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[apstat] Mike's 2Sample Proportion Problem
Posted:
Mar 20, 2007 7:55 PM


Mike, Corey, and Others:
Mike writes:
> I posed this question to my students today. A survey of 1200 potential Democrat > primary voters asked who they would vote for. 444 said Clinton, 390 said Obama. > (these are fabricated numbers). Is there a significant difference in the > proportions? We treated this as a 2sided 2proportion test and got a pvalue > of .0206, therefore concluding that there is pretty strong evidence that Clinton > is ahead. My question is whether this is valid in light of the condition that > the SRS's must be independent. I'm not sure how the independence condition is > met. > Thanks in Advance, > Mike McCarvill > Blind Brook High School > Rye Brook, NY > ====
Let's summarize this as follows:
(1) Clinton: pcap1= 444/1200 = 0.370
(2) Obama: pcap2= 390/1200 = 0.325
(3) Others/Undecided: 366/1200 = 0.305
Not only are these not from independent samples but since we have three categories (and so the 2 df as Corey indicated) this is really a multinomial (trinomial) situation (but, unlike Corey, will ignore using a chisquare test here). For such, the exact [oh, here we go with this again!] standard error of the difference pcap1pcap2 is
sqrt[Var(pcap1) + Var(pcap2)  2Cov(pcap1, pcap2)] = sqrt[(p1)(1p1)/n + (p2)(1p2)/n + 2(p1)(p2)/n]
where that last term under the square root comes from the covariance for such a multinomial (trinomial) distribution. Thus, using the usual point estimates for p1 and p2, namely pcap1 and pcap2, we have for the estimated standard error
sqrt[(0.370)(0.630)/1200 + (0.325)(0.675)/1200 + 2(0.370)(0.325)/1200]
which gives 0.0235
compared with 0.0194 sans the covariance term. Thus, Mike should be using 0.0235 in the denominator of the test stat instead of 0.0194.
Hope this was helpful.
 David Bee
PS: If we were to form 95 percent interval estimates for the difference p1  p2 instead, then with the covariance it would be (0.001, 0.091) and incorrectly without it (0.006, 0.084) [assuming my arithmetic is correct]. PPS: So, which one would Hillary want and which one would Barack want reported in the press? ==== Course related websites: FAQ: http://www.mrderksen.com/faq.htm PLEASE CHECK THIS GREAT SITE FOR FREQUENTLY ASKED QUESTIONS AP Central: http://apcentral.collegeboard.com/stats Archives: http://mathforum.org/kb/forum.jspa?forumID=67  apstat is an Electronic Discussion Group (EDG) of The College Board TO CHANGE YOUR EMAIL ADDRESS, PASSWORD OR SETTINGS, go to http://lyris.collegeboard.com/read/my_account/edit To UNSUBSCRIBE click the unsubscribe button on your Forums page: http://lyris.collegeboard.com/read/my_forums/



