Confidence
Table of contents
Confidence Intervals
There is a fixed true parameter $\theta$ that we want to estimate.
A confidence interval $C_n = (a, b)$ is a random interval, and the calculation of $a$ and $b$ is a function of the sample $X_1, \dots, X_n$.
$C_n$ is a random variable.
Since $C_n$ is a random variable, there can be many realizations of $C_n$ depending on the data.
If the calculation of $C_n$ is repeated many times, and $C_n$ contains $\theta$ with probability $1 - \alpha$, then $C_n$ is a $1 - \alpha$ confidence interval for some significance level $\alpha$.
$$ P(\theta \in C_n) \geq 1 - \alpha $$
Normal Interval
The quantile or the inverse CDF of the standard normal distribution tells us the value of $z$ such that:
\[P(-z \leq Z \leq z) = 1 - \alpha\]where $Z \sim N(0, 1)$.
For a sample mean $\overline{X}$, we know that by CLT:
\[\frac{\overline{X} - \mu}{\sigma / \sqrt{n}} \sim N(0, 1)\]Then, we can use the quantile of the standard normal distribution to say that the following is a $1 - \alpha$ confidence interval for $\mu$:
\[\overline{X} - z \frac{\sigma}{\sqrt{n}} \leq \mu \leq \overline{X} + z \frac{\sigma}{\sqrt{n}}\]