When σ is unknown and the sample is of size n≥ 30, there are two methods for computing confidence intervals forμ. (Notice that, When σ is unknown andthe sample is of size n < 30, there is only one methodfor constructing a confidence interval for the mean by using theStudent's t distribution with d.f. = n -1.)
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 = 30, with samplemean x = 45.2 and sample standard deviation s =5.3.
(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) Compare intervals for the two methods. Would you say thatconfidence intervals using a Student's t distribution aremore conservative in the sense that they tend to be longer thanintervals based on the standard normal distribution?
No. The respective intervals based on the tdistribution are shorter.
Yes. The respective intervals based on the tdistribution are shorter. Â
Yes. The respective intervals based on the tdistribution are longer.
No. The respective intervals based on the tdistribution are longer.
(d) Now consider a sample size of 50. 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 | | | |
(e) 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 | | | |
(f) Compare intervals for the two methods. Would you say thatconfidence intervals using a Student's t distribution aremore conservative in the sense that they tend to be longer thanintervals based on the standard normal distribution?
Yes. The respective intervals based on the tdistribution are longer.
Yes. The respective intervals based on the tdistribution are shorter.  Â
No. The respective intervals based on the tdistribution are longer.
No. The respective intervals based on the tdistribution are shorter.
With increased sample size, do the two methods give respectiveconfidence intervals that are more similar?
As the sample size increases, the difference between the twomethods remains constant.As the sample size increases, thedifference between the two methods is lesspronounced.    As the sample size increases,the difference between the two methods becomes greater.
