Estimatig oulatio roortio: We will cosider a dichotomous categorical variable(s) ( classes: A, ot A) i a large oulatio(s). Poulatio(s) should be at least 0 times larger tha the samle(s). We will discuss a samlig distributio of a estimate of a oulatio roortio =P(A) i our oulatio(s). Suose we take a SRS of size, ad deote X=# of subjects with characteristics A i our samle, the we ca estimate by usig two differet samle statistics: ordiary samle roortio (-hat): = X (Note: sometimes researchers use also Wilso-Adjusted Proortio (-tilde): = X + which gives +4 CI-s more reliable tha those based o -hat, but wee will oly use -hat i our comutatios) Ifereces for a sigle roortio: is a ubiased estimate of. For large samlig distributio of is aroximately ormal, = with mea μ = ad stadard deviatio σ (1 ) we will use that fact i costructig CI for oulatio roortio. The aroximatio gets better with icreasig. ) Oe samle CI ( z-iterval) for : ±z α/ (1 ) Stadard Error of : SE = (1, assume X ad -X both 15 or greater EX1 Durig a 1998 race for state seator a ewsaer coducted a oll ad foud that 607 of 100 registered voters samled would vote for the Reublica cadidate. Let be the oulatio roortio of registered voters who would vote for the Reublica today. a. Give a 90 % level cofidece iterval for. ^= 607 100 =0.5058 0.5058±1.645 0.5058(1 0.5058) gives: (0.48, 0.53) 100 b. Based o your CI from art a, ca you coclude that a Reublica is ow likely to wi if more tha 50% is eeded for a wi. Exlai. Aswer: Not coclusive, CI cotais umbers below ad above 50%. c. What samle size is eeded to cut margi of error i your iterval to 1%?
Samle size cosideratios: =.5 z /,if we have o guess for -hat or E = g g z /,where E g = best guess for -hat Usig secod formula: Aswer:.5058(1.5058)( 1.645.01 ) =6764., =6765 EX A oll was take of 1010 U.S. emloyees, they were asked whether they lay hooky from work at least oce er year, 0 resoded yes. a) Fid 95% CI for =roortio of all U.S. emloyees that lay hooky from work at least oce er year. ^= 0 1010 0.±1.96 =0. 0.(0.8) Gives CI: (0.1753, 0.47) 1010 b) What samle size will esure the margi of error i 95% CI for of o more tha 1%? Assume 0% to be a best guess for hat. =0.(0.8)(1.96/0.01)^=6146.56, roud it u to 6147. Hyothesis Tests for oe Poulatio Proortio := 0 vs H a : 0 or H a : 0 or H a : 0 Samle roortio, ( hat) : = x attribute: Test statistics: z= 0 1 0 0 assume 0, 1 0 both 5 or greater, where x=umber of members i a samle with secified has N(0,1) distributio if is true, so it is a Z-test. Suose i ( EX) we ask followig questio: Is there evidece at 5 % sigificace level that less tha 5 % of U.S emloyees lay hooky from work at least oce er year? Test aroriate hyothesis. : =0.5 H a :<0.5 ^=0., z=-3.67. P=0.0001<0.05
Reject, there is evidece at 5% sigificace level that less tha 5% of U.S. emloyees will lay hooky from work at least oce er year. EX3 A Harris Poll asked 150 U.S. adults their views o baig hadgu sales. Of those samled, 650 favored a ba. At 5% sigificace level, do the data rovide sufficiet evidece that a majority of U.S. adults favor baig hadgu sales? : =0.5 H a :>0.5 ^=0.5 Z=1.41, -value=0.079>0.05 Do ot reject, there is o evidece at 5% sigificace level that majority of U.S. adults favor baig hadgus. The Relatio Betwee Hyothesis Tests ad Cofidece Itervals If Null hyothesis : = 0 agaist two tailed alterative is rejected at sigificace level, the (1- )*100% CI for will ot cotai 0, otherwise (if ot rejected) 0 will be iside of CI. I EX 95% CI for is (0.1753, 0.47), our left tailed test : =0.5 H a :<0.5 had a -value =0.0001, so two tailed test : =0.5 H a : 0.5 would have - value=*0.0001=0.0004<0.05. We ca see that our CI ad test of two sided hyothesis are coected. Our CI does ot cotai 0.5 ad we reject ull hyotheses at 5% sigificace level. I EX3 two tailed test : =0.5 H a : 0.5 would have -value = *0.079=0.158>5%, so ull hyotheses would ot be rejected at α=0.05. At the same time 95% CI for is (0.49, 0.55) ad clearly cotais 0 =0.50 OPTIONAL: Ifereces about Two Poulatios Proortios. Both oulatios are two-category oulatios, ideedet samles give couts with desired attribute i each oulatio. All couts: x 1, 1 x 1, x, x must be 5 or greater 1 ad estimate roortios with desired attribute i each oulatio Two samles z-iterval for 1 : 1 ±z / 1 1 1
EX1. Suose that we wat to kow to what extet is the frequecy of arole violatio related to the tye of crime? Out of 4 ersos who had served time for imulsive murder, 9 violated their arole. Out of 40 ersos who had served time for remeditated murder, 18 violated their arole. Let 1 = roortio of all imulsive murderers who violate arole = roortio of all remeditated murderers who violate arole Obtai 95% CI for 1-. Based o that iterval do you thig there is a differece betwee the roortios? Aswer: (0.0335, 0.4486). Yes, there is a differece, CI has o zero iside. Hyothesis Tests for Two Poulatios Proortios Two samles z test: = vs H a or H a or H a test statistics: z= 1 1 1 1 1 where = x x 1 1 =ooled samle roortio. Test statistics has N(01) distributio if ull hyothesis is true. I the examle about arole violatios we ca ask the followig questio: Are imulsive murderers more likely to violate their arole tha remeditated murderers? Test aroriate hyothesis at 5% sigificace level. =, H a, 1 =.69, =.45 =.573 z =.0 =.014 < 0.05. Reject, evidece for alterative hyothesis. Imulsive murderers will violate arole more ofte tha remeditated oes. 1. The Relatio Betwee Hyothesis Tests ad Cofidece Itervals If = is ot rejected agaist two tailed alterative, at give α level tha (1- α)*100% CI for the differece betwee two meas will cotai 0, otherwise it will ot cotai 0. If CI cotais 0, o evidece that roortios are differet.
Usig Calculator (TI 83, 84) 1 Proortio Z iterval use STAT meu the TESTS otio A is 1-ProZIterval It will use -hat method, just iut x ad If we wat 95% CI usig -tilde, we ca iut x=x+ ad =+4, for other cofidece levels it will ot work Proortios Z iterval use STAT meu the TESTS otio B is -ProZIterval It will use -hat method, just iut x ad for each oulatio Testig other Hyothesis All tests are available i STATS TESTS otio: I each test rocedure Data otio ca be used if samle statistics are ot comuted. It works the same as cofidece itervals rocedures. Test for 1 oulatio roortio: 1-ProZTest Test for oulatios roortios: -ProZTest