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 2 30, use the sample standard deviation 5 as an estimate for 0', and then use the standard normal distribution. This method is based on the fact that for large samples, 5 is a fairly good approximation for 0'. Also, for large n, the critical values for the Student's 1' distribution approach those of the standard normal distribution. Consider a random sample of size n = 36, with sample mean )7 = 45.8 and sample standard deviation 5 = 6.0. (a) Compute 90%, 95%, and 99% condence intervals for [4 using Method 1 with a Student's t distribution. Round endpoints to two digits after the decimal. 90% 95% 99% lower limit 44.16 X upper limit 47.45 x (b) Compute 90%, 95%, and 99% confidence intervals for u using Method 2 with the standard normal distribution. Use 5 as an estimate for a. Round endpoints to two digits after the decimal. 90% 95% 99% lower limit upper limit (c) Compare intervals for the two methods. Would you say that condence 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? ' ' Yes. The respective intervals based on the t distribution are shorter. " \\ No. The respective intervals based on the t distribution are longer. 0 Yes. The respective intervals based on the t distribution are longer. No. The respective intervals based on the t distribution are shorter. u! (d) Now consider a sample size of 81. Compute 90%, 95%, and 99% condence intervals for u using Method 1 with a Student's t distribution. Round endpoints to two digits after the decimal. 90% 95% 99% lower limit upper limit (e) Compute 90%, 95%, and 99% condence intervals for ,4 using Method 2 with the standard normal distribution. Use 5 as an estimate for a. Round endpoints to two digits after the decimal. 90% 95% 99% lower limit upper limit