Confidence

Table of contents

1. Confidence Intervals
2. Normal Interval

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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 $\alpha$.

$$P(\theta \in C_n)\ge 1-\alpha$$

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Normal Interval

The quantile or the inverse CDF of the standard normal distribution tells us the value of $z$ such that:

$$P(-z\le Z\le z)=1-\alpha$$

where $Z\sim N(0,1)$.

For a sample mean $\bar{X}$, we know that by CLT:

$$\frac{\bar{X}-\mu }{\sigma /\sqrt{n}}\sim N(0,1)$$

Then, we can use the quantile of the standard normal distribution to say that the following is a $1-\alpha$ confidence interval for $\mu$:

$$\bar{X}-z\frac{\sigma }{\sqrt{n}}\le \mu \le \bar{X}+z\frac{\sigma }{\sqrt{n}}$$