Chapter 9 & : Comparig Two Treatmets: This chapter focuses o two eperimetal desigs that are crucial to comparative studies: () idepedet samples ad () matched pair samples Idepedet Radom amples from Two Populatios uppose X,,X is a radom sample of size from populatio whose mea is µ ad stadard deviatio is σ ad Y,,Y is a radom sample of size from populatio whose mea is µ ad stadard deviatio is σ, where the samples from the two populatios are idepedet Our goal is to compare µ with µ, that is, we wish to make ifereces about µ -µ Ifereces from Large amples If both ad are large (both greater tha or equal to 30), we ca costruct a 00(-α)% cofidece iterval for µ -µ usig X Y z σ σ α /, X Y z α / stadard deviatios s ad s, respectively σ σ, where σ ad σ ca be replaced by the sample Eample: A sample of 80 light bulbs maufactured by Geeral Electric had a average lifetime of 58 hours with a stadard deviatio of 94 hours A sample of 60 light bulbs maufactured by Noame had a average lifetime of 9 hours with a stadard deviatio of 68 hours Costruct a 95% cofidece iterval for µ -µ Testig two meas: We use the same recipes as we have already discussed for testig oe mea ecept we use the symbol µ istead of µ ad µ istead of µ 0 The possible alterative hypotheses are H a: µ -µ >k; H a: µ -µ <k ; or H a: µ -µ k We also use the followig test statistic: where z σ σ Populatio Populatio ample ize Populatio Mea µ µ Populatio t Dev σ σ ample Mea ample t Dev s s Notes: The assigmet of Populatio ad Populatio is arbitrary If both populatios are approimately ormal ad >30 ad >30, the σ ad σ ca be replaced by s ad s, respectively Eample: A sample of 80 light bulbs maufactured by Geeral Electric had a average lifetime of 58 hours with a stadard deviatio of 94 hours A sample of 60 light bulbs maufactured by Noame had a Page
average lifetime of 9 hours with a stadard deviatio of 68 hours The Noame bulbs are cheaper ad you will buy them if they last as log as or loger tha the GE bulbs Perform the appropriate test at α00 At what sigificace level would you first coclude that the GE bulbs last loger? Ifereces from mall amples If or (or both) is less tha 30, we eed to determie if two additioal assumptios are met before we attempt ay statistical ifereces: Both populatios must be ormal The populatio stadard deviatio for both samples is the same, σ If the above two assumptios, i additio to the assumptios ecessary for large sample ifereces, we ca costruct a 00(-α)% cofidece iterval for µ -µ usig X Y t α / ( pooled ) s ( ) s pooled, X Y t α / pooled, with df -, where Eample: A sample of 80 light bulbs maufactured by Geeral Electric had a average lifetime of 58 hours with a stadard deviatio of 94 hours A sample of 4 light bulbs maufactured by Noame had a average lifetime of 9 hours with a stadard deviatio of 0 hours Costruct a 95% cofidece iterval for µ -µ Page
If we wish to compare two meas, we use the same hypothesis tests for a large sample with the followig modificatio The z-test statistic is replaced with the followig t-test statistic: t, where s s ( ) s ( ) s ad we have ( -) degrees of freedom Eample: Ahab refused to believe the results of his crew I his quest to validate his obsessio, he decided to take four ew measuremets of Moby Dick s weight He foud that Moby Dick s average weight was ow 69 tos with a stadard deviatio of 054 tos Ca Ahab s crew coclude that their computed average weight is less tha Ahab s computed average weight? Use α005 Page 3
Testig two proportios: We use the same recipes as we have already discussed for testig oe proportio ecept we use the symbol p istead of p ad p istead of p 0 We also use the followig test statistic: z where ( ) where Populatio Populatio ample ize Total i ample with Characteristic Proportio i ample p ˆ p ˆ Populatio Proportio p p We ca also calculate a 00(-α)% cofidece iterval for p -p usig ( ) ( ) zα / whe ad are large Eample: I a group of 000 me polled, 850 supported a issue Of 500 wome surveyed, 400 supported the issue Before you implemet ay hypothesis test, what cocers might you have about this problem? Test the hypothesis that the proportio of me supportig the issue equals the proportio of wome supportig the issue agaist the alterative Use α00 Page 4
Matched Pair amples The work we have doe thus far was based o the assumptio that our data was upaired I the situatios where we had two differet populatios, we assumed that the data samples were draw idepedetly from the two populatios How do we make coclusios about two populatios meas, whe the two samples are ot idepedet? Two samples are said to be paired whe for each data value collected from oe sample, there is a correspodig data value collected from the secod sample, ad both of these data values are collected from the same source A perfect eample would be the midterm ad the fial eam marks for a group of studets Whe we have paired data, we ca ru a t-test to test the ull hypothesis that the meas of the two populatios are equal (the same) agaist oe of the usual alterative hypotheses How do we implemet such a test? Cosider we have the followig pairs of ormal data: (, y ), (, y ),, (, y ) ad µ X ad µ Y are the respective populatio meas Now defie d i i -y i, where i ad let µ D be the populatio mea of these differeces The di ( di ) ( di ) d ad D ( ) The ull hypothesis H 0 : µ X -µ Y k becomes H 0 : µ D k ad the test statistic d k t has a studet s t-distributio with (-) degrees of freedom D We ca also calculate a 00(-α)% cofidece iterval for µ X -µ Y usig d z d z D D α /, α / If the umber of pairs is ot greater tha or equal to thirty (ie <30), we ca replace z α/ with t α/ with df- Eample: The followig are the scores out of 00% achieved by a radom sample of 0 Law tudets tudet Midterm Mark (X) Fial Mark (Y) 55 60 60 70 3 65 60 4 85 75 5 75 80 6 70 85 7 75 80 8 65 70 9 65 60 0 85 80 First form a 90% cofidece iterval for the differece i the mea Midterm ad Fial eam marks Page 5
Ca we coclude (at the 95% sigificace level) that the midterm eam was harder tha the fial eam? Eample: The tire wear was recorded for two differet brads A ad B of tires o five differet cars Car Tire A (X) Tire B (Y) 06 0 98 94 3 3 8 4 97 9 5 88 83 Does this data at the 99% sigificace level provide sufficiet evidece to idicate a differece i mea tire wear for the two brads? Page 6
Notes: To use the paired data t-test statistic, we have made the assumptio that DX-Y is a ormal radom variable The setup for the paired data hypothesis test follows the same format as the hypothesis tests we have bee studyig 3 If you have paired data, you ca either implemet a paired t-test or a upaired two-sample t-test, because the paired t-test tests the same hypotheses as the upaired two sample tests that we have studied BUT, if you do ot have paired data, you CAN ONLY implemet a upaired twosample test!!! 4 Whe > 30, a z-test ca be used Page 7