Say you take a sample where \(n\) is not large. Then the CLT doesn’t apply. We must then know/assume a distribution.
Suppose we know it is normal with known \(\mu,\sigma\), both unknown.
The random variable \(T=\frac{\bar{X}-\mu}{S/\sqrt{n}}\) has a pdf called the t-distribution with \(n-1\) degrees of freedom.
The numerator and the denominator are independent random variables (not proven).
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:
- Each \(t_{\nu}\) curve is bell shaped and centered at 0.
- Each \(t_{\nu}\) curve is more spread out than the normal curve.
- As \(\nu\) increases, the spread decreases.
- 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:
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.