2025 SAJC P2 Q10

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

1. The number of hours worked per week by employees of Company \(A\) follow a normal distribution with mean \(40\) hours and standard deviation \(\alpha\) hours, while the number of hours worked per week by employees of Company \(B\) follow a normal distribution with mean \(40\) hours and standard deviation \(\beta\) hours, where \(\alpha > \beta\).
   The diagram shows the probability density function of \(X\), where \(X\) is the random variable denoting the number of hours worked per week by a randomly chosen employee of Company \(A\).

1. On the copy of the diagram in the Printed Answer Booklet, sketch on the same diagram the probability density function of \(Y\), where \(Y\) is the random variable denoting the number of hours worked per week by a randomly chosen employee of Company \(B\).[1]

The probability that a randomly chosen employee of Company \(A\) works less than \(t\) hours per week is represented by the shaded area of the standard normal curve in the diagram below.

2. Write down an equation involving \(t\) and \(\alpha\).[1]
3. Two employees of company \(A\) are randomly chosen. Find the probability that both employees work more than \(t\) hours per week.[1]
4. Given that the probability that a randomly chosen employee of Company \(B\) works more than \(42\) hours per week is at least \(0.25\), find the range of values of \(\beta\).[2]

2. A shop sells small and large bags of rice. The masses of small bags of rice are normally distributed with mean \(1\) kg and standard deviation \(0.16\) kg. The masses of large bags of rice have mean \(5\) kg and standard deviation \(0.5\) kg. The masses of the small and large bags of rice are independent of one another.
   1. In a random sample of \(300\) small bags of rice, find the expected number of bags that have a mass greater than \(1.25\) kg.[2]
   2. Find the probability that the total mass of \(5\) randomly chosen small bags of rice exceed the mean mass of \(30\) randomly chosen large bags of rice by more than \(0.25\) kg.[4]