Analytical Chem istry - DePauw University

104 Analytical Chemistry 2.0Example 4.16Before determining the amount of Na 2 CO 3 in a sample, you decide tocheck your procedure by analyzing a sample known to be 98.76% w/wNa 2 CO 3 . Five replicate determinations of the %w/w Na 2 CO 3 in the standardgave the following results.98.71% 98.59% 98.62% 98.44% 98.58%Using a = 0.05, is there any evidence that the analysis is giving inaccurateresults?So l u t i o nThe mean and standard deviation for the five trials areX = 98.59 s = 0.0973Because there is no reason to believe that the results for the standard mustbe larger (or smaller) than m, a two-tailed t-test is appropriate. The nullhypothesis and alternative hypothesis areThe test statistic, t exp , isH : X =µ H : X ≠µ0 AThere is another way to interpret the resultof this t-test. Knowing that t exp is 3.91and that there are 4 degrees of freedom,we use Appendix 4 to estimate the a valuecorresponding to a t(a,4) of 3.91. FromAppendix 4, t(0.02,4) is 3.75 and t(0.01,4) is 4.60. Although we can reject the nullhypothesis at the 98% confidence level,we cannot reject it at the 99% confidencelevel.For a discussion of the advantages of thisapproach, see J. A. C. Sterne and G. D.Smith “Sifting the evidence—what’swrong with significance tests?” BMJ 2001,322, 226–231.texpu X n= − =s98. 76−98.59 5= 391 .0.0973The critical value for t(0.05,4) from Appendix 4 is 2.78. Since t exp is greaterthan t(0.05,4) we reject the null hypothesis and accept the alternative hypothesis.At the 95% confidence level the difference between X and m istoo large to be explained by indeterminate sources of error, suggesting thata determinate source of error is affecting the analysis.Practice Exercise 4.8To evaluate the accuracy of a new analytical method, an analyst determinesthe purity of a standard for which m is 100.0%, obtaining thefollowing results.99.28% 103.93% 99.43% 99.84% 97.60% 96.70% 98.02%Is there any evidence at a = 0.05 that there is a determinate error affectingthe results?Click here to review your answer to this exercise.Earlier we made the point that you need to exercise caution when interpretingthe results of a statistical analysis. We will keep returning to thispoint because it is an important one. Having determined that a result is