When ? is unknown and the sample is of size n? 30, there are two methods for computing confidence intervals for?.
Method 1: Use the Student's t distribution withd.f. = n ? 1.
This is the method used in the text. It is widely employed instatistical studies. Also, most statistical software packages usethis method.
Method 2: When n ? 30, use the sample standarddeviation s as an estimate for ?, and then usethe standard normal distribution.
This method is based on the fact that for large samples, sis a fairly good approximation for ?. Also, for largen, the critical values for the Student's tdistribution approach those of the standard normaldistribution.
Consider a random sample of size n = 31, with samplemean x = 44.4 and sample standard deviation s =5.9.
(d) Now consider a sample size of 81. Compute 90%, 95%, and 99%confidence intervals for ? using Method 1 with a Student'st distribution. Round endpoints to two digits after thedecimal.
| 90% | 95% | 99% |
| lowerlimit | | | |
| upperlimit | | | |
(e) Compute 90%, 95%, and 99% confidence intervals for ?using Method 2 with the standard normal distribution. Uses as an estimate for ?. Round endpoints to twodigits after the decimal.
| 90% | 95% | 99% |
| lowerlimit | | | |
| upperlimit | | | |
