When σ is unknown and the sample is of size n ≥ 30, there aretwo methods for computing confidence intervals for μ. Method 1: Usethe Student's t distribution with d.f. = n − 1. This is the methodused 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 forσ, and then use the standard normal distribution. This method isbased on the fact that for large samples, s is a fairly goodapproximation for σ. Also, for large n, the critical values for theStudent's t distribution approach those of the standard normaldistribution. Consider a random sample of size n = 41, with samplemean x = 44.9 and sample standard deviation s = 6.5.
(a) Compute 90%, 95%, and 99% confidence intervals for μ usingMethod 1 with a Student's t distribution. Round endpoints to twodigits after the decimal.
90% 95% 99%
lower limit 43.19 42.85 42.16 Â Â
upper limit   46.61 46.95 47.65
(b) Compute 90%, 95%, and 99% confidence intervals for μ usingMethod 2 with the standard normal distribution. Use s as anestimate for σ. Round endpoints to two digits after thedecimal.
90% 95% 99%
lower limit 43.23 42.91 42.28
upper limit 46.57 46.89 47.52
(c) Compare intervals for the two methods. Would you say thatconfidence intervals using a Student's t distribution are moreconservative in the sense that they tend to be longer thanintervals based on the standard normal distribution?
No. The respective intervals based on the t distribution areshorter.
No. The respective intervals based on the t distribution arelonger.
Yes. The respective intervals based on the t distribution areshorter.
Yes. The respective intervals based on the t distribution arelonger. (This answer is correct)
(d) Now consider a sample size of 71. Compute 90%, 95%, and 99%confidence intervals for μ using Method 1 with a Student's tdistribution. Round endpoints to two digits after the decimal.
90% 95% 99%
lower limit
upper limit
(e) Compute 90%, 95%, and 99% confidence intervals for μ usingMethod 2 with the standard normal distribution. Use s as anestimate for σ. Round endpoints to two digits after thedecimal.
90% 95% 99%
lower limit
upper limit
