# The number of coin tosses needed if the proportion of heads is to lie within 0.05 of p with probability at least 0.9?

There's a question I'm not really sure if I did it right or even understand what its trying to say.

There is a coin which produces heads with an unknown probability p. How many times should we throw this coin if the proportion of heads is to lie within 0.05 of p with probability at least 0.9? Hint: Answer is not complete if it relies on p and do not worry about continuity correction.

This question comes from a chapter about binomial normal approximation.

So far I know I have a B~(n,p) and I need to find the number of n so that P ( c/n <= 0.05*p) >=0.9 with c as the number of heads? I'm not sure if this is right.

In turn P( c <= 0.05pn ) >=0.9

So I'll normalize the binomal such that

P( Z <= [(0.05pn - n*p)/(np(1-p)] +0.5 ) >=0.9

with 0.5 as the continuity correction.

Am I on the right track? I thought about using confidence intervals but we haven't discussed about that in the chapter or lecture.

• You want $P[|c/n - p| \leq 0.05] \geq 0.9$ Mar 29, 2014 at 2:14
• Thanks I had trouble trying to understand what the question wants.
– user131516
Mar 29, 2014 at 2:33

I believe you are expected to use the normal distribution approximation. From the normal distribution table, $1.645$ standard deviations around the mean include (just over) $0.9$ of the area. For $N$ flips the standard deviation in the number of heads is $\sqrt{Np(1-p)}$, which is largest at $p=\frac 12$, so $\sigma \leq\frac 12 \sqrt N$. We want $1.645\sigma \le\frac 121.645\sqrt N\lt 0.05N$, so $N \gt 16.45^2\approx 270.6$ Take $N=271$

• Yeah the topic is about Large Sample theory Normal Approximation. However why is 1.645sigma less that 0.05N?
– user131516
Mar 29, 2014 at 2:35
• The allowable error is $0.05N$ from the problem statement. The argument before shows why (at the $90\%$ confidence level) we expect to be within $1.645 \sigma$ Mar 29, 2014 at 2:42
• I think I understand? I'm just having a bit of trouble conceptualizing it.
– user131516
Mar 29, 2014 at 4:34
• Conceptualizing is not a mathematical word. This is a sketch of a proof-what particular step is the problem? We don't really go back to the fundamentals each time-we try to make a statement that the audience will accept. Just like the proofs in your book, you should look at each step and see if you accept it. If not, ask another question until you do. Mar 29, 2014 at 4:42
• I think I do understand, Thanks for your time.
– user131516
Mar 29, 2014 at 6:42