Sample size from a confidence interval? 
Consider a large lot of electronic components where there is interest
  in the proportion of T working components. Suppose it is know for sure
  that $$0.7 < T < 0.9$$ How many components ought to be randomly
  sampled to provide an 80% confidence interval with length less than
  0.01?

How do you approach this problem?

You have a standard deviation $\sqrt{T(1-T)}$ for a sample of size $1$.  If $0<T<1$ then $\sqrt{T(1-T)}\le1/2$.  But if it is known that $0.7<T$, then the standard deviation is $\le\sqrt{(0.7)(1-0.7)}\cong0.46$.  The standard error for a confidence interval based on a sample of size $n$ is therefore $\le0.46/\sqrt{n}$.  The length of the interval is about $2\cdot1.28\cdot0.46/\sqrt{n}$.  (The number $1.28$ is something you get from a table).  So you want to make $n$ big enough so that that is $\le0.01$.
