When \(\displaystyle\sigma\) is unknown and the sample is of size \(\displaystyle{n}\geq{30}\), there are two methods for computing confidence intervals for \(\mu\). Method 1: Use the Student’s t distribution with d.f.=n-1. This is the method used in the text. It is widely employed in statistical studies. Also, most statistical software packages use this method.

Method 2: When \(\displaystyle{n}\geq{30}\), use the sample standard deviation s as an estimate for \(\displaystyle\sigma\), and then use the standard normal distribution. This method is based on the fact that for large samples, s is a fairly good approximation for \(\displaystyle\sigma\). Also, for large n, the critical values for the Student’s t distribution approach those of the standard normal distribution. Consider a random sample of size n=31, with sample mean \(\displaystyle\overline{{{x}}}={45.2}\) and sample standard deviation s=5.3. Compute 90%, 95%, and 99% confidence intervals for \(\displaystyle\mu\) using Method 2 with the standard normal distribution. Use s as an estimate for \(\displaystyle\sigma\). Round endpoints to two digits after the decimal.