Let $X_1,X_2,…,X_n$ be a random sample from the normal distribution with mean $\mu$ and variance $\sigma^2$.

Example of random sample after it is observed:

\[2.73, 1.14, 3.98, 2.15, 5.85, 1.97, 2.54, 2.03\]

Example of random sample before it is observed:

\[X_1,X_2,X_3,X_4,X_5,X_6,X_7,X_8\]

Based on what we are seeing, do we believe that the true population mean $\mu$ is

\[\mu\le3 \quad\text{or}\quad\mu\gt3\]

We have the sample mean is $\overline{x}=2.799$. It’s below $3$, can we say that $\mu\lt3$?

This seem awfully dependent on the random sample we happened to get!

Let try to work with the most generic random sample size of $8$:

\[X_1,X_2,X_3,X_4,X_5,X_6,X_7,X_8\]

Let $X_1,X_2,…,X_n$ be a random sample of size $n$ from the $N(\mu,\sigma^2)$ distribution.

We say that

\[X_1,X_2,...,X_n \stackrel{iid}{\sim}N(\mu,\sigma^2)\]

The sample mean is

\[\overline{X}=\frac{1}{n}\sum_{i=1}^{n}X_i\]

  • We’re going to tend to think that $\mu\lt3$ when $\overline{X}$ is “significant” smaller that 3.

  • We’re going to tend to think that $\mu\gt3$ when $\overline{X}$ is “significant” larger that 3.

  • We’re never going to observe $\overline{X}=3, but we may be able to be convinced that $\mu=3$ if $\overline{X}$ is not too far away.

How do we formalize this? We’re going to set up hypotheses:

\[\begin{align} &H_0: \mu\le3\quad\text{null hypothesis}\\ &H_1: \mu\gt3\quad\text{alternate hypothesis} \end{align}\]
  • The null hypothesis is assumed to be true.
  • The alternate hypothesis is what we are out to show.

Conclusion is either:

\[\text{Reject }H_0\quad\text{OR}\quad\text{Fail to reject }H_0\]

Suppose that $X_1,X_2,…,X_n$ is a random sample from a continous distribution with probability density function (pdf)

\[f(x;\theta)=\begin{cases} \begin{align} &e^{-(x-\theta)}&\quad,x\ge\theta\\ &0&\quad,x\lt\theta \end{align} \end{cases}\]

png

It’s shifted exponential pdf, and the parameter is unknown. Suppose we want to test these hypotheses:

\[H_0:\theta\ge1\quad\text{versus}\quad H_1:\theta\lt1\]

We can look at minimum of $x$, if the minimum is below $1$, then we know that the null hypothesis is not true, so we should go with the alternate. But if the minimum is a little bit above $1$, the null hypothesis may or may not be true and we can’t be sure.

A simple set of hypotheses:

\[\left. \begin{align} H_0:\mu=3\\ H_1:\mu\gt3 \end{align} \right\}\text{all posibilities in the parameter space}\]

Suppose we observe

\[\overline{x} = -59,349,348\]

We might be thinking that the mean is not greater than $3$. Then we probably fail to reject $H_0$.

Let $X_1,X_2,…,X_n$ be a random sample from the normal distribution with mean $\mu$ and variance $\sigma^2$.

Suppose that the variance $\sigma^2$ is known.

\[H_0:\mu=3\]

is called a simple hypothesis.

\[H_0:\mu\le3\]

is called a composite hypothesis.

The definition of a simple hypothesis is if you know the hypothesis is true, do you know the exact distribution that your random sample came from? If you do, the it’s simple.

Let $X_1,X_2,…,X_n$ be a random sample from the normal distribution with mean $\mu$ and variance $\sigma^2$.

\[H_0:\mu=3\]

is a composite hypothesis. Because we don’t know the variance.