(a) [10 marks] If 1K." is the sample mean of a random sample of size n from a normal population with known variance 02, show that the 100(1 00% Condence Interval on the population mean a is given by X _ Zr:r,/'2\"T/I\\/E S P5 i: X'l Za/2J/J where Pia/2 is the upper 100a/ 2 percentage point of the standard normal distribution. (b) [10 marks] Suppose that the life length of an ENERGY light bulb (X; unit: hour) follows the nor mal distribution N01, 02). A random sample of n = 30 bulbs is tested as shown below. 1 2 3 4 5 6 7 8 9 1 :: (i)[2 marks] Compute the sample mean life length. (ii) [5 marks] Construct a 95% two-sided condence interval on the mean life length. (iii) [3 marks] Find a sample size n to construct a two-sided condence interval on a with an error = 20 hours from the true mean life length. Use a = 0.05