Hypothesis Testig () Lecture 8 Barrow, Statistics for Ecoomics, Accoutig ad Busiess Studies, 4 th editio Pearso Educatio Limited 006
Hypothesis Testig () So far we have looked at hypothesis testig about a sigle populatio parameter H 0 : =5000 Now we look at testig hypotheses about differeces betwee two populatio parameters E.g. are wome paid less tha me? Barrow, Statistics for Ecoomics, Accoutig ad Busiess Studies, 4 th editio Pearso Educatio Limited 006
Formal Layout of a Problem. Specify the ull ad alterative hypothesis. Choose sigificace level, e.g. 5% 3. Look up critical value from z or t tables 4. Calculate the test statistic 5. Decisio: i reject or do ot reject th 0 Barrow, Statistics for Ecoomics, Accoutig ad Busiess Studies, 4 th editio Pearso Educatio Limited 006
Testig the Differece of Two Meas To test whether two samples are draw from populatios with the same mea H 0 : = or H 0 : = 0 H : or H 0 : 0 0 The test statistic is z x x s s Barrow, Statistics for Ecoomics, Accoutig ad Busiess Studies, 4 th editio Pearso Educatio Limited 006
Example: Car Compay has Two Factories Average daily output for 30 days is Output the Same? 40 408 Stadard dev of daily output 5 0 H 0 : = or =0 0 H : 0 Sigificace level of % implies z * 0.005=.57 Barrow, Statistics for Ecoomics, Accoutig ad Busiess Studies, 4 th editio Pearso Educatio Limited 006
z x x 40 408 0 s s 5 30 0 30.05 z<z * so it falls i the o-rejectio regio ad therefore we do ot reject the ull hypothesis. There does ot seem to be eough evidece to reject the claim that output is the same i each factory Barrow, Statistics for Ecoomics, Accoutig ad Busiess Studies, 4 th editio Pearso Educatio Limited 006
Testig the Differece of Two Proportios To test whether two sample proportios are equal H 0 : = or H 0 : = 0 0 0 H : or H : 0 The test statistic is z p p ˆ ˆ ˆ ˆ ˆ p p Barrow, Statistics for Ecoomics, Accoutig ad Busiess Studies, 4 th editio Pearso Educatio Limited 006
Example: Customer Satisfactio Are Customers Equally Satisfied with Differet Tours? proportio who say they are satisfied 45/75 48/90 H 0 : =0 H : 0 Sigificace level of 5%, z * 0.05 05=.96 Barrow, Statistics for Ecoomics, Accoutig ad Busiess Studies, 4 th editio Pearso Educatio Limited 006
p p 75*0.6 90*0.533 75 90 ˆ z p p ˆ ˆ ˆ ˆ 0.6 0.533 0 0.564 0.564 0.564 0.564 75 90 0.86 z<z * so we do ot reject the ull hypothesis 0.564 Barrow, Statistics for Ecoomics, Accoutig ad Busiess Studies, 4 th editio Pearso Educatio Limited 006
Small llsamples: Testig the Differece of Two Meas The test statistic is x x t S ~ t( S where S is the pooled ldvariace S s s ) Barrow, Statistics for Ecoomics, Accoutig ad Busiess Studies, 4 th editio Pearso Educatio Limited 006
Idepedet ad Depedet Samples So far assumed our samples are draw idepedetly Ofte we have depedetd samples (e.g. before bf ad after tests, or scores o macro ad micro exams) We use a differet approach to use the ifo. that the data comes from the same observatio Barrow, Statistics for Ecoomics, Accoutig ad Busiess Studies, 4 th editio Pearso Educatio Limited 006
Example: Paired Samples Worker 3 4 5 6 7 8 9 0 Before 4 3 5 8 7 4 4 7 After 3 7 4 8 9 4 5 6 8 Improve met 3 3 4 0 3 x x x B A 3.5, s 3.0, 0 5.5, s Improvemet B.55, 0 A.00, s Improvemet.47, 0 Barrow, Statistics for Ecoomics, Accoutig ad Busiess Studies, 4 th editio Pearso Educatio Limited 006
H H A t 0 : improveme t * 0.05,9 t : improvemet.833.0 0.47 0 0 0 5.07 t>t* t* so we reject the ull hypothesis ad coclude that t traiig has improved worker productivity Barrow, Statistics for Ecoomics, Accoutig ad Busiess Studies, 4 th editio Pearso Educatio Limited 006
Summary The priciples are the same for all tests: calculate the test statistic ad see if it falls ito the rejectio regio The formula for the test statistic depeds upo the problem (mea, proportio, etc) The rejectio regio varies, depedig upo whether it is a oe or two tailed test For large samples we ca always use a z test If is small we ca still use z test if the populatio variace is kow Otherwise use a t (ad pool variaces) Barrow, Statistics for Ecoomics, Accoutig ad Busiess Studies, 4 th editio Pearso Educatio Limited 006