Tail area

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. 

Near uranium tailings area, dissolved vs. total 0.12 FW vs. 0.56 FW 10 ... 

Moffett D, Tellier M. 1978. Radiological investigations of an abandoned uranium tailings area. J Environ Qual 7 310-314. 

McGregor, R.G., Blowes, D.W., Jambor, J.L. and Robertson, W.D. (1998a) The solid-phase controls on the mobility of heavy metals at the Copper Cliff tailings area, Sudbury, Ontario, Canada. Journal of Contaminant Hydrology, 33(3-4), 247-71. 

Bigu J, Grenier NK, Dave TP, et al. 1984. Study of radon gas concentration surface radon flux and other radiation variables from uranium mine tailings areas. Uranium 1 257-277. 

Puro M., KipMe W. B., Knapp R. A., MacDonald T. J., and Stuparyk R. A. (1995) Inco s Copper Cliff tailings area. In Proc. Sudbury 95, Canmur, Natural Resources, Ottawa, Ontario, Canada, Mining and the Environment, vol. 1, pp. 181-191. 

Point your Web browser to http //chemistry.brookscole.com/skoogfac/. From the Chapter Resources menu, choose Web Works, and locate the Chapter 7 section. Click on the link to the statistics on-line textbook. Click on the ANOVA/MANOVA button. Read about the partitioning of the sum of squares in ANOVA procedures. Click on the F-distribution link in this section. Look at the tail areas for an F-dis-tribution with both degrees of freedom equal to 10. Determine the value of F for a significance level of 0.10 with both degrees of freedom equal to 10. 

The p values can be estimated from a table of values from the appropriate t distribution (for example, by finding the tail areas associated with a particular value of the test statistic). More commonly, however, statistical software is used for all statistical analyses and p values are included in the results. The following is a helpful way to interpret p values ... 

Figure 6.7 The f distribution with 9 degrees of freedom, critical region (dashed line) and p value (solid line). Note that the critical region is represented by the tail areas to the left and right of the dashed lines the p value Is represented by the tail areas to the left and right of the solid lines...

In Chapter 6 we described the basic components of hypothesis testing and interval estimation (that is, confidence intervals). One of the basic components of interval estimation is the standard error of the estimator, which quantifies how much the sample estimate would vary from sample to sample if (totally implausibly) we were to conduct the same clinical study over and over again. The larger the sample size in the trial, the smaller the standard error. Another component of an interval estimate is the reliability factor, which acts as a multiplier for the standard error. The more confidence that we require, the larger the reliability factor (multiplier). The reliability factor is determined by the shape of the sampling distribution of the statistic of interest and is the value that defines an area under the curve of (1 - a). In the case of a two-sided interval the reliability factor defines lower and upper tail areas of size a/2. 

It is fairly common to report a p value from such an analysis. As we have seen, the p value is the probability (under the null hypothesis) of observing the result obtained or one that is more extreme. In this analytical strategy we refer to a table of Z scores and the tail areas associated with each to find the sum of the two areas (that is, probabilities) to the left of -5.62 (a result as extreme as the observed or more so) and to the right of 5.62 (the result observed and those more extreme). A Z score of this magnitude is way out in the right-hand tall of the distribution, leading to a p value < 0.0001. 

ZT i- that is, values of with 1 df that cut off the upper tail area of a. 

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