D. In a health study on the physical fitness of elderly people, the researchers asked a...

 

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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)² ≤ x ≤ ≤x≤2 , 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 μ = E[X] = 15 and o² = 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. 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)² ≤ x ≤ ≤x≤2 , 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 μ = E[X] = 15 and o² = 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.