Intrepretation of Bootstrap method in a simple example, with uniform population to infer. In order to understand the functionality of bootstrap, i may use a population with uniform distribution to infer.
We can generate a sample of 50 points from a uniform distribution $U(0, 1)$ with $\mu=0.5$, and $\sigma=0.2887$. An example result (using Matlab) is here :
$$ \bar{x} = 0.5698,\: \: s =0.2952 , \: \: \: (\text{from 50 random points})$$
Using $ \hat{\theta} = \sum_{i=1}^{50} X_{i}/50 $ as the estimator for the populatiok mean $\mu$, the bootstrap result (with $k=1000$ iterations) is here :
$$ \bar{xb}=0.5707, \: \: sb =0.043 , $$
So, by this 1000 resampling, the bootstrap mean will be closer to the sample mean. But this does not infer anything about the population's parameter.
By the CLT, the distribution of the sample mean would be normal $N(\mu=0.5, \: \sigma=\frac{0.2887}{\sqrt{50}}=0.0408)$. The standard deviation of the bootstrap resampling is close to 0.0408, the standard deviation of sample mean distribution of the population.
From this experiment, the only functionality of bootstrap resampling that i can see is that we can infer the standard deviation of the sample mean distribution. Is this statement true? (Is this the true functionality of bootstrap resampling?)
I have read some statements about bootstrap method, they say it is effective and does not require any assumptions about the population's distribution. But i have not really understand how to properly use this method.
Some insights on this will be appreciated. Thanks. Regards, Arief.
 A: Relating to mr.Bruce's @BruceET answer.
Generating the empirical bootstrap distribution $B = \frac{\bar{X_{b}} - \bar{x}}{\sigma/\sqrt{n}}$ (population $\sigma$ is known) using Python :
This time, I generate only $n=10$ samples from the standard uniform distribution.
The sample mean is obtained $ \bar{x} = 0.3327 $, so this is not close enough to the population mean $\mu = 0.5$.


*

*with number of bootstrap resampling $n_{b} = 10,000$


I get the plot and compare it with the empirical standard-t from the uniform population ($9$ degrees of freedom) by generating $30,000$ random samples of standard-t (which should be close to the exact one) :

The result above shows that : if $\sigma$ is known, with only $n=10$ samples we can get nice approximation for the population. Although the sample mean $\bar{x}$ is relatively far from the hidden $\mu = 0.5$.
Generating the empirical bootstrap distribution $B = \frac{\bar{X_{b}} - \bar{x}}{\sigma/\sqrt{n}}$ ($\mu$ and $\sigma$ are not known) :
For this case, I presume we may use $S^{2}$ to substitute the $\sigma$. With different $n=10$ samples from the previous case $ \bar{x}=0.3875$, I get the plots :



in Summary : i may say that the results above is one of main applications of bootstrap method?

Thanks. All the best.
