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 $\mu$. To do so, we compute the sample mean ${\bar {X}}$. It is very certain that ${\bar {X}}\neq \mu$, 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. More precisely, we would like to find an error range $\beta$ such that we have some confidence that

${\bar {X}}-\beta \leq \mu \leq {\bar {X}}+\beta$,

i.e., that $\mu$ lies within ${\bar {X}}\pm \beta$ (or in the range $({\bar {X}}-\beta ,{\bar {X}}+\beta )$).

When computing $\beta$, we usually specify the level of confidence $1-\alpha$ that we want to get. This level of confidence $1-\alpha$ is the probability that if we take the sample $X_{1},X_{2},\ldots ,X_{n}$ of size $n-1$ and compute ${\bar {X}}$, the real mean $\mu$ is in the range $({\bar {X}}-\beta ,{\bar {X}}+\beta )$.

As discussed in the the last section, that the random variable ${\bar {X}}$ is a normal random variable with mean $\mu$ and s.d. $\sigma /{\sqrt {n}}$, i.e.,

${\bar {X}}\sim Normal(\mu ,\sigma ^{2}/n)$

Remarks: When we say that $A\sim Normal(a,b)$ we mean that a random variable $A$ is normally distributed with mean $a$ and variance $b$.

Therefore, we have that

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

is a unit normal random variable.