S3 January 2006 Q2
2. A workshop makes two types of electrical resistor.
The resistance, \(X\) ohms, of resistors of Type A is such that \(X \sim \mathrm{N}(20, 4)\).
The resistance, \(Y\) ohms, of resistors of Type B is such that \(Y \sim \mathrm{N}(10, 0.84)\).
When a resistor of each type is connected into a circuit, the resistance \(R\) ohms of the circuit is given by \(R = X + Y\) where \(X\) and \(Y\) are independent.
Find
(a) \(\mathrm{E}(R)\), (1)
(b) \(\mathrm{Var}(R)\), (2)
(c) \(\mathrm{P}(28.9 \lt R \lt 32.64)\) (6)
| Scheme | Marks |
|---|---|
| \(\mathrm{E}(R) = 20 + 10 = 30\) | B1 |
| (1) |
| Scheme | Marks |
|---|---|
| \(\mathrm{Var}(R) = 4 + 0.84,\ = 4.84\) | M1, A1 |
| (2) |
| Scheme | Marks |
|---|---|
| \(R \sim \mathrm{N}(30, 4.84)\) (Use of normal with their (a), (b)) | B1ft |
| \(\mathrm{P}(28.9 \lt R \lt 32.64) = \mathrm{P}(R \lt 32.64) - \mathrm{P}(R \lt 28.9)\) \(= \mathrm{P}\left(Z \lt \dfrac{32.64 - 30}{2.2}\right) - \mathrm{P}\left(Z \lt \dfrac{28.9 - 30}{2.2}\right)\) Stand their \(\sigma\) and \(\mu\) | M1 |
| \(= \mathrm{P}(Z \lt 1.2) - \mathrm{P}(Z \lt -0.5)\) | A1, A1 |
| \(= 0.8849 - (1 - 0.6915)\) Correct area | M1 |
| \(= 0.8849 - 0.3085 = 0.5764\) (accept AWRT 0.576) | A1 |
| (6) | |
| (9 marks) |