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Let say, I have a population of size $n$ and the probability of death is $p$ which is unknown.

My goal is to estimate that $p$.

I observe that there are $k$ deaths.

I was told that, if I want to estimate the upper limit for $p$ with confidence level 90% then I should solve for $p$ from the equation $P\left[N > k\right] = 0.90$ where $N$ is Binomial distribution with probability $p$.

However I failed to get the insight of above approach. Why I should solve that equation?

Can you please help me to get that insight?

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    $\begingroup$ What is the parameter N?? $\endgroup$
    – Fiodor1234
    Dec 20, 2020 at 19:38
  • $\begingroup$ $N$ is the r.v. which correspond the number of deaths which follows a Binomial distribution $\endgroup$ Dec 20, 2020 at 21:18

1 Answer 1

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Let us say that you want to manage the abundance of an organism, one way to achieve that is by regulating the probability of death. You might regulate the probability of death by making their living environment more rich or secure or whatever.

You go out on the field, and from the $n$ individuals you observe $k$ deaths, so the random variable of the number of death $N$ follows a Binomial distribution.

You may estimate the sampled death probability by $k/n$, and you might find that it is much lower than what you expected.

So you want to regulate things, and want to achieve a higher number of deaths. This can be achieved by determining a probability of death that will result to your desired number of deaths i.e solving $P(N>k)=0.9$ for $p$. This will give you a probability of death that will lead to deaths that are more than what you observed with probability 0.9.

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  • $\begingroup$ But how that estimated value corresponds to the Upper C.I. for the true $p$? $\endgroup$ Dec 20, 2020 at 21:17
  • $\begingroup$ If you want the C.I thought that probability, you have to use the Normal approximation for large number of samples. Check this statisticshowto.com/binomial-confidence-interval/…. You will begging by taking the probability that you displayed, you will substract the mean and divide by the standard deviation of $N$, in both the sides of the inequality. And you will try to solve for $p$. The calculation is not complicated. $\endgroup$
    – Fiodor1234
    Dec 20, 2020 at 22:00

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