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.
616 exam-style questions on Edexcel A Level Maths The Normal Distribution, covering 3.1 The Normal Distribution, 3.2 Finding Probabilities for Normal Distributions, 3.3 The Inverse Normal Distribution Function, 3.4 The Standard Normal Distribution, 3.5 Finding the mean and standard deviation, 3.6 Approximating a Binomial Distribution, and 3.7 Hypothesis Testing with the Normal Distribution. Each one has a worked solution and a mark scheme showing where the marks go.