A fitness enthusiast, Alex, is training for three stages of a workout: a 1200 m row, a 15 km cycle, and a 6 km run. Based on previous records, the times in minutes for each of the three stages follow independent normal distributions:
R∼N(25,3.12)R \sim \text{N}(25, 3.1^2)R∼N(25,3.12) represents the time for the row, C∼N(55,4.82)C \sim \text{N}(55, 4.8^2)C∼N(55,4.82) represents the time for the cycle, S∼N(42,5.52)S \sim \text{N}(42, 5.5^2)S∼N(42,5.52) represents the time for the run.
Find the probability that Alex's total time for a randomly selected row, cycle, and run exceeds 130 minutes.
Find the probability that the time for a randomly selected row will be at least 15 minutes shorter than the time for a randomly selected run.
Given that P(R+C+S<t)=0.025P(R + C + S < t) = 0.025P(R+C+S<t)=0.025, find the value of ttt.
Suppose the stages are performed consecutively in a single session. Alex uses the probability found in part (a) to calculate the probability that in 5 independent workout sessions, their total time exceeds 130 minutes on at least two occasions. Find the probability Alex should obtain.
An observer suggests that the independence assumption between stages within a single session might not be valid. Explain with a reason whether the use of the answer from part (a) in current calculation for part (d) is appropriate.
228 exam-style questions on CCEA A Level Maths 4.5.1 Statistical distributions (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.