AQA A-Level Mathematics Paper 3, June 2025: Question 14

1 mark · Easy difficulty · Short Answer

Identify the values of $a$ and $b$ such that approximately 95% of a normal distribution with mean 9 and standard deviation 1.5 lies between $a$ and $b$.

Practise this question

Question

Question 14 states: It is given that X is normally distributed with mean 9 and variance 1.5 squared, and the probability that X is between a and b is approximately 95%. Identify the correct pair of values of a and b by ticking one box. The choices are: a = 4.5 and b = 13.5; a = 6 and b = 12; a = 7 and b = 11; a = 7.5 and b = 10.5.

Mark scheme

Show the mark scheme Mark scheme for Question 14: Marking instructions state 'Ticks the second box', AO is 1.1b, 1 mark (B1), with typical solution 'a = 6 and b = 12'.

How to answer it

Normal Distribution: Empirical 95% Confidence Interval

📋 What this question tests

This question assesses your understanding of the fundamental properties of the Normal Distribution, specifically:

  • Interpreting the parameters of X ~ N(μ, σ²) to correctly identify the mean (μ) and standard deviation (σ).
  • Applying the empirical rule (the 68–95–99.7 rule) where approximately 95% of the distribution lies within 2 standard deviations of the mean ( μ ± 2σ ).

Question 14

Multiple Choice Question (1 Mark)

✅ Correct Answer

Tick the second box:

a = 6 and b = 12

Mark scheme: [1 mark] — B1 (AO 1.1b) awarded for correctly identifying and ticking the second option.

📐 Step-by-Step Calculation

  1. Identify parameters:
    From X ~ N(9, 1.5²) :
    Mean, μ = 9
    Standard deviation, σ = 1.5
  2. Recall the 95% interval rule:
    For any normal distribution, approximately 95% of data lies within 2 standard deviations:
    P(μ - 2σ ≤ X ≤ μ + 2σ) ≈ 95%
  3. Calculate boundaries:
    Lower bound: a = 9 - 2(1.5) = 9 - 3 = 6
    Upper bound: b = 9 + 2(1.5) = 9 + 3 = 12

💡 Key Knowledge: The Empirical Rule

You are expected to know these approximate symmetrical percentage intervals for X ~ N(μ, σ²) by heart:

  • ≈ 68% lies within 1 standard deviation: [μ - σ, μ + σ]
  • ≈ 95% lies within 2 standard deviations: [μ - 2σ, μ + 2σ]
  • ≈ 99.7% lies within 3 standard deviations: [μ - 3σ, μ + 3σ]

Note: More precisely, 95% corresponds to μ ± 1.96σ, but standard A-Level questions explicitly use "≈ 95%" to refer to the integer approximation 2σ.

🧠 Exam Technique & Option Breakdown

All four options correspond to specific multiples around the mean ( μ = 9 ):

  • 4.5 to 13.5 : 9 ± 3(1.5) = μ ± 3σ (this is ≈ 99.7%)
  • 6 to 12 : 9 ± 2(1.5) = μ ± 2σ (Correct: ≈ 95%)
  • 7 to 11 : 9 ± 2 (subtracted/added 2 directly instead of 2σ)
  • 7.5 to 10.5 : 9 ± 1(1.5) = μ ± 1σ (this is ≈ 68%)

❌ Common Misconceptions & Traps

  • Confusing variance with standard deviation: The distribution is written as N(9, 1.5²) . Remember that the notation is N(μ, σ²) . Here, σ = 1.5 , not 1.5² = 2.25 .
  • Selecting Option 3 ( a = 7, b = 11 ): Students mistakenly add and subtract 2 from 9 (i.e. 9 ± 2 ) forgetting that the interval must be scaled by the standard deviation ( 2 × σ ).
  • Over-relying on the calculator: While you can use the Inverse Normal function on your graphical calculator ( Area = 0.025 and 0.975 giving 6.06 and 11.94), this question tests direct recall of the standard empirical rule approximations.

Topics

Statistics · 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.