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Topic: hypothesis test question on two proportions
Replies: 3   Last Post: Jul 11, 2006 4:06 PM

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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(1-p1)/n1 + p2(1-p2)/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(1-p) (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