When σ is unknown and the sample is of size n≥ 30, there are two methods for computing confidence intervals forμ.
Method 1: Use the Student's t distribution withd.f. = n − 1.
This is the method used in the text. It is widely employed instatistical studies. Also, most statistical software packages usethis method.
Method 2: When n ≥ 30, use the sample standarddeviation s as an estimate for σ, and then usethe standard normal distribution.
This method is based on the fact that for large samples, sis a fairly good approximation for σ. Also, for largen, the critical values for the Student's tdistribution approach those of the standard normaldistribution.
Consider a random sample of size n = 41, with samplemean x = 45.4 and sample standard deviation s =6.0.
(a) Compute a 99% confidence intervals for μ usingMethod 1 with a Student's t distribution. Round endpointsto two digits after the decimal.
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(b) Compute a 99% confidence interval for μ using Method 2with the standard normal distribution. Use s as anestimate for σ. Round endpoints to two digits after thedecimal.
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(d) Now consider a sample size of 71. Compute a 99% confidenceinterval for μ using Method 1 with a Student's tdistribution. Round endpoints to two digits after the decimal.
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(e) Compute a 99% confidence interval for μ using Method 2with the standard normal distribution. Use s as anestimate for σ. Round endpoints to two digits after thedecimal.
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(f) Compare intervals for the two methods. Would you say thatconfidence intervals using a Student's t distribution aremore conservative in the sense that they tend to be longer thanintervals based on the standard normal distribution?
