2024 HCI P2 Q9

In this question, you should state the parameters of any distribution you use.

A ceramic shop sells handmade ceramic cups. The mouths of the cups are assumed to be circular in shape. The diameters of the outer circumferences of the top rim of the cups, \(S\), are assumed to follow a normal distribution with mean \(\mu \) mm and standard deviation \(\sigma \) mm.

1. It is given that \(\mathrm{P}\!\left( S \lt 80.5 \right) = \mathrm{P}\!\left( S \gt 84.5 \right)\) and that the probability of the diameter of the outer circumference of the top rim of a randomly chosen cup being more than \(85\)mm is \(1.15\%\). Find the value of \(\mu \), and show that \(\sigma =1.10\), when corrected to \(3\) significant figures.[3]
2. The shop also makes covers of circular shape that can be fitted over the mouths of the cups. The diameter of any randomly chosen cover, \(C\), in mm, follows a normal distribution with mean \(83\) mm and standard deviation \(1.5\)mm. A cover would be considered to be well-fitted over the mouth of a cup if the diameter of the cover is larger than that of the outer circumference of the top rim of the cup by not more than \(2\) mm. Find the probability that a randomly chosen cover is well-fitted over the mouth of a randomly chosen cup.[3]
3. A cover and a cup are randomly chosen. If the cover is well-fitted over the mouth of the cup, find the probability that the diameter of the cover is larger than that of the cup by more than \(1.5\) mm.[3]