SOLUTION: About 80% of property crimes are never solved. How many crimes will need to be under investigation in order to be 93% confident of solving more than 1?

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Question 978080: About 80% of property crimes are never solved. How many crimes will need to be under investigation in order to be 93% confident of solving more than 1?
Answer by Boreal(15235) (Show Source):
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Confidence intervals for binomial functions are very difficult to use. The problem is that the normal approximation doesn't work well with small numbers. This is a very long way around but does arrive at the answer.

Suppose n=20
p=0.2
Expected value is 4. Need to have a CI containing 2. z-value for z.93=1.81
This is close enough where a normal approximation isn't too bad (np>5 is better).
CI=+/- 1.81*SE
0.20-(1.81*SE) *n >2; Let's look at the first part.
{ (0.20)-1.81*sqrt [0.16/n]}=0.20-1.81*0.4/sqrt (n)=0.20- (0.724/sqrt (n))
Now put it together.
{0.20-[0.724/n^(1/2)]} *n >2
I'd substitute numbers for n, starting with 25
0.20-[0.724/5]*25=1.38, which is too small. I would increase in increments of 5 to avoid fractional people.
Then 30
0.20-(0.724/sqrt (30)) *30=2.03
One would need at least 30.