Whenis unknown and the sample is of sizen30, there are two methods for computing confidence intervals for.

Method 1: Use the Student'stdistribution withd.f.=n1.

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: Whenn30, use the sample standard deviationsas an estimate for, and then use the 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'stdistribution approach those of the standard normal distribution.

Consider a random sample of sizen=36, with sample meanx=45.3and sample standard deviations=5.0.(a)

(a) Compute a 99% confidence intervals forusing Method 1 with a Student'stdistribution. Round endpoints to two digits after the decimal.

(b)Compute a 99% confidence interval forusing Method 2 with the standard normal distribution. Usesas an estimate for. Round endpoints to two digits after the decimal.

(c) Compare intervals for the two methods. Would you say that confidence intervals using a Student'stdistribution are more conservative in the sense that they tend to be longer than intervals based on the standard normal distribution?

(d) Now consider a sample size of81. Compute a 99% confidence interval forusing Method 1 with a Student'stdistribution. Round endpoints to two digits after the decimal.

(e) Compute a 99% confidence interval forusing Method 2 with the standard normal distribution. Usesas an estimate for. Round endpoints to two digits after the decimal.

(f) Compare intervals for the two methods. Would you say that confidence intervals using a Student'stdistribution are more conservative in the sense that they tend to be longer than intervals based on the standard normal distribution?

With increased sample size, do the two methods give respective confidence intervals that are more similar?