Amy records the times that she takes to complete \(3\)-hour practice papers. The times taken, in minutes, to complete Chemistry practice papers and Mathematics practice papers have normal distributions with means and standard deviations as shown in the table below.
| Mean | Standard deviation |
|---|
| Chemistry practice paper | \(160\) | \(7.5\) |
| Mathematics practice paper | \(170\) | \(3.5\) |
Amy completes one Chemistry and one Mathematics practice papers. Find the probability that only one of the papers is completed within its \(3\)-hour time constraint.[2]
Find the probability that the time taken to complete a randomly chosen Chemistry practice paper is within \(20\) minutes of the time taken to complete a randomly chosen Mathematics practice paper.[3]
Find the probability that the total time taken to complete two randomly chosen Chemistry practice papers is more than \(1.5\) times the time taken to complete a randomly chosen Mathematics practice paper by more than half an hour.[3]
Amy adjusts her mean time to complete a Mathematics practice paper while keeping the standard deviation at \(3.5\) minutes. She wishes to complete at least \(90\%\) of the Mathematics practice papers in less than \(2.5\) hours. Using a non-graphical method, find the largest new mean time. Give your answer correct to the nearest minutes.[3]
State an assumption needed for your calculations in all the parts above.[1]