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0580/42 May/Jun 2023, Question 4: Estimating the Mean for Grouped Data

Cambridge IGCSE Mathematics 0580 · Paper 42 (Extended) · Calculator · 18 marks

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The question, in short

Using a grouped frequency table for heights of 80 children, find the median interval, estimate the mean, work out probabilities, complete cumulative frequency, draw the curve, and read off IQR and 60th percentile.

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

Part (ai) · 1 mark

State which height interval contains the median for the 80 children.

  1. 1Median is the 40th−41st value
  2. 2Cumulative: 2, 15, 39, 71
  3. 3The 40th value is in 1.65 < h ≤ 1.8

Answer1.65 < h ≤ 1.8

Part (aii) · 4 marks

Calculate an estimated mean height from the grouped frequencies.

  1. 1Midpoints: 1.3, 1.45, 1.575, 1.725, 1.85
  2. 2Σfh = 2 × 1.3 + 13 × 1.45 + 24 × 1.575 + 32 × 1.725 + 9 × 1.85
  3. 3= 2.6 + 18.85 + 37.8 + 55.2 + 16.65 = 131.1
  4. 4Mean = 131.1 ÷ 80 = 1.63875 m

Answer1.63875

Part (bi) · 1 mark

Find the probability a randomly chosen child is 1.4 m or shorter.

  1. 12 children have h ≤ 1.4
  2. 2P = 2/80 = 1/40

Answer1/40

Part (bii) · 3 marks

For two children chosen at random, find the probability both exceed 1.5 m with exactly one above 1.8 m.

  1. 1Taller than 1.5 m but not taller than 1.8 m: 24 + 32 = 56; taller than 1.8 m: 9
  2. 2P = 56/80 × 9/79 + 9/80 × 56/79
  3. 3= 2 × 504/6320 = 1008/6320
  4. 4= 63/395

Answer63/395

Part (ci) · 2 marks

Fill in the missing cumulative frequency values for each height boundary.

  1. 12 + 13 = 15
  2. 215 + 24 = 39
  3. 339 + 32 = 71
  4. 471 + 9 = 80

Answer15, 39, 71, 80

Part (cii) · 3 marks

Plot the cumulative frequency diagram on the given grid.

  1. 1Plot each cumulative frequency at the upper boundary of its class
  2. 2Points (1.4, 2), (1.5, 15), (1.65, 39), (1.8, 71), (1.9, 80)
  3. 3Join with a smooth increasing curve

AnswerCorrect curve through (1.4, 2), (1.5, 15), (1.65, 39), (1.8, 71), (1.9, 80)

Part (di) · 2 marks

Use your cumulative frequency diagram to estimate the interquartile range.

  1. 1LQ at cumulative frequency 20: about 1.53 m
  2. 2UQ at cumulative frequency 60: about 1.75 m
  3. 3IQR ≈ 1.75 − 1.53 = 0.22 m

Answer0.22 (accept 0.2 to 0.24)

Part (dii) · 2 marks

Use your cumulative frequency diagram to estimate the 60th percentile height.

  1. 160% of 80 = 48
  2. 2Read across from 48 to the curve and down
  3. 360th percentile ≈ 1.69 m

Answer1.69 (accept 1.68 to 1.70)

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Also searched as: 0580/42/M/J/23 · 0580_s23_qp_42 · 0580 s23 42 · question 4.
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