Question 3 - A-Level Maths

Edexcel S3 2012 June — Question 3 11 marks

Exam Board	Edexcel
Module	S3 (Statistics 3)
Year	2012
Session	June
Marks	11
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Central limit theorem
Type	Justifying CLT for confidence intervals
Difficulty	Standard +0.3 This is a straightforward S3 question requiring standard CLT explanation, routine confidence interval calculation (σ known, large n=100), and basic interpretation. The CLT application is explicit and the calculations are mechanical with no conceptual challenges beyond typical A-level statistics.
Spec	5.01a Permutations and combinations: evaluate probabilities 5.05d Confidence intervals: using normal distribution

3. (a) Explain what you understand by the Central Limit Theorem. 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 \(\sum x = 1740000\) (b) Find a 95\% confidence interval for the mean lifetime of a brake pad.
(c) Explain the relevance of the Central Limit Theorem in part (b). Brake pads are made to be changed every 20000 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.

Show mark scheme Show mark scheme source

Question 3:

Part (a)

Answer	Marks	Guidance
Answer/Working	Marks	Guidance
\((X_1, X_2, X_3, \ldots, X_n\) is a random) sample of size \(n\), for \(n\) is large	B1
(from a population with mean \(\mu\) and variance \(\sigma^2\)) then \(\bar{X}\) is (approximately) Normal	B1

(2 marks)

Part (b)

Answer	Marks	Guidance
Answer/Working	Marks	Guidance
\(\bar{x} = \frac{1740000}{100} = 17400\)	B1
\(\bar{x} \pm z\frac{\sigma}{\sqrt{n}} = 17400 \pm 1.96 \times \frac{5000}{\sqrt{100}}\)	M1, B1	Recognisable \(z\) value required for method; 2nd B1: 1.96 or better seen
\([16420, 18380]\)	A1A1	Accept 3sf if correct expression seen; 5/5 for \([16420, 18380]\)

(5 marks)

Part (c)

Answer	Marks	Guidance
Answer/Working	Marks	Guidance
\(\bar{X}\): Normal (approx) by CLT, and normal needed to find CI	B1, B1

(2 marks)

Part (d)

Answer	Marks	Guidance
Answer/Working	Marks	Guidance
20000 above upper confidence limit (not just outside)	B1ft
Complaint justified	dB1ft

(2 marks)

# Question 3:

## Part (a)

| Answer/Working | Marks | Guidance |
|---|---|---|
| $(X_1, X_2, X_3, \ldots, X_n$ is a random) **sample** of size $n$, for $n$ is **large** | B1 | |
| (from a population with mean $\mu$ and variance $\sigma^2$) then $\bar{X}$ is (approximately) Normal | B1 | |

**(2 marks)**

## Part (b)

| Answer/Working | Marks | Guidance |
|---|---|---|
| $\bar{x} = \frac{1740000}{100} = 17400$ | B1 | |
| $\bar{x} \pm z\frac{\sigma}{\sqrt{n}} = 17400 \pm 1.96 \times \frac{5000}{\sqrt{100}}$ | M1, B1 | Recognisable $z$ value required for method; 2nd B1: 1.96 or better seen |
| $[16420, 18380]$ | A1A1 | Accept 3sf if correct expression seen; 5/5 for $[16420, 18380]$ |

**(5 marks)**

## Part (c)

| Answer/Working | Marks | Guidance |
|---|---|---|
| $\bar{X}$: Normal (approx) by CLT, and normal needed to find CI | B1, B1 | |

**(2 marks)**

## Part (d)

| Answer/Working | Marks | Guidance |
|---|---|---|
| 20000 **above** upper confidence limit (**not** just outside) | B1ft | |
| Complaint justified | dB1ft | |

**(2 marks)**

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3. (a) Explain what you understand by the Central Limit Theorem.

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 $\sum x = 1740000$\\
(b) Find a 95\% confidence interval for the mean lifetime of a brake pad.\\
(c) Explain the relevance of the Central Limit Theorem in part (b).

Brake pads are made to be changed every 20000 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.\\

\hfill \mbox{\textit{Edexcel S3 2012 Q3 [11]}}

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Q1 12 Q2 8 Q3 11 Q4 10 Q5 9 Q6 14 Q7 11