S3 June 2012 Q3
3.
(a) Explain what you understand by the Central Limit Theorem. (2)
A garage services hire cars on behalf of a hire company. The garage knows that the lifetime of the brake pads has a standard deviation of 5000 miles. The garage records the lifetimes, \(x\) miles, of the brake pads it has replaced. The garage takes a random sample of 100 brake pads and finds that \(\sum x = 1\,740\,000\)
(b) Find a 95% confidence interval for the mean lifetime of a brake pad. (5)
(c) Explain the relevance of the Central Limit Theorem in part (b). (2)
Brake pads are made to be changed every 20 000 miles on average.
The hire car company complain that the garage is changing the brake pads too soon.
(d) Comment on the hire company’s complaint. Give a reason for your answer. (2)
| Scheme | Marks |
|---|---|
| (\(X_1, X_2, X_3, \ldots, X_n\) is a random) sample of size \(n\), for \(n\) is large, | B1 |
| (from a population with mean \(\mu\) and variance \(\sigma^2\) ) then \(\overline{X}\) is (approximately) Normal. | B1 |
| (2) |
| Scheme | Marks |
|---|---|
| \(\bar{x} = \dfrac{1740000}{100} = 17400\) | B1 |
| \(\bar{x} \pm z\dfrac{\sigma}{\sqrt{n}}, = 17400 \pm 1.96 \times \dfrac{5000}{\sqrt{100}}\) | M1, B1 |
| [16420,18380] | A1A1 |
| (5) |
Notes
Recognisable \(z\) value required for method.
2nd B1 1.96 or better seen award
Final A1s accept 3sf if correct expression seen.
5/5 for [16420,18380]
| Scheme | Marks |
|---|---|
| \(\overline{X}\) : Normal (approx) by CLT, and normal needed to find CI. | B1,B1 |
| (2) |
| Scheme | Marks |
|---|---|
| 20000 above upper confidence limit (not just outside) | B1ft |
| Complaint justified. | dB1ft |
| (2) | |
| (11 marks) |