Analytical Chem istry - DePauw University

100 Analytical Chemistry 2.0The four steps for a statistical analysis ofdata:1. Pose a question, and state the null hypothesisand the alternative hypothesis.3. Choose a confidence level for the statisticalanalysis.3. Calculate an appropriate test statisticand compare it to a critical value.4. Either retain the null hypothesis, orreject it and accept the alternative hypothesis.In this textbook we use a to represent theprobability of incorrectly rejecting thenull hypothesis. In other textbooks thisprobability is given as p (often read as “pvalue”).Although the symbols differ, themeaning is the same.the null hypothesis, which we either retain or reject. If we reject the nullhypothesis, then we must accept the alternative hypothesis, concludingthat the difference is significant and that it cannot be explained by randomerror.Failing to reject a null hypothesis is not the same as accepting it. Weretain a null hypothesis because there is insufficient evidence to prove itincorrect. It is impossible to prove that a null hypothesis is true. This is animportant point that is easy to forget. To appreciate this point let’s returnto our sample of 100 pennies in Table 4.13. After looking at the data youmight propose the following null and alternative hypotheses.H 0 : The mass of any U.S. penny in circulation is in the range of 2.900g–3.200 g.H A : A U.S. penny in circulation may have a mass less than 2.900 g or amass of more than 3.200 g.To test the null hypothesis you reach into your pocket, retrieve a penny, anddetermine its mass. If the penny’s mass is 2.512 g then you reject the nullhypothesis, and accept the alternative hypothesis. Suppose that the penny’smass is 3.162 g. Although this result increases your confidence in the nullhypothesis, it does not prove it is correct because the next penny you pullfrom your pocket might weigh less than 2.900 g or more than 3.200 g.After stating the null and alternative hypotheses, the second step is tochoose a confidence level for the analysis. The confidence level defines theprobability that we will reject the null hypothesis when it is, in fact, true.We can express this as our confidence in correctly rejecting the null hypothesis(e.g. 95%), or as the probability that we are incorrectly rejecting thenull hypothesis. For the latter, the confidence level is given as a, whereα= 1−confidence level(%)100For a 95% confidence level, a is 0.05.The third step is to calculate an appropriate test statistic and to compareit to a critical value. The test statistic’s critical value defines a breakpointbetween values that lead us to reject or to retain the null hypothesis. Howwe calculate the test statistic depends on what we are comparing, a topicwe cover in section 4F. The last step is to either retain the null hypothesis,or to reject it and accept the alternative hypothesis.4E.3 One-Tailed and Two-Tailed Significance TestsSuppose we want to evaluate the accuracy of a new analytical method. Wemight use the method to analyze a Standard Reference Material containinga known concentration of analyte, m. We analyze the standard severaltimes, obtaining an mean value, X , for the analyte’s concentration. Ournull hypothesis is that there is no difference between X and m