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4024/22 Oct/Nov 2024, Question 2: Estimating the Mean for Grouped Data

Cambridge O Level Mathematics 4024 · Paper 22 (Calculator) · Calculator · 10 marks

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The question, in short

Use a table of 10 cars' ages and values to plot points, fit a line, estimate a value at 7 years, and judge extrapolation to 12 years.

Summarised in our own words. Use the official paper for the exact wording and the diagram.

Part (ai) · 2 marks

Plot the four remaining age-value pairs from the table onto the given scatter diagram.

  1. 1Plot (5, 4500), (6, 3000), (8, 1500) and (8, 2000)

Answer4 points plotted correctly

Part (aii) · 1 mark

Add a straight line of best fit through the plotted points.

  1. 1Rule a straight line with negative gradient through the middle of the points, e.g. through the mean point (5.1, 3860)

AnswerRuled line of best fit

Part (aiii) · 1 mark

Read off the line the estimated value for a 7-year-old car.

  1. 1Go up from age 7 to the line and read across to the value axis
  2. 2Value ≈ $2400

Answer2400 (accept 2100 to 2700)

Part (aiv) · 1 mark

Explain why the diagram cannot reliably estimate a 12-year-old car's value.

  1. 112 years is outside the range of the data (ages 3 to 8), so the line would be extrapolated and may even give a negative value

AnswerOutside range of data OR 12 would give a negative value (which is impossible) OR The line of best fit does not go as far as 12

Part (bi) · 1 mark

Work out the fraction of 50 cars travelling over 50 000 km, simplified.

  1. 1More than 50 000 km: 11 + 17 = 28 cars
  2. 228/50 = 14/25

Answer14/25

Part (bii) · 1 mark

Identify which distance interval holds the median for the 50 cars.

  1. 1Median is the 25th/26th value; cumulative totals 8, 22, 33, 50
  2. 2Median interval: 50 < d ≤ 60

Answer50 < d ≤ 60

Part (biii) · 3 marks

Estimate the mean distance using the grouped frequency table's midpoints.

  1. 1Midpoints 25, 45, 55, 80
  2. 2Σfx = 8 × 25 + 14 × 45 + 11 × 55 + 17 × 80 = 200 + 630 + 605 + 1360 = 2795
  3. 3Mean = 2795 ÷ 50 = 55.9 thousand km

Answer55.9

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