D. In a health study on the physical fitness of elderly people, the researchers asked a group of senior citizens to ride an electronic bike over a certain trajectory of 10 km. The time needed to finish this trajectory X was recorded (in hours) and had the following continuous distribution: 24k (x - 1) x x2 , otherwise f(x)=k(2-x) 0 a. Find the constant k such that f(r) is a density function. b. What is the time by which 60% of the elderly persons have finished this trajectory? c. What is the probability that in a group of 50 elderly people the average time to finish this trajectory is larger than 1 hour? You can use that = E[X] = 15 and o = Var(X) = 120 723 d. In an extra part of this study, the researchers tested a group of 60 senior citizens (80+) from an elderly home in ty and found that the average time was equal to 1.02 hours with a standarddeviation of 0.32 hour. Verify on 5% significance level whether the average time to finish this trajectory for this subgroup is significantly larger than in the general population of elderly people. Provide sufficient details in your calculations.