AQA AS Level Mathematics Paper 2, June 2025: Question 19

✅ Correct Answer

1

💡 Key Knowledge

Pure electric vehicles (propulsion type: Electric) do not produce direct carbon emissions. Therefore, their CO₂ emissions value is identically x = 0 g/km.

Looking at the table, the frequency for x = 0 is 1.

❌ Common Errors

• Summing all frequencies (19) assuming all cars in the table are electric.
• Confusing pure electric cars with hybrid/petrol-electric cars (which have low but non-zero emissions, e.g. 13, 45, 49 g/km).

🧠 Exam Technique

Read the question header carefully: it states the distribution contains "electric and electric/petrol BMW cars". A value of 0 g/km is the defining signature of pure electric propulsion.

✅ Correct Answer

Mean = 48 (g/km)

📐 Step-by-Step Calculation

1. Find total frequency (n):
   n = Σf = 1 + 2 + 3 + 9 + 2 + 2 = 19
2. Find sum of emissions (Σfx):
   Σfx = (0 × 1) + (13 × 2) + (45 × 3) + (49 × 9) + (77 × 2) + (78 × 2)
   Σfx = 0 + 26 + 135 + 441 + 154 + 156 = 912
3. Compute mean (x̄):
   x̄ = Σfx / n = 912 / 19 = 48

🧠 Calculator Shortcut

Do not calculate this by hand! Put your Casio calculator into Statistics Mode (1-Variable), ensure Frequency is ON, enter the table values, and read x̄ directly to save time and prevent arithmetic errors.

❌ Common Errors

• Dividing by 6 (the number of distinct rows/values) instead of 19 (the total frequency).
• Omitting the row x = 0 from the total frequency count.

✅ Correct Answer

Standard Deviation = 20.9 (g/km)

Accepts AWRT 20.9 (population σ) or AWRT 21.4 (sample s).

📐 Step-by-Step Calculation

1. Calculate Σfx²:
   Σfx² = (0²×1) + (13²×2) + (45²×3) + (49²×9) + (77²×2) + (78²×2)
   Σfx² = 0 + 338 + 6075 + 21609 + 11858 + 12168 = 52048
2. Population Standard Deviation (σ):
   σ = √[(Σfx² / n) - (x̄)²]
   σ = √[(52048 / 19) - 48²] = √[2739.368... - 2304] = √435.368... ≈ 20.865... ≈ 20.9
3. Sample Standard Deviation (s):
   s = √[52048 - (912²/19)] / 18 ≈ 21.43... ≈ 21.4

❌ Common Errors

• Forgetting to take the square root at the end, writing the variance (435.4) instead.
• Premature rounding during intermediate steps.

🧠 Exam Technique

Both population standard deviation (σx = 20.9) and sample standard deviation (sx = 21.4) are accepted on mark schemes for 1-mark standard deviation questions. Using the calculator's built-in stats output guarantees full marks.

✅ Correct Answer

The 2002 data does not contain any electric or electric/petrol cars (or: neither propulsion type existed in the 2002 data set).

💡 Large Data Set Fact File

• The AQA Large Data Set contains car registration data from two separate years: 2002 and 2016.
• In 2002, all recorded cars were powered only by traditional internal combustion engines (Petrol and Diesel).
• Alternative propulsion types (Electric and Electric/Petrol hybrids) were only introduced in the 2016 data set.

❌ Common Errors (Zero Marks Awarded)

• "Sample size is too small" — Misses the core factual reality of the LDS.
• "Technology has changed since 2002" — Too vague; fails to address whether data exists to calculate stats.
• "Data was collected in different years so it cannot be compared" — Comparing across years is valid if the categories exist!
• Mentioning only one fuel type: you must be clear that neither electric nor hybrid cars were present in 2002.

🧠 Examiner Insight

Examiners deliberately write LDS questions to check if you have explored the actual spreadsheet. If a comparison is declared "not valid", check whether the subset you are asked to compare even exists in both years! You cannot calculate a mean or standard deviation of an empty set (frequency = 0).