Two-tailed probability

The P resulting from these computations will be the exact one- or two-tailed probability depending on which of these two approaches is being employed. This value tells us if the groups differ significantly (with a probability less than 0.05, say) and the degree of significance. 

The critical value of the t test can be abbreviated as fa(2),v, where a(2) refers to the two-tailed probability of a and v = n — 1 (degree of freedom). For the two-tailed t test, compare the calculated t value with the critical value from the t distribution table. In general, if t > ta(2),v, then reject the null hypothesis. When comparing the means of replicate determinations, it is desirable that the number of replicates be the same in each case. 

The statistical parameters generated in the process of fitting the data to the equation are also used to determine the significance of the equation. A common criterion is to retain coefficients if their two-tailed probability is less than 0.05 P(2-tail) < 0.05. A two-tailed probability smaller than 0.05 means that the deviation from the true value lies in the positive or negative regions of the normal error curve corresponding to less than 5% of the area. It... 

An unnecessary complication, which was possibly once introduced to make life easier, is the distinction between one- and two-tailed Student /-values (tails are also used in other statistics). Two-tailed probabilities are spread over the two ends of the distribution with half the given probability in each tail, and are denoted by putting a double prime (") after the probability value. One-tailed probabilities are shown as a single prime ( ) and refer to just one tail of the distribution. For example, for a 95% confidence interval and 10 degrees of freedom, 0.025, 10 is equal to o.o5",io> as can be seen from figure 2.9. Annoyingly, in Excel the z values obtained from the normal distribution are always one tailed (=—NORMSINV(p)1) but the Student /-values... 

Figure 3.6 One- and two-tailed probabilities. The extreme 5% of the distribution can be shared equally between each end (two tailed), or the entire 5% can be located at one end or the other.

The intention here is to state that the count or peak area we have measured lies, within a defined degree of confidence, between the two limits, Al —and A -l-fc. o iM. In this case, the factor for the two-tailed probability distribution should be used (see Table 5.3) and for 95 % confidence we might chose to present the result as ... 

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. 

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%. 

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. 

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. 

An algorithm for calculating the symmetrical (two-tailed) /-factors for p = 0.1 is incorporated its use corresponds to the statement that the probability that measurements on a future batch, given the linear trend already established, will inadvertently be found to be below the specification limits of Y% of nominal, at a shelf-life that would lead one to expect a residual content at or above the specification limit, is p = 0.05. ... 

An example is shown in Table 16.14. A two-factor analysis of variance for the covariate, as shown in Table 16.15, clearly indicates that the two sexes started with approximately the same means (p = 0.5598). Moreover, there were no differences between the group means in either sex as indicated by the large tail probabilities for treatment (p = 0.8823) and sexxtreatment interaction (p = 0.6532). These facts justify using sex as a factor in the analysis, as was done here. 

The probability resulting from a two-tailed test is exactly double that of a one-tailed from the same data. 

Fisher s probabilities are not necessary symmetric. Although some analysts will double the one-tailed p value to obtain the two-tailed result, this method is usually overly conservative. 

For common statistics, such as the Student s t value, chi-square, and Fisher F, Excel has functions that return the critical value at a given probability and degrees of freedom (e.g., =TINV (0.05,10) for the two-tailed Lvalue at a probability of 95% and 10 degrees of freedom), or which accept a calculated statistic and give the associated probability (e.g., =TDIST( t, 10, 2 ) for 10 degrees of freedom and two tails). Table 2.3 gives common statistics calculated in the course of laboratory quality control. 

Table 2.4. Values of the critical two-tailed Grubbs s G statistic at 95% and 99% probability for a single outlier...

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. 

Most tables of / distributions look at the critical value of the / statistics for different degrees of freedom. Table A.4 relates to the two-tailed /-test, which we employ in this text, and asks whether a parameter differs significantly from another. If there are 10 degrees of freedom and the / statistic equals 2.32, then the probability is fairly high, slightly above 95 % (5 % critical value), that it is significant. 

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