Section9.4Confidence Intervals for the Mean (\(\sigma\) unknown)
Up to this point we have assumed that the population standard deviation, \(\sigma\text{,}\) was known. This is unrealistic -- since we are creating an interval to estimate the population mean, \(\mu\text{,}\) we likely don’t know the population standard deviation either! Hence we will estimate \(\sigma\) with the value of the sample standard deviation, \(s\text{.}\) But this introduces another source of unreliability, especially in small samples. To keep the confidence interval at the desired level, we make the intervals wider by replacing the critical values in our confidence interval formula, \(z_{\alpha/2}\text{,}\) with larger critical values, \(t_{\alpha/2}\text{.}\)
The larger critical values come from the Student’s \(t\)-distribution developed in 1908 by an Irish brewing employee, William S. Gosset. He was a Guinness Brewery employee researching new methods of manufacturing ale. He needed a distribution that could be used with small samples. Employees were not allowed to publish research results, so he published under the pseudonym, “Student”.
It is a family of curves based on the concept of degrees of freedom (df), which refer to the number of values that are free to vary. As the degrees of freedom increases, the shape of the \(t\)-distribution becomes similar to the normal distribution.
The \(t\)-distribution is flatter and wider than the normal distribution. This means that the critical score for the \(t\)-distribution is therefore higher than the critical \(z\)-score for the same confidence level. This results in wider confidence intervals when using the \(t\)-distribution.
How do we find the critical values, \(t_{\alpha/2}\text{?}\)
Table: Use this link https://www.brockport.edu/live/files/6866-studentstdistributiontablepdf (or Table 5 in Appendix A of the textbook). Note that you will locate the degrees of freedom along the left column and the confidence level across the top of the table -- the desired critical value is located where the two meet inside the table.
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