comparisons among multiple treatment means were made by Fisher s LSD procedure, and differences between individual treatments and controls were determined by one-tailed Dunnett s or Wilcoxon tests. Statements of statistical significance refer to a probability of type 1 error of 5% or less (p s 0.05). Median lethal concentrations (LCjq) were determined by the Trimmed Spearman-Karber method using TOXSTAT software (version 3.5 Lincoln Software Associates, Bisbee, AZ). 

In part this is because of the way we set up the hypotheses in our earlier discussion we asked is the coin fair or is the coin not fair We could have asked a different set of questions is the coin fair or are heads more likely than tails in which case we could have been justified in calculating the p-value only in the tail corresponding to heads more likely than tails . This would have given us a one-tailed or a one-sided test. Under these circumstances, had we seen 17 tails and 3 heads then this would not have led to a significant p-value, we would have discounted that outcome as a chance finding, it is not in the direction that we are looking for. 

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