# Hypothesis test with simulated data

A simulation gave me the following density of data (grey histogram, blue kernel density estimation). Having an alternative input $$I^*$$ with the same distribution (green line), I would like to check whether $$I^*$$ belongs to my simulated data (hypothesis testing). Is it correct that I have to look at the quantiles of my approximately normal distributed data in order to verify my hypothesis? Say for $$\alpha = 0.05$$ I calculate the $$\alpha$$-quantiles and if $$I^*$$ is located in this area, I reject the hypothesis?

I'm a stranger to hypothesis-testing and appreciate any help. Thank you!

• I'm using simulated data, because the aim is to visualize some theoretical laws of convergence. But maybe some background: The data is created out of a specifal central limit theorem $n^{-1/2} (X_k - \mathbb{E}X_k) \rightarrow \mathcal{N}(0,\sigma^2)$. Given that I use the continuuos mapping theorem with $h: \mathbb{R}^k \rightarrow \mathbb{R}$. Therefore my random variables converge under $h$ to a normal distribution as well. Next I approach $\mathbb{E}X_k$ with a lot of simulations and then calculate $h(n^{-1/2} (X_k - \mathbb{E}X_k)) \rightarrow h(\mathcal{N}(0,\sigma^2))$ – nomeal Apr 10 at 18:39
• Now $I^*$ from above is one extra calculation $h(n^{-1/2} (\hat{X}_k - \mathbb{E}X_k))$. My aim is to show with $I^*$, whether I accept or reject $H_0 : X_k \text{ is poisson data}$. – nomeal Apr 10 at 18:44
• So to answer your second argument directly: Which seems like a single data point, is in fact $k$ data points, before beeing mapped by $h$. Do you understand my point? – nomeal Apr 10 at 18:48