2019 TJC P2 Q9

A Wheel Set refers to a set of wheel rim and tyre. The three types of wheel sets are the Clincher Bike Wheel Set, Tubular Bike Wheel Set and Mountain Bike Wheel Set. The weight of a rim of a Clincher Bike Wheel Set follows a normal distribution with mean \(1.5\) kg and standard deviation \(0.01\) kg. The weight of its tyre follows a normal distribution with mean \(110\) g and standard deviation \(5\) g

1. Let \(C\) be the total weight in grams of a randomly chosen Clincher Bike Wheel Set in grams. Find \(\mathrm{P}\left( C>1620 \right)\).[3]
2. State, in the context of the question, an assumption required in your calculation in (i).[1]

Let \(T\) be the total weight in grams of a Tubular Bike Wheel Set, where \(T\sim \mathrm{N}\left( \mu ,{{15}^{2}} \right)\).

3. The probability that the weight of a randomly chosen Clincher Bike Wheel Set exceeds a randomly chosen Tubular Bike Wheel Set by more than 150 g is smaller than 0.70351correct to 5 decimal places. Find the range of values that \(\mu\) can take.[5]

Let \(M\) be the total weight in grams of a randomly chosen Mountain Bike Wheel Set with mean \(1800\) g and standard deviation \(20\) g.

4. Find the probability that the mean weight of \(50\) randomly chosen Mountain Bike Wheel Sets is more than \(1795\) g.[3]