An opinion poll based on a sample of 50 subjects estimated p, the proportion of the population in favor of the proposition, as 0.72.

(i) Estimate the true proportion, θ, with a 95% con?dence interval. State any assumptions you may have to make in answering this question.

(ii) If the true population proportion is suspected to be θ = 0.8, and the estimate from an opinion poll is to be determined to within ±0.05 with 95% con?dence, how many people, n, should be sampled?

(iii) If the proportion is to be estimated to within the same margin of ±0.05, but with 90% con?dence, what is the value of n required? Comment on the effect that reducing the con?dence level has on the sample size, n required to achieve the desired precision.