AQA Further Mathematics Paper 3s 2025 Mark Scheme Pdf

Download the official AQA A-Level Further Mathematics Paper 3 (Statistics) mark scheme for June 2025 (7367/3S). This **final version (v1.0)** includes full worked solutions, examiner guidance, and detailed mark allocation (M, A, B, R, E marks) used during standardisation.

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## **Mark Scheme Overview (pages 2–4)**

The mark scheme explains how marks are awarded:

- **M (Method):** Correct process
- **A (Accuracy):** Correct answer
- **B (Independent):** Standalone marks
- **R (Reasoning):** Logical justification
- **E (Explanation):** Clear explanation

👉 According to **page 3**, examiners:
- Credit **valid alternative methods**
- Apply a **best-fit marking approach**
- Use standardisation to ensure consistent marking

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## **Section A: Multiple Choice Answers (Q1–Q3)** *(page 6)*

- **Q1:** **6400**
- **Q2:** **–6**
- **Q3:**
✔ *“He could have made a Type I error but not a Type II error.”*

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## **Key Worked Answers & Methods**

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### **Continuous Distribution (Q4, pages 7)**

- **Q4(a):**
\[
P(0.1 < X < 0.3) = 0.488
\]

- **Q4(b):**
\[
\text{Median} = \frac{\pi}{18}
\]

- **Q4(c):**
\[
f(x) =
\begin{cases}
3\cos(3x) & 0 \le x \le \frac{\pi}{6} \\
0 & \text{otherwise}
\end{cases}
\]

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### **Variance & Expectation (Q5, page 8)**

- **Q5(a):**
\[
\text{Var}(D + R) = \frac{n^2}{12}
\]

- **Q5(b):**
\[
n = 16
\]

👉 The **marking grid on page 8** shows that method marks require showing both
Var(D) and Var(R) before combining them.

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### **Confidence Interval & Hypothesis Test (Q6, pages 9–10)**

- Sample mean:
\[
\bar{x} = 91.2
\]

- Variance:
\[
s^2 = 25.6
\]

- Critical value:
\[
t_{10} = 1.812
\]

- **90% confidence interval:**
\[
(88.44,\ 93.96)
\]

- **Conclusion (page 10):**
- 90 lies inside interval →
- **Do not reject \( H_0 \)**
- *Insufficient evidence to suggest the mean ≠ 90*

---

### **Probability Modelling (Q7, pages 11–12)**

- Original mean:
\[
E(X) = 7.7
\]

- New condition:
\[
E(X^n) = 50 \times 7.7
\]

- Solving gives:
\[
10^n \approx 94
\]

- **New scores:**
\[
0,\ 94,\ 8759
\]

- Refinement (Q7b):
- Example valid answer:
\[
0,\ 100,\ 8760
\]

---

### **Chi-Squared Test (Q8, pages 13–14)**

- Test statistic:
\[
\chi^2 = 3.638
\]

- Critical value:
\[
\chi^2_{(1)} = 3.841
\]

- Conclusion:
- \( 3.638 < 3.841 \)
- **Do not reject \( H_0 \)**
- *No significant association between preferences*

👉 The **table on page 13** shows observed vs expected frequencies used in the calculation.

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### **Poisson & Exponential Models (Q9, pages 15–16)**

- **Q9(a):**
\[
\text{Mean time} = 0.5\ \text{minutes}
\]

- **Q9(b):**
\[
P(X > 130) = 0.169
\]

- **Q9(c):**
- Model:
\[
X \sim \text{Binomial}(7,\ 0.169)
\]
- Final probability:
\[
P(X = 2) \approx 0.24
\]

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## **Assessment Objectives (page 5)**

- **AO1:** Methods & accuracy
- **AO2:** Reasoning & proof
- **AO3:** Modelling & interpretation

👉 The **table on page 5** shows how statistical reasoning and modelling are assessed.

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## **Why This Mark Scheme Matters**
- Shows **exact final answers + full working**
- Explains **method marks (crucial for partial credit)**
- Reveals **examiner expectations**
- Essential for mastering **Statistics Paper 3**

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Fully updated for the 2025/2026 academic year and optimised for instant download on markscheme.net.

This mark scheme is designed for Year 13 students studying AQA A-Level Further Mathematics, particularly the Statistics option.

It is ideal for revision, checking answers, understanding mark allocation, and improving exam technique for top grades.