Analytical Chem istry - DePauw University

Chapter 4 Evaluating Analytical Data107Practice Exercise 4.9To compare two production lots of aspirin tablets, you collect samplesfrom each and analyze them, obtaining the following results (in mg aspirin/tablet).Lot 1: 256 248 245 245 244 248 261Lot 2: 241 258 241 244 256 254Is there any evidence at a = 0.05 that there is a significant difference inthe variance between the results for these two samples?Click here to review your answer to this exercise.4F.4 Comparing Two Sample MeansThree factors influence the result of an analysis: the method, the sample,and the analyst. We can study the influence of these factors by conductingexperiments in which we change one of the factors while holding the othersconstant. For example, to compare two analytical methods we can havethe same analyst apply each method to the same sample, and then examinethe resulting means. In a similar fashion, we can design experiments tocompare analysts or to compare samples.Before we consider the significance tests for comparing the means oftwo samples, we need to make a distinction between unpaired data andpaired data. This is a critical distinction and learning to distinguish betweenthe two types of data is important. Here are two simple examples thathighlight the difference between unpaired data and paired data. In eachexample the goal is to compare two balances by weighing pennies.• Example 1: Collect 10 pennies and weigh each penny on each balance.This is an example of paired data because we use the same 10 penniesto evaluate each balance.• Example 2: Collect 10 pennies and divide them into two groups offive pennies each. Weigh the pennies in the first group on one balanceand weigh the second group of pennies on the other balance.Note that no penny is weighed on both balances. This is an exampleof unpaired data because we evaluate each balance using a differentsample of pennies.In both examples the samples of pennies are from the same population. Thedifference is how we sample the population. We will learn why this distinctionis important when we review the significance test for paired data; first,however, we present the significance test for unpaired data.It also is possible to design experimentsin which we vary more than one of thesefactors. We will return to this point inChapter 14.One simple test for determining whetherdata are paired or unpaired is to look atthe size of each sample. If the samplesare of different size, then the data mustbe unpaired. The converse is not true. Iftwo samples are of equal size, they may bepaired or unpaired.Un p a i r e d Da t aConsider two analyses, A and B with means of X Aand X B, and standarddeviations of s A and s B . The confidence intervals for m A and for m B are