Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

A sample of size $ n = 20 $ is drawn from a population with population proportion $ p = 0.40 $. Assume that the sample size is less than or equal to $ 5 \% $ of the population. Let $ \hat{P} $ be the sample proportion.

Question: Find the mean.

I have no clue how to find the mean in this case.

share|cite|improve this question
is this binomial distribution?check it if is helpful – dato datuashvili Jan 13 '13 at 7:41
@OP Please do not deface your post like this. – user38268 Jan 13 '13 at 8:53
@Joe: I have rolled back your question. Do not deface your post especially when someone has invested time in your question and answered it. – user17762 Jan 13 '13 at 8:55

Let $ X $ be the number of successes recorded in the sample, which can be viewed as a collection of $ 20 $ Bernoulli trials. Then $ X \sim \text{Binom}(20,0.4) $.

The sample proportion, by definition, is $ \hat{P} = \dfrac{X}{20} $; its mean (expectation) is calculated as follows: \begin{align} \text{E}[\hat{P}] &= \text{E} \left[ \frac{X}{20} \right] \\ &= \frac{1}{20} \cdot \text{E}[X] \\ &= \frac{1}{20} \cdot (20)(0.4) \\ &= 0.4. \end{align}

If you increase the sample size $ n $, then by the Central Limit Theorem, the probability distribution of $ \hat{P} $ is approximately $ \text{N} \left( p,\dfrac{p(1 - p)}{n} \right) $.

share|cite|improve this answer
do we need to know that sample size is less by $5$% from population size? – dato datuashvili Jan 13 '13 at 8:12
Usually, one applies the Central Limit Theorem to a very large sample. However, we only have $ n = 20 $ in this case. – Haskell Curry Jan 13 '13 at 8:17
generally we can't directly calculate population proportion from sample size and sample proportion right? – dato datuashvili Jan 13 '13 at 8:19
Yes, that’s true. However, if the sample size is large, then we can construct confidence intervals for $ p $ using the Central Limit Theorem. This is basically what you have shown. :) – Haskell Curry Jan 13 '13 at 8:24
However, the OP has already specified $ p $ as $ 0.4 $, so I don’t think that is what he’s trying to do. – Haskell Curry Jan 13 '13 at 8:26

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.