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When ? is unknown and the sample is of size n ? 30, there are two...

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Basic Math

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

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 ?, and then usethe standard normal distribution.
This method is based on the fact that for large samples, sis a fairly good approximation for ?. Also, for largen, the critical values for the Student's tdistribution approach those of the standard normaldistribution.

Consider a random sample of size n = 31, with samplemean x = 44.4 and sample standard deviation s =5.9.

(d) Now consider a sample size of 81. Compute 90%, 95%, and 99%confidence intervals for ? using Method 1 with a Student'st distribution. Round endpoints to two digits after thedecimal.

90%95%99%
lowerlimit    
upperlimit    


(e) Compute 90%, 95%, and 99% confidence intervals for ?using Method 2 with the standard normal distribution. Uses as an estimate for ?. Round endpoints to twodigits after the decimal.

90%95%99%
lowerlimit    
upperlimit    

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