When σ is unknown and the sample is of size n≥ 30, there are two methods for computing confidence intervals forμ.
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 σ, and then usethe standard normal distribution.
This method is based on the fact that for large samples, sis a fairly good approximation for σ. Also, for largen, the critical values for the Student's tdistribution approach those of the standard normaldistribution.
Consider a random sample of size n = 41, with samplemean x = 45.7 and sample standard deviation s =6.4.
(a) Compute 90%, 95%, and 99% confidence intervals forμ using Method 1 with a Student's t distribution.Round endpoints to two digits after the decimal.
| 90% | 95% | 99% |
| lower limit    | | | |
| upper limit    | | | |
(b) Compute 90%, 95%, and 99% confidence intervals for μusing Method 2 with the standard normal distribution. Uses as an estimate for σ. Round endpoints to twodigits after the decimal.
| 90% | 95% | 99% |
| lower limit    | | | |
| upper limit    | | | |
(c) Now consider a sample size of 71. Compute 90%, 95%, and 99%confidence intervals for μ using Method 1 with a Student'st distribution. Round endpoints to two digits after thedecimal.
| 90% | 95% | 99% |
| lower limit    | | | |
| upper limit    | | | |
(d) Compute 90%, 95%, and 99% confidence intervals for μusing Method 2 with the standard normal distribution. Uses as an estimate for σ. Round endpoints to twodigits after the decimal.
| 90% | 95% | 99% |
| lower limit    | | | |
| upper limit    | | | |
