One-tailed tests

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. 

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. 

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.)... 

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

Two-tailed tests require larger sample sizes than one-tailed tests. Assessing two directions at the same time requires a greater investment. 

For example, if a study were to investigate the potential of a new antihypertensive drug, a one-tailed test maybe used to look for a decrease but not an increase in BP. 

Tables la to Id represent the value of the ratio of the variances which would occur owing to chance if the two variances were in fact from normal distributions which have the same variance. The probabilities given in the tables (0 20 for la, 0 05 for lb, etc.) cover what is called the one-tailed test situation. In using the table with this probability level, we are interested only in determining whether the F ratio obtained is larger than that which could be attributed to chance.

For one-tailed tests on observed data from unknown sources, the critical values in the table are for (1 - a). For two-tailed tests, the critical values are for a/2 and l-(a/2) for the low and high ends of the distribution. 

For the one-tailed test, the table is to be read from the top, with the column headings representing the value of 1 - a. 

In the analysis of variance, the comparisons are made using the a values for the one-tailed situation. In comparing two observed variances, the one-tailed test is used when we are asking whether the population variance represented by Sf is larger than that represented by Stwo-tailed test when we are asking, Are they equal ... 

The 95% critical value from the standard normal distribution for this one-tailed test would be 1.645. Therefore, we would reject the hypothesis of no autocorrelation. 

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 second type of probability sum calculated from the set of Ps contains only those trial tables that are more extreme in the same direction as the measured contingency table. As with the chi-square test, if this probability is less than or equal to the significance level, a, chosen for the study, then the null hypothesis of no effect is rejected otherwise, the null hypothesis is accepted. This P is referred to as a one-tail or one-sided P-value and its associated test, a one-tailed test. The difficulty in this type of test is to correctly identify the trial tables from the set that are more extreme in the same direction as the measured contingency table. 

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. 

If the null hypothesis is retained, i.e., there is no statistical significant difference between the two variances, then the calculated F value will approach 1. Some critical values of F can be found in Table 2.2a and Table 2.2b. The test can be used in two ways to test for a significant difference in the variances of the two samples or to test whether the variance is significantly higher or lower for either of the two data sets, hence two tables are shown, one for the one-tailed test and one for the two-tailed test. 

From tables, we obtain the critical value of t= 1.83 for a one-tailed test (i.e., the result from our method is significantly higher than the reference sample value at the 95% confidence limit). Comparing the calculated value of t with the critical value of t, we observe that the null hypothesis is rejected and there is a significant difference between the experimentally determined mean compared with the reference result. Clearly, the high precision of the method (0.6) compared with the deviation between mean result and the accepted or true value (85-83), contributes to the rejection of the null hypothesis. 

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