In a mass production of items produced indepedently of eachother the probability of an item being defect is $p$. An unbiased estimator for $p$ is
$\hat p = \frac{X}{n}$ where $X$ = amount of items with a defect and $n$ is amount of produced items.
Assume $p = 0.12$ and $n = 55$. Use that $\hat p$ under these conditions is approximately normally distributed to calculate the probability that the unbiased estimator $\hat p$ deviates less than $0.05$ from the true $p$.
So I want to find $P(|\hat p -p| < 0.05)$. If i want to use the standard normal distribution I have to know $z = \frac{X-\mu}{\sigma}$.
Given $p=0.12$, I find $E(X) = 6.6$. And since we want $\hat p$ to deviate less than $0.05$, I set $\sigma = 0.05$.
if $X$ deviates less than $0.05$ the maximum amount of defects can be $6.05$ and the minimum $5.95$. Using this to find what values $Z$ has to be between gives me big numbers, which can't be true... What have I done wrong?
Using $\hat{p}$ as an estimate makes us $100(1-\alpha)\%$ confident that the error will not exceed $$z_{\alpha / 2} \sqrt{\hat{p} \hat{q} / n}$$
If we want the error to be $.05$, then we have $$.05 = z_{\alpha / 2} \sqrt{(.12)(.88)/ 55}$$
When I solved I got $z_{\alpha /2} = 1.141$, which implies that $\alpha \approx 0.254$
So the probability that the estimate $\hat{p}$ doesn't deviate from the actual value of $p$ by more than $.05$ is $100(1-\alpha)\% = 74.6\%$