Realistically, the value of \(\sigma\) is almost never known.

• We know that we can estimate \(\sigma\) using \(s\).
• Can we plug in \(s\) for \(\sigma\)? Sometimes!

• For samples of size \(n \ge 30\), \(\bar{X}\) will be approximately normal even if \(X\) isn’t.

• In this case, we can plug in \(s\) for \(\sigma\): \[\bar{x} \pm z_{\alpha/2}\frac{s}{\sqrt{n}}.\]

…but what about when \(n < 30\)?

If \[Z = \frac{\bar{X}-\mu}{\sigma/\sqrt{n}}\] has a standard normal distribution (for \(X\) normal or \(n\ge30\)), the slightly modified \[T = \frac{\bar{X}-\mu}{s/\sqrt{n}}\] has what we call the t-distribution with \(n-1\) degrees of freedom.

The only thing we need to know about degrees of freedom is that \(\text{df}=n-1\) is the t-distribution’s only parameter.

• The t-distribution is symmetric and always centered at 0.
• When \(n\ge30\), the t-distribution is approximately equivalent to the standard normal distribution.
• For smaller sample sizes, the t-distribution has more area in the tails.

• Plug in \(s\) for \(\sigma\) and use a t critical value \(t_{\text{df}, \, \alpha/2}\).
  • The t critical value is the \([1- \alpha/2]\)th percentile of the t-distribution with \(n-1\) degrees of freedom.
• The resulting 95% confidence interval is \[\bar{x} \pm t_{\text{df}, \, \alpha/2}\frac{s}{\sqrt{n}}.\]

Rossman and Chance t Probability Calculator

• For this applet, enter the degrees of freedom \(n-1\) next to “df”.
• Then check the top box under “t-value probability” and make sure the inequality is clicked to “>” .
• Enter the value of \(\alpha/2\) for the probability.
• Click anywhere else on the page and the applet will automatically fill in the box under “t-value”.
• This is your t critical value.