According to some folklore, if you examine n clover plants in a field and do not find any with four leaves, you may be reasonably confident that the fraction of four-leafed clovers is less than 3/n. Justify the folklore. You may assume n is reasonably large, but state your assumptions.
I assumed n is reasonably large, with n = 1000. Lets say the parameter is $\theta$, and we can assume $X_{i}$ $\sim Ber(\theta)$. Thus we can also assume that because n is approximately large, we can invoke a binomial approx. to the normal.
Thus:
$H_{0}: \theta = 0.003$
$H_{a}: \theta < 0.003$
Y = $\sum X_{i}$ is binomially distributed with $u = np = 1000(0.003) = 3$
$\sigma = \sqrt{1000 *0.003*0.997} = 1.73$
$P(Y \leq 0.5) = \frac{0.5 - 3}{1.73} \sim N(0,1)$
$P(Z \leq -1.445) = .074$
For $\alpha$ level of $0.10$ you may reject the null hypothesis and reasonably conclude that $\theta < 0.003$
The continuity correction factor shifts the observed Y to $0.5$
Is this correct? Are my assumptions valid?
The normal approximation to the binomial is not justified here as you are at the end of the tail of the binomial distribution.
A Poisson approximation to the binomial would be justifiable, or you could just say the probability of seeing zero four-leaf clovers is $(1-\theta)^n$ which for small $\theta$ and large $n$ is about $e^{-n/\theta}$.
If we want the probability of seeing seeing zero four-leaf clovers to be $5\%$ or more under the null hypothesis that the parameter is $\theta$, then we want $e^{-n/\theta} \ge 0.05$ so $\theta \le -\dfrac{\log_e(0.05)}{n}\approx \dfrac{2.996}{n}\approx \dfrac{3}{n}$.
This is known as the rule of three.