S160 #12 Samplig Distributio of the Proportio, Part 2 JC Wag February 25, 2016
Outlie 1 Estimatig Proportio Usig Itervals Cofidece Iterval for the Populatio Proportio iclicker Questios 2 JC Wag (WMU) S160 #12 S160, Lecture 12 2 / 16
Review of Large Sample Result for Sample Proportio Recall that for large sample (i.e., 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 ˆp(1 ˆp) SE = SEˆp =. ). JC Wag (WMU) S160 #12 S160, Lecture 12 3 / 16
Estimatig Populatio Proportio Usig Itervals Sice, for large sample size, ( ˆp N p, p(1 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 1.96SE about p: ˆp p 1.96SE or that (mathematically) ). p is iside the iterval ˆp ± 1.96SE 95% of the time That is, with 95% certaity, the iterval ˆp ± 1.96SE cotais the true value p. JC Wag (WMU) S160 #12 S160, Lecture 12 4 / 16
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(1 ˆp) ˆp ± 1.96 1.96SE = 1.96 ˆp(1 ˆp) is called 95% margi of error = ME JC Wag (WMU) S160 #12 S160, Lecture 12 5 / 16
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.15.85 (95%) margi of error = ME = 1.96 40 ad hece a 95% cofidece iterval for p is 0.11 0.15 ± 0.11 = (0.15 0.11, 0.15 + 0.11) = (0.04, 0.26) 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) 0.15 ± 1.96.15.85 360 = 0.15 ± 0.037 = (0.113, 0.187) JC Wag (WMU) S160 #12 S160, Lecture 12 6 / 16
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 ˆp(1 ˆp) SE =. 3 Calculate the margi of error ME = 1.96 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. JC Wag (WMU) S160 #12 S160, Lecture 12 7 / 16
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 ˆp(1 ˆp) SE =. 3 Calculate the margi of error ME = 1.96 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. JC Wag (WMU) S160 #12 S160, Lecture 12 7 / 16
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 ˆp(1 ˆp) SE =. 3 Calculate the margi of error ME = 1.96 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. JC Wag (WMU) S160 #12 S160, Lecture 12 7 / 16
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 ˆp(1 ˆp) SE =. 3 Calculate the margi of error ME = 1.96 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. JC Wag (WMU) S160 #12 S160, Lecture 12 7 / 16
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 ˆp(1 ˆp) SE =. 3 Calculate the margi of error ME = 1.96 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. JC Wag (WMU) S160 #12 S160, Lecture 12 7 / 16
Iterpretatio of a 95% Cofidece Iterval Whe a sample becomes available (i.e, 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: (0.113, 0.187). We are 95% cofidet that the true proportio of busiess studets plaig to atted graduate school is betwee 0.113 ad 0.187. Or that we are 95% cofidet that the true percetage of busiess studets plaig to atted graduate school is betwee 11.3% ad 18.7%. JC Wag (WMU) S160 #12 S160, Lecture 12 8 / 16
Iterpretatio of a 95% Cofidece Iterval cotiued Suppose that a 95% cofidece iterval for the true populatio proportio is (0.23,0.47) the the correct iterpretatio is With 95% cofidece, the true populatio proportio is betwee 0.23 ad 0.47. The followig gives a example list of icorrect iterpretatios: There is 95% chace that true populatio proportio is betwee 0.23 ad 0.47. We are 95% cofidet that the true sample proportio is betwee 0.23 ad 0.47. JC Wag (WMU) S160 #12 S160, Lecture 12 9 / 16
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 0.10. The 95% cofidece iterval for the true populatio proportio from the survey yielded (0.113,0.187). The cojectured value is implausible. JC Wag (WMU) S160 #12 S160, Lecture 12 10 / 16
iclicker Questio 12.1 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 = 0.32. The survey showed that X = 405 adults believe i astrology. Cosequetly, a 95% cofidece iterval for the true proportio is (0.248,0.292). Is the cojectured value plausible accordig to the cofidece iterval? A. Yes. B. No. C. Isufficiet iformatio to judge. Note: ˆp = X/ = 405/1500 = 0.27, SE = ˆp(1 ˆp)/ = 0.27(1 0.27)/1500 = 0.011, ME = 1.96 SE = 1.96 0.011 = 0.022. Hece, a 95% cofidece iterval for the true proportio is (0.27 0.022, 0.27 + 0.022) = (0.248, 0.292). JC Wag (WMU) S160 #12 S160, Lecture 12 11 / 16
iclicker Questio 12.2 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 (0.248,0.292). Which of the followig is true? A. We are 95% cofidet that the sample proportio is betwee 0.248 ad 0.292. B. The chace that the true proportio is betwee 0.248 ad 0.292 is 95%. C. We are 95% cofidet that the true proportio is betwee 0.248 ad 0.292. D. The chace that the true proportio is betwee 0.248 ad 0.292 is 5%. E. Noe of the previous. JC Wag (WMU) S160 #12 S160, Lecture 12 12 / 16
Outlie 1 Estimatig Proportio Usig Itervals Cofidece Iterval for the Populatio Proportio iclicker Questios 2 JC Wag (WMU) S160 #12 S160, Lecture 12 13 / 16
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 99.7% reliability i estimatio is to be esured with a margi of error (ME) for the estimatio of proportio at 0.02. What is the miimum sample size? JC Wag (WMU) S160 #12 S160, Lecture 12 14 / 16
Electio Poll Example, cotiued ( ) p(1 p) Recall that ˆp N p, Accordig to the empirical rule, with 99.7% chace the estimate ˆp will be p(1 p) withi 3 about the mea p. That is, to get a margi of error at ME=0.02 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. JC Wag (WMU) S160 #12 S160, Lecture 12 15 / 16
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=0.02 with 99.7% reliability, the sample size should be set at = 32 1 2 1 2 ME 2 = 9 4ME 2 = 9 4 (.02) 2 = 5625. JC Wag (WMU) S160 #12 S160, Lecture 12 16 / 16