Why does the mean of the sample proportion equal the population proportion. $E(\hat{P})=p$
The reasons that I can think of:
- As most of samples proportion you take lie on the range of the population proportion for a desired characteristic. (assuming large sample size)
But as most samples lie on the range of $p,$ why is it the average of sample proportion that equals $p$ and not the mode maybe?
I am really confused if anyone can explain this to me with examples would really help.
And why do we always talk about average/mean ?
An intuitive approach is to imagine flipping a biased coin that comes up $0\ \frac 23$ of the time and $1\ \frac 13$ of the time. The mode of the distribution is $0$, but you would not expect that if you flipped a lot of times you would always get $0$. You would expect to get $0\ \frac 23$ of the time and $1\ \frac 13$ of the time, so the average is $\frac 13$, the mean of the distribution.
For a real proof, look up the central limit theorem.