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0580/42 Oct/Nov 2024, Question 5: Estimating the Mean for Grouped Data

Cambridge IGCSE Mathematics 0580 · Paper 42 (Extended) · Calculator · 15 marks

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The question, in short

Use a cumulative frequency graph for 80 motorists' monthly distances, a lap speed calculation, and a histogram of 160 cars' top speeds to find averages and probabilities.

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

Part (aia) · 1 mark

Estimate the median distance from the cumulative frequency curve.

  1. 1Median at cumulative frequency 80 ÷ 2 = 40
  2. 2Read across from 40 to the curve and down: median = 1480 km

Answer1480

Part (aib) · 2 marks

Estimate the interquartile range from the cumulative frequency curve.

  1. 1Upper quartile at CF 60: UQ = 1600 km
  2. 2Lower quartile at CF 20: LQ = 1160 km
  3. 3IQR = 1600 − 1160 = 440 km

Answer440

Part (aii) · 2 marks

Find the probability a randomly chosen motorist travels more than 1800 km.

  1. 1From 1800 km the curve reads a cumulative frequency of 72
  2. 2Number travelling more than 1800 km = 80 − 72 = 8
  3. 3P = 8/80

Answer8/80

Part (b) · 3 marks

Calculate the car's average speed in km/h from a 5.104 km lap in 1 min 18 s.

  1. 1Time = 1 min 18 s = 1.3 min
  2. 2Speed = 5.104 ÷ 1.3 × 60
  3. 3= 235.56… ≈ 236 km/h

Answer236 or 235.5 to 235.6

Part (ci) · 1 mark

Show that 8 cars have top speed in the interval 120 < v ≤ 160 using the histogram.

  1. 1Frequency = class width × frequency density
  2. 2= (160 − 120) × 0.2 = 8

Answer(160 − 120) × 0.2 = 8

Part (cii) · 6 marks

Estimate the mean top speed of the 160 cars, showing all working.

  1. 1Frequencies: 8, 1.1 × 20 = 22, 1.8 × 20 = 36, 1.6 × 40 = 64, 0.5 × 60 = 30 (total 160)
  2. 2Midpoints: 140, 170, 190, 220, 270
  3. 3Σfx = 8 × 140 + 22 × 170 + 36 × 190 + 64 × 220 + 30 × 270
  4. 4Σfx = 1120 + 3740 + 6840 + 14 080 + 8100 = 33 880
  5. 5Mean = 33 880 ÷ 160 = 211.75 km/h

Answer211.75

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