D. Harvey - Modern Analytical Chemistry (794078), страница 27
Текст из файла (страница 27)
Three possible outcomes are shown in Figure 4.9. In Figure 4.9a, the probability distribution curvesare completely separated, strongly suggesting that the samples are significantly different. In Figure 4.9b, the probability distributions for the two samples are highlyoverlapped, suggesting that any difference between the samples is insignificant. Figure 4.9c, however, presents a dilemma. Although the means for the two samples appear to be different, the probability distributions overlap to an extent that a significant number of possible outcomes could belong to either distribution. In this casewe can, at best, only make a statement about the probability that the samples aresignificantly different.*The topic of detection limits is discussed at the end of this chapter.1400-CH04 9/8/99 3:54 PM Page 83Chapter 4 Evaluating Analytical Data83The process by which we determine the probability that there is a significantdifference between two samples is called significance testing or hypothesis testing.Before turning to a discussion of specific examples, however, we will first establish ageneral approach to conducting and interpreting significance tests.4E.2 Constructing a Significance TestA significance test is designed to determine whether the difference between twoor more values is too large to be explained by indeterminate error.
The first stepin constructing a significance test is to state the experimental problem as a yesor-no question, two examples of which were given at the beginning of this section. A null hypothesis and an alternative hypothesis provide answers to the question. The null hypothesis, H0, is that indeterminate error is sufficient to explainany difference in the values being compared. The alternative hypothesis, HA, isthat the difference between the values is too great to be explained by randomerror and, therefore, must be real. A significance test is conducted on the null hypothesis, which is either retained or rejected. If the null hypothesis is rejected,then the alternative hypothesis must be accepted.
When a null hypothesis is notrejected, it is said to be retained rather than accepted. A null hypothesis is retained whenever the evidence is insufficient to prove it is incorrect. Because of theway in which significance tests are conducted, it is impossible to prove that a nullhypothesis is true.The difference between retaining a null hypothesis and proving the null hypothesis is important.
To appreciate this point, let us return to our example on determining the mass of a penny. After looking at the data in Table 4.12, you mightpose the following null and alternative hypothesesH0: Any U.S. penny in circulation has a mass that falls in the range of2.900–3.200 gHA: Some U.S. pennies in circulation have masses that are less than2.900 g or more than 3.200 g.To test the null hypothesis, you reach into your pocket, retrieve a penny, and determine its mass.
If the mass of this penny is 2.512 g, then you have proved that thenull hypothesis is incorrect. Finding that the mass of your penny is 3.162 g, however, does not prove that the null hypothesis is correct because the mass of the nextpenny you sample might fall outside the limits set by the null hypothesis.After stating the null and alternative hypotheses, a significance level for theanalysis is chosen. The significance level is the confidence level for retaining the nullhypothesis or, in other words, the probability that the null hypothesis will be incorrectly rejected. In the former case the significance level is given as a percentage (e.g.,95%), whereas in the latter case, it is given as α, where α is defined asα = 1−confidence level100Thus, for a 95% confidence level, α is 0.05.Next, an equation for a test statistic is written, and the test statistic’s criticalvalue is found from an appropriate table.
This critical value defines the breakpointbetween values of the test statistic for which the null hypothesis will be retained orrejected. The test statistic is calculated from the data, compared with the criticalvalue, and the null hypothesis is either rejected or retained. Finally, the result of thesignificance test is used to answer the original question.significance testA statistical test to determine if thedifference between two values issignificant.null hypothesisA statement that the difference betweentwo values can be explained byindeterminate error; retained if thesignificance test does not fail (H0).alternative hypothesisA statement that the difference betweentwo values is too great to be explained byindeterminate error; accepted if thesignificance test shows that nullhypothesis should be rejected (HA).1400-CH04 9/8/99 3:54 PM Page 8484Modern Analytical Chemistry4E.3 One-Tailed and Two-Tailed Significance Tests(a)ValuesConsider the situation when the accuracy of a new analytical method is evaluated byanalyzing a standard reference material with a known µ.
A sample of the standard isanalyzed, and the sample’s mean is determined. The null hypothesis is that the sample’s mean is equal to µ–H0: X = µIf the significance test is conducted at the 95% confidence level (α = 0.05), then the–null hypothesis will be retained if a 95% confidence interval around X contains µ. Ifthe alternative hypothesis is–HA: X ≠ µ(b)Valuesthen the null hypothesis will be rejected, and the alternative hypothesis accepted if µlies in either of the shaded areas at the tails of the sample’s probability distribution(Figure 4.10a). Each of the shaded areas accounts for 2.5% of the area under theprobability distribution curve.
This is called a two-tailed significance test becausethe null hypothesis is rejected for values of µ at either extreme of the sample’s probability distribution.The alternative hypothesis also can be stated in one of two additional ways–HA: X > µ–HA: X < µ(c)ValuesFigure 4.10Examples of (a) two-tailed, (b) and (c) onetailed, significance tests.
The shaded areas ineach curve represent the values for whichthe null hypothesis is rejected.two-tailed significance testSignificance test in which the nullhypothesis is rejected for values at eitherend of the normal distribution.one-tailed significance testSignificance test in which the nullhypothesis is rejected for values at onlyone end of the normal distribution.type 1 errorThe risk of falsely rejecting the nullhypothesis (α).type 2 errorThe risk of falsely retaining the nullhypothesis (β).for which the null hypothesis is rejected if µ falls within the shaded areas shown inFigure 4.10(b) and Figure 4.10(c), respectively. In each case the shaded area represents 5% of the area under the probability distribution curve.
These are examples ofone-tailed significance tests.For a fixed confidence level, a two-tailed test is always the more conservative test–because it requires a larger difference between X and µ to reject the null hypothesis.Most significance tests are applied when there is no a priori expectation about therelative magnitudes of the parameters being compared. A two-tailed significance test,therefore, is usually the appropriate choice.
One-tailed significance tests are reservedfor situations when we have reason to expect one parameter to be larger or smallerthan the other. For example, a one-tailed significance test would be appropriate forour earlier example regarding a medication’s effect on blood glucose levels since webelieve that the medication will lower the concentration of glucose.4E.4 Errors in Significance TestingSince significance tests are based on probabilities, their interpretation is naturallysubject to error.
As we have already seen, significance tests are carried out at a significance level, α, that defines the probability of rejecting a null hypothesis that istrue. For example, when a significance test is conducted at α = 0.05, there is a 5%probability that the null hypothesis will be incorrectly rejected. This is known as atype 1 error, and its risk is always equivalent to α. Type 1 errors in two-tailed andone-tailed significance tests are represented by the shaded areas under the probability distribution curves in Figure 4.10.The second type of error occurs when the null hypothesis is retained eventhough it is false and should be rejected.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
