5. Suppose that the random variables X., Xin, are IID N (,), i = 1,2, and that X,'s are independent of X2, 's. Assume that , are unknown but , are known. Derive a 2j 100(1-)% confidence interval for - based on the sufficient statistics for (M,M). 6. Two types of cars were compared for their braking distances. Test runs were made for each car in a driving range. Once a car reached the stable speed of 60 miles per hour, the brakes were applied. The distance (feet) each car travelled from the moment the brakes were applied to the moment the car came to a complete stop was recorded. The summary statistics are shown below: Car Make A Make B Sample size A = 12 X S 37.1 3.1 n = 10 39.6 4.3 Construct a 95% confidence interval for -. Assume that the elapsed times are distributed as N(,) and N(, ), respectively for Make A and Make B with all the parameters unknown. 7. A sample of 10 pine trees grown on the north side of a hill has a mean of 25.4 metres and a standard deviation of 2.1 metres. A second sample of 12 trees from the south side has a mean of 23.2 metres and a standard deviation of 1.7 metres. Find the 99% confidence interval for the difference in the mean heights of the two populations of trees. Interpret the confidence interval.