2026 HCI P2 Q10

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

Two machines, A and B, produce bolts. The masses in grams of bolts from machines A and B follow the distributions \(A\sim\mathrm N(\mu,\sigma_1^2)\) and \(B\sim\mathrm N(\mu,\sigma_2^2)\) respectively. It is also given that \(\sigma_1<\sigma_2\).

1. On the same axes, sketch the distribution curves for the masses of the bolts produced by machines A and B.[2]
2. It is given that \(1.08\big[\mathrm P(|B-\mu|\leq\sigma_1)\big]=\mathrm P(|B-\mu|\leq\sigma_2)\).
   Show that \(1.08\big[\mathrm P(|B-\mu|\leq\sigma_1)\big]=\mathrm P(|Z|\leq1)\) and find \(\dfrac{\sigma_1}{\sigma_2}\), correct the 4 decimal places.[3]

It is now given that \(A\sim\mathrm N(50,2^2)\).

3. A bolt is accepted if its mass is between 47 g and 53 g.
   1. Find the probability that the mass of a randomly chosen bolt from machine A is acceptable.[1]
   2. Find the probability that, out of 15 randomly chosen bolts from machine A, the last bolt is the third bolt that is rejected.[3]

The mass of an empty box is normally distributed with mean 500 g and standard deviation \(\sigma\) g.

4. 100 bolts are randomly selected from machine A and placed into a randomly selected empty box. The total mass of the empty box and the 100 bolts is denoted by \(T\). Given that the probability that \(T\) differs from its mean by less than 0.25 kg is at least 0.95, find the range of values of \(\sigma\).[4]