1. (40 marks) Hognose is a programming language that is designed to implement stochastic algorithms, which are the methods to solve computational problems by random numbers. It is known that the execution times of Hognose programs for a certain class of problems are normally distributed. Hognose was redesigned to x some design aws. A simple random sample of 14 problems in the class was selected to test the performance of the new version of Hognose. The execution times of the old and new versions were recorded (in seconds) in the following table. Problem ID 1 2 3 4 5 6 7 8 9 Old version 104.7 93.4 93.8 54.0 96.6 102.8 70.1 79.8 81.0 New version 105.0 89.5 92.5 93.4 102.4 69.7 79.3 80.2 Problem ID 10 11 12 14 Old version 90.1 96.5 91.9 136.9 107.9 New version 96.0 91.9 (a) Test whether the mean change of the execution time is greater than 0 seconds. Assume that the change of the execution times is also normally distributed. (12 marks) (b) Test whether the standard deviation of the execution time of the new version equals 20 seconds. (12 marks) (c) Test whether the execution times of more than 50% of the problems in the class are improved in the new version? (12 marks) (d) What is the p-value of the test in (c)? (4 marks)