Confidence Intervals

(\(\sigma\) unknown)

To find a confidence interval when the population standard deviation is unknown one uses the formula:

\[\overline{x}-t_\alpha\cdot\left(\frac{s}{\sqrt{n}}\right)<\mu<\overline{x}+t_\alpha\cdot\big(\frac{s}{\sqrt{n}}\big)\]

The degrees of freedom are \(n-1\).

As before \(\alpha\) is the percent you want the confidence level to be (for example 90%). To find the correct \(t\)-value one looks for (\alpha\) at the bottom of the \(t\)-table, then follow up to get the \(t\)-value for two-tails!

Just like before there are some needed assumptions.

Assumptions for finding confidence interval for a mean when (\sigma\) is unknown.

  • The sample is a random sample.
  • Either \(n\geq30\) or the population is normally distributed when \(n<30\).

Confidence Intervals

(\(\sigma\) unknown)