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prof
Posts:
5
Registered:
7/10/06


Re: hypothesis test question on two proportions
Posted:
Jul 11, 2006 10:15 AM


m00es wrote: > prof wrote: > > In the two population proportion test of H0: p1 = p2 vs Ha: p1 not > > equal p2 with a los of 0.05, one way to conduct the test is to > > construct a 95% CI on p1  p2. Then, if the interval contains "0", > > accept H0; otherwise reject H0. An alternate way is to calculate the z > > statistic under the assumption that H0 is true and compare to the 5% > > critical region. If the statistic is in the critical region, reject H0; > > otherwise accept H0. The question is: do the two methods give identical > > conclusions? The answer is not obvious to me since in the CI approach > > an assumption of H0 being true is not used; thus there are separate > > estimates for the two sample proportions which appear in the CI > > formula. whereas, in the test statistic approach, H0 being true is > > assumed; thus giving a combined estimate for the common proportion p > > which is used in the test statistic.. It is not obvious that identical > > conclusions would always be reached. Help please!!! > > Let's make this question more concrete. > > Assume you have two independent samples of sizes n1 and n2 and the > number of people with some outcome are observed in each group. Let x1 > and x2 denote these numbers. Then p1 = x1/n1 and p2 = x2/n2 are the > observed proportions in each group. We now want to test whether H0: pi1 > = pi2, where pi1 and pi2 are the true proportions in each group. > > For an approximate 95% confidence interval for pi1  pi2, we calculate: > > (p1  p2) + 1.96 sqrt( p1(1p1)/n1 + p2(1p2)/n2 ) > > and if 0 is not included in the confidence interval, we reject H0: pi1 > = pi2. > > The hypothesis test assumes that H0: pi1 = pi2 is true. Therefore, p1 > and p2 are estimates of the same proportion. These estimates are then > commonly pooled together into p = (x1 + x2)/(n1 + n2), yielding a > single estimate for pi = pi1 = pi2. Then: > > z = (p1  p2) / sqrt( p(1p) (1/n1 + 1/n2) ) > > can be compared against +1.96, the critical values for a normal > distribution with alpha = .05. > > It is now possible to construct an example where the confidence > interval and the hypothesis test will yield a different conclusion. > > Example: > > x1 = 40, n1 = 50, p1 = .8 > x2 = 340, n2 = 500, p2 = .68 > p = .69 > > 95% CI = (.002, .238) and therefore we conclude pi1 not equal to pi2. > z = 1.75 and therefore we do not reject H0: pi1 = pi2. > > Those "pathological" cases are rare though and the group sizes have to > be substantially different for this to happen. > > Also note that the CI and the hypothesis test as given here are > approximations anyway. > > m00es
Thank you very much, Your explanation was very helpful. One more question. Which method is usually preferred??? Burt



