An industrial chemical process consists of three sequential stages: preparation, reaction, and filtration. The durations in minutes for these stages are modeled as independent normal random variables:
P∼N(40,2.52)P \sim \text{N}(40, 2.5^2)P∼N(40,2.52) represents the preparation time, R∼N(110,6.42)R \sim \text{N}(110, 6.4^2)R∼N(110,6.42) represents the reaction time, F∼N(75,4.22)F \sim \text{N}(75, 4.2^2)F∼N(75,4.22) represents the filtration time.
Determine the probability that the total time for a randomly selected sequence of all three stages exceeds 240 minutes.
Calculate the probability that the filtration stage is at least 30 minutes longer than the preparation stage for a randomly selected batch.
Given that P(P+R+F<t)=0.05P(P + R + F < t) = 0.05P(P+R+F<t)=0.05, find the value of ttt.
In a single manufacturing shift, 6 independent batches are processed. A quality control officer uses the probability calculated in part (a) to determine the probability that the total processing time exceeds 240 minutes in at least two of these batches. Find the value the officer should obtain.
An engineer, Dr. Aris, suggests that the assumption of independence between stages within the same batch is likely flawed. Explain with a reason whether the use of the answer from part (a) in the calculation for part (d) is appropriate.
Practise Edexcel A Level Maths The Normal Distribution with exam-style questions for A Level Maths. 439 questions, matched to the Edexcel A Level Maths (9MA0) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.
