Let $X_{1}, X_{2}, X_{3}, X_{4}, X_{5}$ be a random sample from a binomial distribution with $n=10$ and $p$ unknown.
How do I show that $\frac{\bar{X}}{10}$ is an unbiased estimator for $p$ and then estimate $p$ based on the data: 3,4,4,5,6?
My try
In the book I am reading the following definition is made: An estimator $\hat{\theta}$ is an unbiased estimator for a parameter $\theta$ if and only if $E[\hat{\theta}]=\theta$. Can I use this to show the above?
In this case we have $E[\frac{\bar{X}}{10}] = \frac{1}{10}E[\bar{X}] = \frac{\mu}{10}$, is that equal to $p$ and if so, why? If it is then the estimated $p$ based on the data 3,4,4,5,6 becomes $\frac{\frac{3+4+4+5+6}{2}}{10} = \frac{11}{10} $.
Unbiased does not mean, that one obtains the correct estimate for all realizations. It means that the average of the estimator is equal to the correct value. Practically speaking, if you were to repeat the experiment of estimating $p$ using $\bar{X}$ many times, the average of all estimates would be close to $p$. However, a single estimate can very well be far from the actual correct value.
In your case, the estimator is unbiased, since
$$ E\left[\frac{\bar{X}}{10}\right] = \frac{1}{10} E\left[\bar{X}\right] = \frac{1}{10} E\left[\frac{X_1+X_2+X_3+X_4+X_5}{5}\right] = \frac{1}{50} \sum_{i=1}^5 E[X_i]. $$ With $E[X_i] = 10p$ as the mean of the binomial distribution,
$$ E\left[\frac{\bar{X}}{10}\right] = \frac{50}{50}p = p.$$
This is a special case of a general result on mean estimators, see, e.g., https://www.statlect.com/fundamentals-of-statistics/mean-estimation