p = fraction of large population that smokes
n = sample size
y = # in sample that smoke
The maximum likelihood estimate of p is p-hat = y/n
Consider the random variable Y and estimator F = Y/n
How could you work out P(-0.03 < F - p < 0.03) knowing n and p?
This is what I have so far, I'm just not sure how to get rid of the st.dev variable:
let x be the standard deviation
Y ~ G(p,x)
Y-bar ~ G(p, x/ sqrt n)
P (-0.03 < F - p < 0.03) = P(p - 0.03 < Y-bar < p + 0.03)
= P[ ((p - 0.03) -p)/(x/sqrtn) < (Y-bar - p) / (x/sqrt n) < ((p + 0.03)-p)/(x/sqrt n)]
= P (-0.03*sqrtn/x < Z < 0.03*sqrtn/x)
So how would I figure this out without knowing x (standard deviation)?