full step plx

Problem 5 [23 points] We want to do some research on post office queue system. With traditional single line to each windows system, the standard deviation of the waiting times for customers is 6.5 minutes. Now we change it to the individual lines at its various window and collect a sample for 15 customers. The sample mean is 14.0 minutes and the sample deviation is 5.0 minutes. Assume that the waiting time follows normal distribution. (a) [3 points] Construct a 90% Confidence Interval for the Variance of waiting time o2 based on the sample mean and sample variance. (b) [3 points] Test if the new lining system (individual lines at its various window) has larger variation of waiting time for customers than the traditional one at a = 0.05. Suppose that the mean waiting time of the new lining system is 15 minutes (Known population mean). (c) [4 points] Construct an unbiased estimator for the Variance of the new lining system based on the Known population mean. Which the distribution will this estimator follow? Please also calculate the parameter of it. (d) [4 points] Construct a 90% Confidence Interval for the Variance of waiting time o2 based on the population mean and sample variance. Hints: (n - 1)$2 = >, (X; - x) ? = En=(Xi - H+ H- x)2 (e) [4 points] Please compare the Confidence interval in Part (a) and Part (d), which is more accurate? Why? (f) [5 points] Test if the new lining system (individual lines at its various window) has larger variation of waiting time for customers than the traditional one at a = 0.05 based on the new information. Is there any different between Part (f) and Part(b)