# Intervals Based on a Normal Population Distribution: The T-Distribution

Posted by Beetle B. on Sun 16 July 2017

$$\newcommand{\Cov}{\mathrm{Cov}}$$ $$\newcommand{\Corr}{\mathrm{Corr}}$$ $$\newcommand{\Sample}{X_{1},\dots,X_{n}}$$

What if $$n$$ is not large? Then the CLT doesn’t apply. We must then know/assume a distribution.

Suppose it is normal with known $$\mu,\sigma$$, both unknown.

The rv $$T=\frac{\bar{X}-\mu}{S/\sqrt{n}}$$ has a pdf called the t-distribution with $$n-1$$ degrees of freedom.

The numerator and the denominator are independent random variables.

Note that now that $$n$$ is small, we cannot use $$S\approx\sigma$$.

## Properties of t-Distributions

The degrees of freedom are denoted by $$\nu$$. Let $$t_{\nu}$$ be the pdf:

1. Each $$t_{\nu}$$ curve is bell shaped and centered at 0.
2. Each $$t_{\nu}$$ curve is more spread out than the normal curve.
3. As $$\nu$$ increases, the spread decreases.
4. As $$\nu\rightarrow\infty,t_{\nu}\rightarrow$$ normal curve.

Notation: Let $$t_{\alpha,\nu}$$ be the number such that the area of the curve to the right is $$\alpha$$. It is called the t-critical value.

Then:

\begin{equation*} P\left(-t_{\alpha/2,n-1}<T<t_{\alpha/2,n-1}\right)=1-\alpha \end{equation*}

And the $$100(1-\alpha)$$ CI for $$\mu$$ is $$\bar{x}\pm t_{\alpha/2,n-1}s/\sqrt{n}$$

If you want a one sided CI, it is $$\bar{x}\pm t_{\alpha,n-1}s/\sqrt{n}$$

Figuring out the needed $$n$$ is hard.