Expand Your Knowledge: Alternate Method for Confidence Intervals When s is unknown and the sample is of size n $ 30, there are two methods for computing confidence intervals for m.

Method 1: Use the Student’s t distribution with d.f.5n 2 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 n $ 30, use the sample standard deviation s as an estimate for s , 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 s . 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 5 31, with sample mean x 5 45.2 and sample standard deviation s 5 5.3.

(a) Compute 90%, 95%, and 99% confidence intervals for m using Method 1 with a Student’s t distribution. Round endpoints to two digits after the decimal.

(b) Compute 90%, 95%, and 99% confidence intervals for m using Method 2 with the standard normal distribution. Use s as an estimate for s . Round endpoints to two digits after the decimal.

(c) Compare intervals for the two methods. Would you say that confidence intervals using a Student’s t distribution are more conservative in the sense that they tend to be longer than intervals based on the standard normal distribution?

(d) Repeat parts

(a) through

(c) for a sample of size n 5 81. With increased sample size, do the two methods give respective confidence intervals that are more similar?

AppendixLO1