2 1. The r.s., of size n2, from population 2 will be. 2 and 2. 2) The two populations are independent. This implies that all of the n1 n2

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1 Chapter 8 Comparig Two Treatmets Iferece about Two Populatio Meas We wat to compare the meas of two populatios to see whether they differ. There are two situatios to cosider, as show i the followig examples: ) I a experimet desiged to study the effects of illumiatio level o task performace ( Performace of Complex Tasks Uder ifferet Levels of Illumiatio, J. Illumiatig Egieerig, 976: 35-4), subjects were required to isert a fie-tipped probe ito the eyeholes of te eedles i rapid successio both for a low-light-level with a black backgroud ad for a higher level with a white backgroud. It is of iterest to compare the mea times for completio of the task uder the two differet coditios. ) Compare the mea lifetime,, for trasistors produced by productio lie to the mea lifetime,, for trasistors produced by productio lie. We wat to kow whether these two meas differ. I the first case, we are comparig related meas, usig depedet samples. For each member of oe sample, there is a matched member of the other sample. I the secod case, we are comparig urelated meas, usig idepedet samples. There is o atural way to match each member of oe sample with a member of the other sample. We will use somewhat differet procedures for hypothesis tests, depedig o whether our samples are depedet or idepedet. There is aother issue to be cosidered. Are the variaces of the two populatios equal or uequal. This issue, of course, did ot arise with iferece about a sigle populatio. We will see that the procedure for iferece about the differece betwee the meas depeds o the compariso of the variaces. Comparig Two Meas, Idepedet amples We will assume the followig: ) We have selected a radom sample from each of the two populatios. The r.s., of size, from populatio will be deoted by i.i.d. with mea ad variace X, X,, X. These r.v. s are assumed to be. The r.s., of size, from populatio will be deoted by X, X,, X. These r.v. s are assumed to be i.i.d. with mea ad variace. ) The two populatios are idepedet. This implies that all of the r.v. s listed above are idepedet of each other. 3) Either both populatios are ormal, or the coditios of the Cetral Limit Theorem apply. (We may also check for ormality of each populatio usig ormal probability plots with the samples of data.)

2 We wat to estimate the differece,, betwee the populatio meas. A logical poit estimator X X of this parameter is. It is easily show that this statistic is a ubiased estimator of the parameter. It is also easily show that the variace of the estimator is V X X Give these results ad the assumptios listed above, it is clear that the radom variable X X. has a approximate stadard ormal distributio. We wat to use this fact to do iferece about the differece betwee the two populatio meas. However, the radom variable give above depeds o two other ukow parameters. We eed to estimate the two populatio variaces. Testig Hypotheses About the ifferece Betwee Two Meas Assume that we wat to test whether the meas of two populatios, populatio ad populatio, differ. I other words, our alterative hypothesis has oe of the followig forms: H a : - 0 H a : - < 0 H a : - > 0 There are two cases to cosider: Either the populatios have the same variability, or they do ot. The form of the test statistic will deped o whether we ca make the assumptio that the populatios have equal variaces. If we ca assume equal variaces, the we wat to use the followig statistic: P t X X s s 0. Here the quatity is the pooled variace estimate. If we caot assume equal variaces, the we wat to use the followig statistic: t X X s s 0. Uder the ull hypothesis, this statistic has a approximate t-distributio with degrees of freedom give by

3 3 / / I either case, our hypothesis proceeds as follows: tep : tate the ull ad alterative hypotheses. tep : tate the chose sample sizes ad sigificace level. tep 3: tate the test statistic (substitutig 0 for - ), ad statig the distributio of the test statistic uder the ull hypothesis. tep 4: Fid the rejectio regio ad critical value(s). tep 5: Choose the samples, collect the data, calculate the value of the test statistic. tep 6: If we reject the ull hypothesis, the coclusio should be stated i the followig form: We reject H 0 at the () level of sigificace. We have sufficiet evidece to coclude that (statemet of alterative hypothesis). If we fail to reject the ull hypothesis, the coclusio should be stated i the followig form: We fail to reject H 0 at the () level of sigificace. We do ot have sufficiet evidece to coclude that (statemet of alterative hypothesis). I the followig examples, we assume that the populatio variaces are equal. We ca also do a simple graphical check for equality, by costructig side-by-side boxplots of the two data sets. We could also check for ormality usig probability plots, if we had the data sets available. We will lear later how to test for equality of the variaces. Example: Let ad deote true average tread lives for two competig brads of size P05/65R5 radial tires. We wat to test whether the average tread lives are differet. We choose a radom sample of 45 tires of the first type ad a radom sample of 45 tires of the secod type. We test each tire uder idetical coditios util the tread wears out. We obtai the followig data: x 4,500 mi., s 00 mi., x 40,400 mi., s 900 mi. Example: The accompayig table gives summary data o cube compressive stregth (N/mm ) for cocrete specimes made with a pulverized fuel-ash mix ( A study of twety-five-year-old pulverized fuel ash cocrete used i foudatio structures, Proceedigs of the Istitute of Civil Egieers, Mar. 985, 49-65). We wat to test whether the true mea 7-day stregth is less tha the true mea 8-day stregth. Age (days) ample ize ample Mea ample

4 Cofidece Itervals for iffereces Betwee Populatio Meas We ca fid cofidece iterval estimates for the differeces betwee two populatio meas (idepedet samples) usig the followig formulas, depedig o whether the populatio variaces are equal or uequal: ) For equal populatio variaces, use s s X X t, d. f.. I this case, d.f. = +. ) For uequal populatio variaces, use X s s X t. I this case, d.f. = the smaller of the values ad., d. f. Example: Estimate the differece i mea tread lives from the first example above. Iterpret this iterval estimate. Example: Estimate the differece, 7 8, i mea compressive stregths from the secod example above. Iterpret this iterval estimate. 4 Choice of ample izes, Whe Variaces Are Equal For the two-sided alterative hypothesis, H A : 0, with equal sample sizes ad equal populatio variaces, we may use Charts Va ad Vb i the Appedix, together with d fid the appropriate sample size. The sample size read from the curve will be *. 0, to Tests of Hypotheses Cocerig the ifferece Betwee Two Populatio Meas, Uequal Variaces If we do ot have reaso to believe that the populatios have equal variability, we should check for equal variability by some method. Oe way to do this is to do a hypothesis test i which the ull hypothesis is that the populatio variaces are equal. Aother way is to graphically compare the two data sets. We will look at the secod method ow, ad look at testig equality of the variaces later. Example: The void volume withi a textile fabric affects comfort, flammability, ad isulatio properties. Permeability of a fabric refers to the accessibility of void spaces to the flow of a gas or liquid. The paper The relatioship betwee porosity ad air permeability of wove textile fabrics (Joural of Testig ad Evaluatio, 997: 08-4) gave summary iformatio o air permeability (cm 3 /cm /sec) for a umber of differet fabric types. Cosider the followig data o two differet types of plai-weave fabric: Fabric Type ample ize ample Mea ample Cotto Triacetate We wat to test whether plai-weave triacetate has a higher mea permeability tha plai-weave cotto. We also wat a 95% cofidece iterval estimate of the differece betwee the meas.

5 Tests of Hypotheses Cocerig the ifferece Betwee Two Populatio Meas, epedet amples Whe there is a atural pairig betwee each member of oe populatio ad a member of the other populatio, the test for differeces betwee the populatio meas must be doe somewhat differetly. For depedet samples, ifereces are performed based o the differeces betwee the scores for each pair. Let X, X, X 3,, X be the observatios made o the members of the first sample, ad let Y, Y, Y 3,, Y be the observatios made o members of the secod sample. The radom variables we will use i this test will be = X Y, = X Y, 3 = X 3 Y 3,, = X Y. Usig the set of differece radom variables, we coduct a oe-sample t-test. The sample mea is the i i, ad the sample variace is i i. We may either use the raw data of differece scores to coduct our t-test, or we may use these sample statistics based o the differece scores. The alterative hypotheses have oe of the followig forms: H a : - 0 or H a : 0 H a : - < 0 or H a : < 0 H a : - > 0 or H a : > 0 Here, the differece betwee the populatio meas. 5 The steps i the hypothesis test are similar to those i previous tests: tep : tate the two hypotheses to be tested. tep : tate the sample size (ote that both samples must have the same size), ad the chose sigificace level. d tep 3: The test statistic is t 0, which uder the ull hypothesis has a t distributio with d.f. s =. tep 4: Fid the rejectio regio ad critical value(s). tep 5: Choose samples, collect data, calculate the value of the test statistic. tep 6: tate the coclusio, i terms of the alterative hypothesis, ad beig sure to iclude the sigificace level of the test. Cofidece Itervals for iffereces Betwee Related Populatio Meas If we wat to obtai a iterval estimate of the differece betwee two populatio meas, whe there is a pairwise relatioship betwee members of oe populatio ad members of the other, we first compute the differece scores

6 6 Note that, if the samples are either from ormal distributios or are large eough that we may use the Cetral Limit Theorem, the T has a (approximate) t distributio with d.f. =. The a ( α)00% cofidece iterval estimate for is t, t,., Example: I a experimet desiged to study the effects of illumiatio level o task performace ( Performace of Complex Tasks Uder ifferet Levels of Illumiatio, J. Illumiatig Egieerig, 976: 35-4), subjects were required to isert a fie-tipped probe ito the eyeholes of te eedles i rapid successio both for a low-light-level with a black backgroud ad for a higher level with a white backgroud. It is of iterest to compare the mea times for completio of the task uder the two differet coditios. The data are give i the table below. We wat to test whether the higher level of illumiatio yields a decrease of more tha 5 secods i true mea task completio time. ubject Black White We also wat a 95% cofidece iterval estimate of the differece betwee the mea times to complete the task uder low illumiatio v. high illumiatio.