第9章1標本による仮説検定

Central Limit Theorem

Given a random variable (RV) with known mean $\mu$ and known standard deviation σ. We are sampling with size n and we are interested in two new RVs - the sample mean, $\overline{X}$, and the sample sum, $\Sigma X$. If the size n of the sample is sufficiently large, then $\overline{X}~N\left(\mu \text{,}\frac{\sigma }{\sqrt{n}}\right)$ and $\Sigma X~N(n\mu ,\sqrt{n}\sigma )$. If the size n of the sample is sufficiently large, then the distribution of the sample means and the distribution of the sample sums will approximate a normal distribution regardless of the shape of the population. The mean of the sample means will equal the population mean and the mean of the sample sums will equal n times the population mean. The standard deviation of the distribution of the sample means, $\frac{\sigma }{\sqrt{n}}$, is called the standard error of the mean.