2. (20 points) Comparison of condence intervals on Normal mean ,u (60 Given that the asymptotic distribution for the sample median, M of a random sample drawn from a N( #102) population is x/M p.) N N(0, 30?), approximately/ asymptotically Use the Pivot method to derive and show that the 100(1 00% asymptotic interval estimator for )4 based on m is l 71' m i- 2% $0: (1) (Hint: use Tm, X) : ill2;\"; ~ N(0,1) then use 1 or : P(z% S M;'\" S 2%) to start the derivation) Recall the 100(1 0:)% CI. on ,u. in using it is :Ejzzgj. (2) Comment on the difference between these two 100(1 00% condence intervals (the one in (1) and (2) in terms of the radius. Which is shorter? A sample of 49 observations is drawn from a N(,u, 25) population. The sample median is found to be m = 12.456 and the sample mean is :E = 11.050. Construct the 95% condence intervals for n, [1) in using :E, (2) in using result in (a)