AQA A-Level Mathematics Paper 3, June 2025: Question 20

9 marks · Medium difficulty · Multi-step Problem

Use a normal distribution to calculate probabilities for car emissions, explain properties of continuous models, compare distributions using mean and standard deviation, and answer questions based on the Large Data Set.

Practise this question

Question

Question 20 introduces a model where oxides of nitrogen emissions, F in g/km, of a car registered in 2000 follow a normal distribution with mean 0.41 and standard deviation 0.07. Parts (a), (b), and (c)(i) ask to find probabilities: P(F < 0.39), P(0.3 < F < 0.5), and P(F > 0.6). Part (c)(ii) asks to explain why P(F >= 0.6) = P(F > 0.6). Part (d) introduces a 2010 model with mean 0.36 and standard deviation 0.09 and asks to compare the emissions between 2000 and 2010. Parts (e) and (f) ask questions referencing the Large Data Set regarding data cleaning and why comparing a sample of Nissan cars to the LDS cannot be made.

Mark scheme

Show the mark scheme Mark scheme for Question 20 detailing marks for each part totaling 9 marks. (a) B1 for 0.3875 (AWFW [0.387, 0.39]). (b) B1 for 0.8427 (AWFW [0.84, 0.843]). (c)(i) B1 for 0.0033 (AWFW [0.003, 0.0034]). (c)(ii) E1 for deducing P(F = 0.6) = 0 and E1 for explaining the continuous nature of the normal distribution. (d) Two E1 marks for correctly comparing means and standard deviations in context (on average/typically and variation/spread). (e) E1 for noting missing data/blanks in the LDS. (f) E1 for stating there are no Nissan cars in the LDS.

How to answer it

Normal Distribution & Large Data Set (LDS) Analysis

📋 What this question tests

This 9-mark question assesses fundamental statistics skills across two distinct domains:

  • Continuous Normal Distributions: Calculating cumulative probabilities on a calculator, understanding continuity property (P(X = c) = 0), and interpreting mean and standard deviation in a real-world comparative context.
  • Large Data Set (LDS) Familiarity: Practical data-cleaning reasons (missing data/blanks) and knowing specific categorical values present (makes of cars included in the AQA LDS).
Parts (a), (b) & (c)(i) • Calculator Probabilities • 3 Marks Total

Calculating Normal Cumulative Probabilities

Given: Oxides of nitrogen emissions F ~ N(0.41, 0.07²) for year 2000 cars, where mean μ = 0.41 and standard deviation σ = 0.07.

📐 Step-by-Step Calculator Inputs

  1. Part (a): Find P(F < 0.39)
    Lower: -10000 (or -9999 )
    Upper: 0.39 , σ = 0.07, μ = 0.41
  2. Part (b): Find P(0.3 < F < 0.5)
    Lower: 0.3 , Upper: 0.5 , σ = 0.07, μ = 0.41
  3. Part (c)(i): Find P(F > 0.6)
    Lower: 0.6 , Upper: 10000 (or 9999 ), σ = 0.07, μ = 0.41

✅ Correct Answers & Ranges

  • (a): 0.3875 [Accept AWFW: 0.387 to 0.39]
  • (b): 0.8427 [Accept AWFW: 0.84 to 0.843]
  • (c)(i): 0.0033 [Accept AWFW: 0.003 to 0.0034]
Award B1 for each correct probability within tolerance.

🧠 Exam Technique

Always write down at least 3 or 4 decimal places unless specified. Use the exact standard deviation σ = 0.07 , never variance 0.07² = 0.0049 .

❌ Common Errors

Squaring or square-rooting the standard deviation when entering values into your Casio ClassWiz (it prompts for σ, not σ²).

Part (c)(ii) • Conceptual Understanding • 2 Marks

Continuity Property of the Normal Distribution

"Explain why P(F ≥ 0.6) = P(F > 0.6) in this model."

✅ Model Answer (Full 2 Marks)

P(F = 0.6) = 0 [E1 mark]
because the normal distribution is a continuous model (or F is a continuous variable) [E1 mark].

🧠 Mark Breakdown

  • 1st Mark (E1, AO 2.2a): Stating explicitly that the point probability is zero: P(F = 0.6) = 0 .
  • 2nd Mark (E1, AO 2.4): Explicitly using the word "continuous" to justify why individual point probabilities equal zero.

❌ Common Misconception

Vague statements like "the area of a line is very thin" or "it's too small to matter" lose marks. You must state P(F = 0.6) = 0 and identify the distribution as continuous.

Part (d) • Comparative Analysis in Context • 2 Marks

Comparing Distributions in Context

Comparing Year 2000 (μ = 0.41, σ = 0.07) with Year 2010 (μ = 0.36, σ = 0.09).

✅ Model Answer

Comparison of Central Tendency (Mean):
The oxides of nitrogen emissions by cars in 2000 are greater on average (or typically higher) than those in 2010.

Comparison of Spread (Standard Deviation):
The emissions in 2000 have less variation (or are less spread out / more consistent) than those in 2010.

💡 Mandatory Context Rules

  • Must mention "on average", "typically", or "in general" when comparing the means. Simply writing "2000 is bigger" gets 0 marks!
  • Must use variation words: "spread", "varied", "dispersion", or "consistent" for standard deviation.
  • Must refer to context: at least once name "oxides", "emissions", and mention the years 2000 and 2010.

❌ Examiner Trap

Do NOT use the words "range" or "variety" when interpreting standard deviation. The mark scheme explicitly states: "Do not allow comparison that only includes 'range' or 'variety'", as standard deviation is not the range!

Parts (e) & (f) • Large Data Set (LDS) Knowledge • 2 Marks

Large Data Set Contextual Questions

✅ Part (e): Why clean the data? (1 Mark)

Acceptable responses:

  • There are some blanks / missing data for oxides of nitrogen emissions in the LDS.
  • The emissions values are not known/recorded for every car.
Award E1 (AO 2.4) for referencing missing data/blanks.

✅ Part (f): Why comparison cannot be made? (1 Mark)

Acceptable responses:

  • There are no Nissan cars in the Large Data Set.
  • The LDS only contains data for BMW, Ford, Toyota, Vauxhall, and Volkswagen cars.
Award E1 (AO 2.4) for stating Nissan is not in the LDS (or listing the only 5 makes present).

🧠 Exam Technique for LDS Questions

LDS questions test whether you have actually worked with the spreadsheet during your course. You are expected to know which car makes are present and that real datasets contain incomplete records / blank fields (often marked n/a or left blank).

❌ What NOT to say

  • For (e): "To remove outliers" — While generally true in statistics, data cleaning in this specific LDS context is required due to missing entries.
  • For (f): "The sample is too small" or "A local town isn't representative" — While sensible in abstract statistics, the exact hurdle is that the make Nissan does not exist in the dataset!

Topics

Statistics · K: Statistical sampling · L: Data presentation and interpretation · N: Statistical distributions

Question and mark scheme from the AQA A-Level Mathematics examination, Paper 3, June 2025. QuestionVault is an independent revision resource; questions remain the copyright of the awarding body.