S160 #12 Samplig Distributio of the Proportio, Part 2 JC Wag February 25, 2016 Review of Large Sample Result for Sample Proportio Recall that for large sample (ie, sample size is large, say p > 5 ad (1 p) > 5), the sample proportio ˆp of successes has a approximate ormal distributio: ( ˆp N p, p(1 p) Hece, the sample proportio ˆp estimates the populatio proportio p with a stadard error estimated by ) SE = SEˆp =
Estimatig Populatio Proportio Usig Itervals Sice, for large sample size, ( ) p(1 p) ˆp N p, The estimate ˆp ted to miss the expected value p by 1SD which is estimated by 1SE Hece, approximately 95% of time ˆp will fall withi 196SE about p: or that (mathematically) ˆp p 196SE p is iside the iterval ˆp ± 196SE 95% of the time That is, with 95% certaity, the iterval ˆp ± 196SE cotais the true value p 95% Cofidece Iterval for Populatio Proportio 95% Cofidece Iterval for p: A 95% cofidece iterval estimate for the populatio proportio p is give by ˆp ± 196 196SE = 196 is called 95% margi of error = ME
Busiess Graduates Example If 6 out of 40 studets pla to go to graduate school, the proportio p of all studets who pla to go to graduate school is estimated as ˆp = 6 40 = 15 with a (95%) margi of error = ME = 196 15 85 40 011 ad hece a 95% cofidece iterval for p is 015 ± 011 = (015 011, 015 + 011) = (004, 026) If 54 out of 360 studets pla to go to graduate school, the a 95% cofidece iterval for p is (ote: ˆp = 54/360 = 15) 015 ± 196 15 85 360 = 015 ± 0037 = (0113, 0187) Calculatio of a 95% cofidece iterval for the true proportio Give a sample of size, the umber of successes X is observed A 95% cofidece iterval for the true proportio p is costructed as follows: 1 Calculate the (poit estimate) ˆp = X 2 Calculate the stadard error SE = 3 Calculate the margi of error ME = 196 SE 4 Costruct the cofidece iterval (poit estimate ME, poit estimate+me) = (ˆp ME, ˆp+ME) 5 If a 95% cofidece iterval for the true percetage is requested, covert the iterval above to percetages by shiftig the umbers 2 decimal places to the right
Iterpretatio of a 95% Cofidece Iterval Whe a sample becomes available (ie, has bee observed), the cofidece iterval is completely specified Cautios should be exercised cocerig the iterpretatio of its result We say that, with 95% cofidece, the true proportio p is betwee ˆp ME ad ˆp + ME sice the iterval either cotais or misses etirely the true proportio For the Busiess Graduates Example, the 95% cofidece iterval has bee obtaied: (0113, 0187) We are 95% cofidet that the true proportio of busiess studets plaig to atted graduate school is betwee 0113 ad 0187 Or that we are 95% cofidet that the true percetage of busiess studets plaig to atted graduate school is betwee 113% ad 187% Iterpretatio of a 95% Cofidece Iterval cotiued Suppose that a 95% cofidece iterval for the true populatio proportio is (023,047) the the correct iterpretatio is With 95% cofidece, the true populatio proportio is betwee 023 ad 047 The followig gives a example list of icorrect iterpretatios: There is 95% chace that true populatio proportio is betwee 023 ad 047 We are 95% cofidet that the true sample proportio is betwee 023 ad 047
Usig Cofidece Iterval Suppose that it is desired to check if a cojectured value p 0, say, for the populatio proportio is plausible A sample is take If the 95% cofidece iterval for the true populatio proportio cotais p 0, the it is judged plausible Otherwise, it is judged implausible For example it is cojectured, before the survey, that the true proportio of busiess studets plaig to atted graduate school is 010 The 95% cofidece iterval for the true populatio proportio from the survey yielded (0113,0187) The cojectured value is implausible iclicker Questio 121 A survey of = 1500 America adults was coducted to check if they believe i astrology It was cojectured that the true proportio of America adults believig i astrology is p 0 = 032 The survey showed that X = 405 adults believe i astrology Cosequetly, a 95% cofidece iterval for the true proportio is (0248,0292) Is the cojectured value plausible accordig to the cofidece iterval? A Yes B No C Isufficiet iformatio to judge Note: ˆp = X/ = 405/1500 = 027, SE = / = 027(1 027)/1500 = 0011, ME = 196 SE = 196 0011 = 0022 Hece, a 95% cofidece iterval for the true proportio is (027 0022, 027 + 0022) = (0248, 0292)
iclicker Questio 122 A survey of = 1500 America adults was coducted to check if they believe i astrology The survey showed that X = 405 adults believe i astrology Cosequetly, a 95% cofidece iterval for the true proportio is (0248,0292) Which of the followig is true? A We are 95% cofidet that the sample proportio is betwee 0248 ad 0292 B The chace that the true proportio is betwee 0248 ad 0292 is 95% C We are 95% cofidet that the true proportio is betwee 0248 ad 0292 D The chace that the true proportio is betwee 0248 ad 0292 is 5% E Noe of the previous Sample Size Determiatio to esure estimatio accuracy Electio Poll Example Cosider p the proportio of votes a cadidate will get i a presidetial electio A electio poll is istalled ad a 997% reliability i estimatio is to be esured with a margi of error (ME) for the estimatio of proportio at 002 What is the miimum sample size?
Electio Poll Example, cotiued ( Recall that ˆp N p, ) p(1 p) Accordig to the empirical rule, with 997% chace the estimate ˆp will be withi 3 p(1 p) about the mea p That is, to get a margi of error at ME=002 with the said reliability, the sample size is to be set so that 3 p(1 p) = ME or = 32 p(1 p) ME 2 Electio Poll Example, cotiued Two ways to resolve the paradox i comig up with a sample size for the estimatio of the ukow populatio proportio p usig = 32 p(1 p) ME 2 is to replace p by : a prelimiary estimate p the coservative estimate 1 2 If the secod way is used, the to get a margi of error of ME=002 with 997% reliability, the sample size should be set at = 32 1 2 1 2 ME 2 = 9 4ME 2 = 9 4 (02) 2 = 5625