Analytical Chem istry - DePauw University

Chapter 4 Evaluating Analytical Data103alternative hypothesis, H : X ≠µ A, is that the difference between X andm is too large to be explained by indeterminate error.The test statistic is t exp , which we substitute into the confidence intervalfor m (equation 4.12).µ= X ±Rearranging this equation and solving for t exptexpsn4.14texp =µ − X n4.15sgives the value of t exp when m is at either the right edge or the left edge ofthe sample’s confidence interval (Figure 4.14a).To determine if we should retain or reject the null hypothesis, we comparethe value of t exp to a critical value, t(a,n), where a is the confidencelevel and n is the degrees of freedom for the sample. The critical valuet(a,n) defines the largest confidence interval resulting from indeterminateerrors. If t exp > t(a,n), then our sample’s confidence interval is too large tobe explained by indeterminate errors (Figure 4.14b). In this case, we rejectthe null hypothesis and accept the alternative hypothesis. If t exp ≤ t(a,n),then the confidence interval for our sample can be explained indeterminateerror, and we retain the null hypothesis (Figure 4.14c).Example 4.16 provides a typical application of this significance test,which is known as a t-test of X to m.Values for t(a,n) are in Appendix 4.Another name for the t-test is Student’st-test. Student was the pen name for WilliamGossett (1876-1927) who developedthe t-test while working as a statisticianfor the Guiness Brewery in Dublin, Ireland.He published under the name Studentbecause the brewery did not wantits competitors to know they were usingstatistics to help improve the quality oftheir products.(a) (b) (c)Xt− expsnXt+ expsnXt− expsnt s t sX + exp X − expXnnt+ expsnt sX − ( αν , )nt sX + ( αν , )nt sX − ( αν , )nt sX + ( αν , )nFigure 4.14 Relationship between confidence intervals and the result of a significance test. (a) The shaded areaunder the normal distribution curve shows the confidence interval for the sample based on t exp . Based on thesample, we expect m to fall within the shaded area. The solid bars in (b) and (c) show the confidence intervals form that can be explained by indeterminate error given the choice of a and the available degrees of freedom, n. For(b) we must reject the null hypothesis because there are portions of the sample’s confidence interval that lie outsidethe confidence interval due to indeterminate error. In the case of (c) we retain the null hypothesis because theconfidence interval due to indeterminate error completely encompasses the sample’s confidence interval.