In a city, 20% of the people smoke but I don't know this value. To estimate it, you conduct a survey to 1000 people if they smoke or not. Use the central limit theorem to estimate the probability that your estimate is larger than 25%.
So far, I only figured out that the random variable in this problem should be Bernoulli's random variable. (Because it's either. Smoke or not on each survey?) I have no idea how to go to the next step. Please help!!
Suppose X~Binomial(n,p). $\hat{p}=\frac{X}{n}$ is an unbiased estimator of p, with $E[\hat{p}]=p$ and $Var[\hat{p}]=\frac{p(1-p)}{n}$.
By central limit theorem, we have $Z=\frac{\hat{p}-p}{\sqrt{\frac{p(1-p)}{n}}}$ ~ $N(0,1)$
So, $Pr(\hat{p}>0.25)=Pr(Z>\frac{0.25-0.2}{\sqrt{\frac{0.2(1-0.2)}{1000}}})=3.86\times 10^{-5}$