# Most Conservative Estimate : $p = 0.5$

Question :

An Electronics Company has just installed a new machine that makes a part that is used in clocks. The company wants to estimate the proportion of these parts produced by this machine that are defective. The company manager wants this estimate to be within 0.02 of the population proportion for a 95% confidence level. What is the most conservative estimate of the sample size that will limit the maximum error to within 0.02 of the population proportion?

Ans : ... For the most conservative estomate of the sample size, we use $p=0.50$ and $1-p=0.50$. ...

I want to ask why $p=0.5$ is used for the most conservative estimate. When it comes to the most conservative estimate, I always use $p=0.5$ is OK?

• Because, for every n, the variance of the binomial (n,p) is maximal when p=0.5. – Did May 2 '14 at 10:45

## 1 Answer

The variance of a binomial random variable (the number of defectives) is given by $np(1-p).$

When is this a maximum? Take the derivative with respect to $p$ and set equal to zero:

$$n-2np=0$$

$$1-2p=0$$

$$p=0.5$$

Checking the second derivative shows this is indeed the maximum. So, yes, you are OK.