I am fresh to statistics, and I have been asked a question to aid my understanding of the concept of "bias" in statistical sampling, which I require some help in clarifying.
Suppose $p$ is the proportion of a population with a certain characteristic. We take a simple random sample $\{y_1,y_2,...y_n\}$ without replacement from the population. We can use $\hat{p}=\sum^n_{i=1}\frac{y_i}{n}$ to estimate the sample proportion. (For reference, $y_i=1$ if the $i^{th}$ element has the characteristic, and $y_i = 0$ otherwise.)
I am given that $E(y_i)=E(y_i^2)=p$, and $\sigma^2=Var(y_i)=p-p^2$. Suppose now I am to use $\tilde{\sigma}^2=\hat{p}-\hat{p}^2$ as an estimator to $\sigma^2$. Can I say if this would be an unbiased estimator, and if not, what would be the bias and how can I construct an unbiased estimator?
My understanding of bias is how far the estimated value "deviates" from the true value. In the case of variance, that would mean $\tilde{\sigma}^2=\hat{p}-\hat{p}^2$ does not match the population variance due to the sample size vs population size difference, in this case by a factor of $\frac{n}{N}$. Can someone kindly point out what is wrong with my understanding here, or guide me further along?
Thank you very much in advance!