# A sample of size n = 20 is drawn from a population with population proportion, p = 0.40. Find the mean

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.

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is this binomial distribution?check it if is helpful classweb.gmu.edu/tkeller/HANDOUTS/Handout10.pdf – dato datuashvili Jan 13 '13 at 7:41
– dato datuashvili Jan 13 '13 at 7:47
@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 $\text{N} \left( p,\dfrac{p(1 - p)}{n} \right)$.
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
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