2025 JPJC P2 Q10

In this question, you should state clearly all the distributions that you use, together with the values of the appropriate parameters.

A company produces two types of yarn balls with two machines. Machine A produces balls of 2-ply yarn. Over a period of time, it is found that \(80\%\) of the balls of yarn from Machine A have lengths less than \(340\) metres and \(10\%\) of the balls of yarn from Machine A have lengths less than \(280\) metres.

1. Assuming that the lengths, in metres, of yarn in balls of 2-ply yarn from Machine A follows the distribution \(N(\mu, \sigma^2)\), find \(\mu\) and show that \(\sigma = 28.3\).[3]

For the rest of the question let \(\mu = 300\) and \(\sigma = 20\).

Machine B produces balls of 3-ply yarn. The lengths, in metres, of yarn in balls of 3-ply yarn from Machine B follows the distribution \(N(220, 6^2)\).

2. Five balls of 3-ply yarn from Machine B are randomly chosen. Find the probability that two of them have lengths more than 230 m and three of them have lengths less than 215 m.[2]
3. The probability that two randomly chosen balls of 2-ply yarn from Machine A and three randomly chosen balls of 3-ply yarn from Machine B have total length exceeding \(k\) metres is 0.385. Find the value of \(k\), correct to 2 decimal places.[3]

4. A technician changed the settings on the two machines. The lengths of yarn in balls of 2-ply yarn from Machine A is reduced by \(9\%\) and the lengths of yarn in balls of 3-ply yarn from Machine B is reduced by \(5\%\). Find the probability that the total length of yarn in three randomly chosen balls of 2-ply yarn from Machine A is within 50 metres of the total length of yarn in four randomly chosen balls of 3-ply yarn from Machine B.

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