Probstat/notes/confidence intervals

This is part of probstat

Suppose that we take a sample of size $n$, $X_{1},X_{2},\ldots ,X_{n}$ from a population which is normally distributed. Also suppose that the population has mean $\mu$ and variance $\sigma ^{2}$. In this section, we assume that we do not know $\mu$ but we know the variance $\sigma ^{2}$. The case we the variance is unknown will be discussed here.

We would like to estimate the mean. Therefore, we compute the sample mean ${\bar {X}}$. It is very certain the ${\bar {X}}\neq \mu$, but we hope that it will be close to $\mu$. In this section, we try to quantify how close the sample mean to the real mean.

Recall that the random variable ${\bar {X}}$ is a normal random variable with mean $\mu$ and s.d. $\sigma /{\sqrt {n}}$ Therefore,

${\sqrt {n}}({\bar {X}}-\mu )/\sigma$

will be a unit normal random variable.

We consider how ${\bar {X}}$ deviates from the true mean $\mu$.

To be added