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?

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): You can put this solution on YOUR website!
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.

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