# When σ is unknown and the sample size is n\geq30, there are tow methods for computing confidence intervals for μμ. Method 1: Use the Student's t distr

When σ is unknown and the sample size is $$\displaystyle{n}\geq{30}$$, there are tow methods for computing confidence intervals for μμ. 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 $$\displaystyle{n}\geq{30}$$, use the sample standard deviation s as an estimate for σσ, 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 σσ. 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 x¯=45.2 and sample standard deviation s = 5.3. (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?

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Result part (a):
90%: 43.5846 to 46.8154
95%: 43.2562 to 47.1438
99%: 42.5823 to 47.8177
Result part (b):
90%: 43.6341 to 46.7659
95%: 43.3343 to 47.0657
99%: 42.7441 to 47.6559
The confidence intervals using a Student's t distribution are more conservative, because the confidence intervals are a little wider than the confidence intervals using the standard normal distribution.