Expand Your Knowledge: Alternate Method for Confidence

Chapter , Problem 21

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Expand Your Knowledge: Alternate Method for Confidence Intervals When \(\sigma\) is unknown and the sample is of size \(n\ \geq\ 30\), there are two methods for computing confidence intervals for \(\mu\).

Method 1: Use the Student’s t distribution with d.f. = n - 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 \(n\ \geq\ 30\), use the sample standard deviation s as an estimate for \(\sigma\), 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 \(\sigma\). 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 = 31, with sample mean \(\overline{x} = 45.2\) and sample standard deviation s = 5.3.

(a) Compute 90%, 95%, and 99% confidence intervals for \(\mu\) 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 \(\mu\) using Method 2 with the standard normal distribution. Use s as an estimate for \(\sigma\). 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 = 81. With increased sample size, do the two methods give respective confidence intervals that are more similar?