Question

When σ is unknown and the sample is of size n ≥ 30, there are two methods for computing confidence intervals for μ.

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 ≥ 30, use the sample standard deviation s as an estimate for σ, 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 σ. 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 = 41, with sample mean x = 45.0 and sample standard deviation s = 5.7.

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

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

	90%	95%	99%
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(c) Now consider a sample size of 81. Compute 90%, 95%, and 99% confidence intervals for μ using Method 1 with a Student's t distribution. Round endpoints to two digits after the decimal.

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

	90%	95%	99%
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Answer 1