Chapter10 COMPARING TWO TREATMENTS

1、Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.13.1Chapter 13Inference About ComparingTwo PopulationsCopyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.13.2Comparing Two PopulationsPreviously we looked at techniques to estimate and test parameters for one population:Pop

2、ulation Mean , Population Variance , andPopulation Proportion pWe will still consider these parameters when we are looking at two populations, however our interest will now be: The difference between two means. The ratio of two variances. The difference between two proportions.Copyright 2005 Brooks/

3、Cole, a division of Thomson Learning, Inc.13.3Difference of Two MeansIn order to test and estimate the difference between two population means, we draw random samples from each of two populations. Initially, we will consider independent samples, that is, samples that are completely unrelated to one

4、another.(Likewise, we consider for Population 2)Sample, size: n1Population 1Parameters:Statistics:Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.13.4Difference of Two MeansIn order to test and estimate the difference between two population means, we draw random samples from each of

5、two populations. Initially, we will consider independent samples, that is, samples that are completely unrelated to one another.Because we are compare two population means, we use the statistic:Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.13.5Sampling Distribution of 1. is normall

6、y distributed if the original populations are normal or approximately normal if the populations are nonnormal and the sample sizes are large (n1, n2 30)2. The expected value of is3. The variance of is and the standard error is:Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.13.6Makin

7、g Inferences About Since is normally distributed if the original populations are normal or approximately normal if the populations are nonnormal and the sample sizes are large (n1, n2 30), then:is a standard normal (or approximately normal) random variable. We could use this to build test statistics

8、 or confidence interval estimators for Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.13.7Making Inferences About except that, in practice, the z statistic is rarely used since the population variances are unknown.Instead we use a t-statistic. We consider two cases for the unknown p

9、opulation variances: when we believe they are equal and conversely when they are not equal.?Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.13.8When are variances equal? How do we know when the population variances are equal? Since the population variances are unknown, we cant know f

10、or certain whether theyre equal, but we can examine the sample variances and informally judge their relative values to determine whether we can assume that the population variances are equal or not.Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.13.9Test Statistic for (equal variance

11、s) Calculate the pooled variance estimator asand use it here:degrees of freedomCopyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.13.10CI Estimator for (equal variances) The confidence interval estimator for when the population variances are equal is given by:degrees of freedompooled va

12、riance estimatorCopyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.13.11Test Statistic for (unequal variances) The test statistic for when the population variances are unequal is given by:Likewise, the confidence interval estimator is:degrees of freedomCopyright 2005 Brooks/Cole, a divi

13、sion of Thomson Learning, Inc.13.12Which case to use?Which case to use? Equal variance or unequal variance?Whenever there is insufficient evidence that the variances are unequal, it is preferable to perform theequal variances t-test.This is so, because for any two given samples:The number of degrees

14、 of freedom for the equal variances caseThe number of degrees of freedom for the unequal variances caseLarger numbers of degrees of freedom have the same effect as having larger sample sizesCopyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.13.13Example 13.1Do people who eat high-fiber

15、cereal for breakfast consume, on average, fewer calories for lunch than people who do not eat high-fiber cereal for breakfast?What are we trying to show? What is our research hypothesis?The mean caloric intake of high fiber cereal eaters ( )is less than that of non-consumers ( ), i.e. is ?Copyright

16、2005 Brooks/Cole, a division of Thomson Learning, Inc.13.14Example 13.1The mean caloric intake of high fiber cereal eaters ( )is less than that of non-consumers ( ), translates to:(i.e. )Thus, H1:Hence our null hypothesis becomes:H0:IDENTIFYPhrase H0 & H1 as a“difference of means”Copyright 2005

17、Brooks/Cole, a division of Thomson Learning, Inc.13.15Example 13.1A sample of 150 people was randomly drawn. Each person was identified as a consumer or a non-consumer of high-fiber cereal. For each person the number of calories consumed at lunch was recorded. The data:Independent Popns;Either you e

18、at high fibercereal or you dontn1+n2=150There is reason to believethe population variancesare unequalRecall H1:Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.13.16Example 13.1Thus, our test statistic is:The number of degrees of freedom is:Hence the rejection region isCOMPUTECopyrigh

19、t 2005 Brooks/Cole, a division of Thomson Learning, Inc.13.17Example 13.1Our rejection region:Our test statistic:Since our test statistic (-2.09) is less than our critical value of t (-1.658), we reject H0 in favor of H1 that is, there is sufficient evidence to support the claim that high fiber cere

20、al eaters consume less calories at lunch.COMPUTECompareCopyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.13.18Example 13.1Likewise, we can use Excel to do the calculationsCOMPUTERecall H0:Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.13.19Example 13.1however, we still

21、 need to be able to interpret the Excel output:INTERPRETCompareBeware! Excel gives a right tail critical value!i.e. 1.6573 vs. 1.6573 !or look at p-valueCopyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.13.20Confidence IntervalSuppose we wanted to compute a 95% confidence interval esti

22、mate of the difference between mean caloric intake for consumers and non-consumers of high-fiber cerealsThat is, we estimate that non-consumers of high fiber cereal eat between 1.56 and 56.86 more calories than consumers.Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.13.21Confidence

23、 IntervalAlternatively, you can use the Estimators workbookvalues in bold face are calculated for youCopyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.13.22Example 13.2Two methods are being tested for assembling office chairs. Assembly times are recorded (25 times for each method). At

24、a 5% significance level, do the assembly times for the two methods differ?That is, H1: Hence, our null hypothesis becomes: H0:Reminder: since our null hypothesis is a “not equals” type, it is a two-tailed test.IDENTIFYCopyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.13.23Example 13.2T

25、he assembly times for each of the two methods are recorded and preliminary data is preparedCOMPUTEThe sample variances are similar, hence we will assume that the population variances are equalCopyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.13.24Example 13.2Recall, we are doing a two-

26、tailed test, hence the rejection region will be:The number of degrees of freedom is:Hence our critical values of t (and our rejection region) becomes:COMPUTECopyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.13.25Example 13.2In order to calculate our t-statistic, we need to first calcul

27、ate the pooled variance estimator, followed by the t-statisticCOMPUTECopyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.13.26Example 13.2Since our calculated t-statistic does not fall into the rejection region, we cannot reject H0 in favor of H1, that is, there is not sufficient evidenc

28、e to infer that the mean assembly times differ.INTERPRETCopyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.13.27Example 13.2Excel, of course, also provides us with the informationINTERPRETCompareor look at p-valueCopyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.13.28Confi

29、dence IntervalWe can compute a 95% confidence interval estimate for the difference in mean assembly times as:That is, we estimate the mean difference between the two assembly methods between .36 and .96 minutes. Note: zero is included in this confidence intervalCopyright 2005 Brooks/Cole, a division

30、 of Thomson Learning, Inc.13.29TerminologyIf all the observations in one sample appear in one column and all the observations of the second sample appear in another column, the data is unstacked. If all the data from both samples is in the same column, the data is said to be stacked.Copyright 2005 B

31、rooks/Cole, a division of Thomson Learning, Inc.13.30Identifying Factors IFactors that identify the equal-variances t-test and estimator of :Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.13.31Identifying Factors IIFactors that identify the unequal-variances t-test and estimator of

32、:Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.13.32Matched Pairs ExperimentPreviously when comparing two populations, we examined independent samples.If, however, an observation in one sample is matched with an observation in a second sample, this is called a matched pairs experim

33、ent.Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.13.33Example 13.4Is there a difference between starting salaries offered to MBA grads going into Finance vs. Marketing careers? More precisely, are Finance majors offered higher salaries than Marketing majors?In this experiment, MBA

34、s are grouped by their GPA into 25 groups. Students from the same group (but with different majors) were selected and their highest salary offer recorded.Heres how the data looksCopyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.13.34Example 13.4The numbers in black are the original sta

35、rting salary data; the number in blue were calculated.although a student is either in Finance OR in Marketing (i.e. independent), that the data is grouped in this fashion makes it a matched pairs experiment (i.e. the two students in group #1 are matched by their GPA rangethe difference of the means

36、is equal to the mean of the differences, hence we will consider the “mean of the paired differences” as our parameter of interest:Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.13.35Example 13.4Do Finance majors have higher salary offers than Marketing majors?Since:We want to resear

37、ch this hypothesis: H1:(and our null hypothesis becomes H0: )IDENTIFYCopyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.13.36Test Statistic for The test statistic for the mean of the population of differences ( ) is:which is Student t distributed with nD1 degrees of freedom, provided th

38、at the differences are normally distributed.Thus our rejection region becomes:Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.13.37Example 13.4From the data, we calculatewhich in turn we usefor our t-statisticwhich we compare to our critical value of t:COMPUTECopyright 2005 Brooks/Co

39、le, a division of Thomson Learning, Inc.13.38Example 13.4Since our calculated value of t (3.81) is greater than our critical value of t (1.711), it falls in the rejection region, hence we reject H0 in favor of H1; that is, there is overwhelming evidence (since the p-value = .0004) that Finance major

40、s do obtain higher starting salary offers than their peers in Marketing.INTERPRETCompareCopyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.13.39Confidence Interval Estimator forWe can derive the confidence interval estimator for algebraically as:In the previous example, what is the 95%

41、confidence interval estimate of the mean difference in salary offers between the two business majors?That is, the mean of the population differences is between LCL=2,321 and UCL=7,809 dollars.Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.13.40Identifying FactorsFactors that identif

42、y the t-test and estimator of :Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.13.41Inference about the ratio of two variancesSo far weve looked at comparing measures of central location, namely the mean of two populations.When looking at two population variances, we consider the rat

43、io of the variances, i.e. the parameter of interest to us is:The sampling statistic: is F distributed withdegrees of freedom.Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.13.42Inference about the ratio of two variancesOur null hypothesis is always:H0:(i.e. the variances of the two

44、populations will be equal, hence their ratio will be one)Therefore, our statistic simplifies to:Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.13.43Example 13.6In example 13.1, we looked at the variances of the samples of people who consumed high fiber cereal and those who did not a

45、nd assumed they were not equal. We can use the ideas just developed to test if this is in fact the case.We want to show: H1:(the variances are not equal to each other)Hence we have our null hypothesis: H0:IDENTIFYCopyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.13.44Example 13.6Since

46、our research hypothesis is: H1:We are doing a two-tailed test, and our rejection region is:CALCULATEFCopyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.13.45Example 13.6Our test statistic is:Hence there is sufficient evidence to reject the null hypothesis in favor of the alternative; th

47、at is, there is a difference in the variance between the two populations.CALCULATEF.58Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.13.46Example 13.6We may need to work with the Excel output before drawing conclusionsINTERPRETOur research hypothesisH1:requires two-tail testing, but

48、 Excel only gives us valuesfor one-tail testingIf we double the one-tail p-value Excel gives us, we have the p-value ofthe test were conducting (i.e. 2 x 0.0004 = 0.0008). Refer to the text and CD Appendices for more detail.Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.13.47Example

49、 13.6If we wanted to determine the 95% confidence interval estimate of the ratio of the two population variances in Example 13.1, we would proceed as followsThe confidence interval estimator for , is:CALCULATECopyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.13.48Example 13.6The 95% co

50、nfidence interval estimate of the ratio of the two population variances in Example 13.1 is:That is, we estimate that lies between .2388 and .6614Note that one (1.00) is not within this intervalCALCULATECopyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.13.49Identifying FactorsFactors th

51、at identify the F-test and estimator of :Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.13.50Difference Between Two Population ProportionsWe will now look at procedures for drawing inferences about the difference between populations whose data are nominal (i.e. categorical).As menti

52、oned previously, with nominal data, calculate proportions of occurrences of each type of outcome. Thus, the parameter to be tested and estimated in this section is the difference between two population proportions: p1p2.Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.13.51Statistic a

53、nd Sampling DistributionTo draw inferences about the the parameter p1p2, we take samples of population, calculate the sample proportions and look at their difference. is an unbiased estimator for p1p2.x1 successes in a sample of size n1 from population 1Copyright 2005 Brooks/Cole, a division of Thom

54、son Learning, Inc.13.52Sampling DistributionThe statistic is approximately normally distributed if the sample sizes are large enough so that:Since its “approximately normal” we can describe the normal distribution in terms of mean and variancehence this z-variable will also be approximately standard

55、 normally distributed:Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.13.53Testing and Estimating p1p2Because the population proportions (p1 & p2) are unknown, the standard error:is unknown. Thus, we have two different estimators for the standard error of , which depend upon the

56、null hypothesis. Well look at these cases on the next slideCopyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.13.54Test Statistic for p1p2There are two cases to considerCopyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.13.55Example 13.8A consumer packaged goods (CPG) compa

57、ny is test marketing two new versions of soap packaging. Version one (bright colors) is distributed in one supermarket, while version two (simple colors) is in another. Since the first version is more expensive, it must outsell the other design, that is its market share, p1, must be greater than tha

58、t of the other soap package design, i.e. p2.That is, we want to know, is p1 p2? or, using the language of statistics:H1: (p1p2) 0Hence our null hypothesis will be H0: (p1p2) = 0 case 1IDENTIFYCopyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.13.56Example 13.8Here is the summary dataOur null hypothesis is H0: (p1p2) = 0, i.e. is a “case 1” type problem, hence we need to calculate the pooled proportion:IDENTIFYCopyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.13.57Example 13.8At a 5% significan