Weaknesses of Wald confidence interval for binomial distributionA survey samples 1000 people, among which 500 say they will vote for A, 400 for B e 100 for C. Calculate a confidence interval for the proportion of people that will vote for A. What are the major weaknesses of the confidence interval you just calculated?Let An∼Binom(n,p). By the CLT,n(A―n−p)x→{d}N(0,p(1−p))hence np(1−p)(A―n−p)x→{d}N(0,1)Since p is unknown, we approximate it with A―n. It follows that, at 1−α confidence,p∈A―n±z1−α2A―n(1−A―n)nSince this confidence interval relies on the CLT, it gives poor results when n is small or p is very close to 0 or 1. But this is not the case here, as n=1000 and A―n=0.5, so I fail to see what weaknesses the question is talking about?

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Vegljamzt6 · Accepted Answer

On theoretical grounds there are two reasons for suspicion that a (say 95%) Wald CI may not actually have 95% coverage probability. First, it uses the normal approximation to the binomial. Second, it estimates the standard error p1−pn by (in your notation) A―n1−A―nn.Immediately obvious undesirable consequences are that the Wald CI degenerates to a point at 0 or 1 if the observed proportion of Successes is 0 or 1, respectively.SNKPerhaps more seriously, the actual coverage probability of the Wald interval differs greatly from one value of p to another, and that coverage probability is often below the intended 100(1−α) percent. An easily accessible article discussing this point is Brown, Cai, and DasGupta (2001) in Statistical Science.The graphs shown in the second link are for n=25. You have used n=1000 for your example. The Wald interval is based on asymptotic theory and so works better for a thousand observations than for a few dozen, but in general the weaknesses remain.

Weaknesses of Wald confidence interval for binomial distribution A survey samples 1000 people, a

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