# Thread: binomial distributions and limit theorems!!!

1. ## binomial distributions and limit theorems!!!

find th smallest value of n in a binomial distribution for which we can assert:
P{ |(Xn/n) – p| < .1} ≥ .9

2. Originally Posted by lauren2988
find th smallest value of n in a binomial distribution for which we can assert:
P{ |(Xn/n) – p| < .1} ≥ .9
Well this translates into: find the smallest $\displaystyle n$ such that:

$\displaystyle P(p-0.1<\frac{X_n}{n} <p+0.1) \ge 0.9$

where the rv $\displaystyle X_n$ the number of successes in $\displaystyle n$ independent trials with probability of success in a single trial of $\displaystyle p$. Now given the title of this thread I assume we are supposed to use a normal approximation to the binomial to do this, in which case we have:

$\displaystyle \frac{X_n}{n} \sim N(p, p(1-p)/n)$.

$\displaystyle 90\%$ of the probabilty mass of a normal RV is contained within $\displaystyle \pm 1.645$ standard deviations of the mean, so if $\displaystyle p>0.1$ we have:

$\displaystyle 0.1 \ge 1.645 \sqrt{p(1-p)/n}$

which will allow you to find the smallest $\displaystyle n$ for which this is true.

Now you can do the case where $\displaystyle p<0.1$ yourself.

CB