Suppose we are conducting an electoral poll. Let X1, ..., Xn Bern(p) random variables that indicate whether (randomly sampled) voter i intends to vote for the Republican Candidate, George.

If we are able to truly randomly sample the voting population, we can estimate the proportion p of the voting population that intends to vote for the Republican candidate!using the sample mean. It will have expectation

p and standard deviation p(1p). n

The population, though, can be split into two distinct subgroups (lets say men and women), who represent proportions r1 and r2 of the overall population, respectively. So r1 + r2 = 1. In addition, they are likely to vote for Governor George at different rates: p1 and p2 respectively, so the overall population proportion is p = r1p1 +r2p2. Let's suppose our samplingprocess has issues, and on average, produces proportions q1 of men and q2 of women.

Let us suppose thatn=400,r1 =r2 =0.5,q1 =0.25q2 =0.75,p1 =0.4 and p2 = 0.6. What is the probability that, if we use the regular sample mean to estimate p, we overestimate the support for George by more than 0.05?