when is unknown and the sample is of size n 30, there are two methods for comput
ID: 3350592 • Letter: W
Question
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 with df. = 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 s as an estimate for o, 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 . 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 = 44.6 and sample standard deviation s = 5.5. (a) Compute 90%, 95%, and 99% confidence intervals for using Method 1 with a Student's t distribution. Round endpoints to two digits after the decimal lower limit 35.27 upper limit 53.93 90% 95% 9996 X 33.37 55.83 29.48 59.72 (b) Compute 90%, 95%, and 99% confidence intervals for using Method 2 with the standard normal distribution. Use s as an estimate for . Round endpoints to two digits after the decimal. 90% 95% 99% lower limit 35.55 upper limit 53.65 X33.82 X 55.38 30.43 X58.77 (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? No. The respective intervals based on the t distribution are shorter Yes, The respective intervals based on the t distribution are shorter No. The respective intervals based on the t distribution are longer Yes, The respective intervals based on the t distribution are longer (d) Now consider a sample size of 71, Compute 90%, 95%, and 99% confidence intervals for using Method 1 with a Student's t distribution. Round endpoints to two digits after the decimal. lower limit 35.43 upper limit 53.77 (e) Compute 90%, 95%, and 99% confidence intervals for using Method 2 with the standard normal distribution. Use s as an estimate for . Round endpoints to two digits after the decimal lower limit 35.55 upper limit 53.65 90% 95% 99% X 33.63 X55.57 X30.04 X59.16 90% 95% 99% X33.82 X55.38 X30.43 58.77Explanation / Answer
formula used
in t-distribution
CI = (Xbar - t * s/sqrt(n) , Xbar + t * s/sqrt(n))
z-dist.
(Xbar - z * sigma/sqrt(n) , Xbar + z*sigma /sqrt(n))
a) 90 95 99 t -1.69726 -2.04227 -2.75 lower 42.9234 42.58258 41.88347 upper 46.2766 46.61742 47.31653 b) z -1.64485 -1.95996 -2.57583 lower 42.97517 42.66389 42.05552 upper 46.22483 46.53611 47.14448 c) option D) is correct d) 90 95 99 t -1.66691 -1.99444 -2.6479 lower 43.51196 43.29817 42.87163 upper 45.68804 45.90183 46.32837 z -1.64485 -1.95996 -2.57583 lower 43.52635 43.32067 42.91868 upper 45.67365 45.87933 46.28132Related Questions
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