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4024/22 May/Jun 2024, Question 9: Estimating the Mean for Grouped Data

Cambridge O Level Mathematics 4024 · Paper 22 (Calculator) · Calculator · 13 marks

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The question, in short

Use a cumulative frequency curve for 80 Variety A trees to estimate median and percentile, find a Class I height boundary, complete a frequency table, and use midpoints to find p and q for 50 Variety B trees.

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Part (aia) · 1 mark

Read the median height in metres from the cumulative frequency curve for the 80 trees.

  1. 1Median at cumulative frequency 80 ÷ 2 = 40
  2. 2Read across from 40 to the curve and down: ≈ 1.33 m

Answer1.33 to 1.335

Part (aib) · 2 marks

Use the curve to estimate the 30th percentile height in metres.

  1. 130% of 80 = 24
  2. 2Read across from 24 to the curve and down: ≈ 1.28 m

Answer1.28 to 1.285

Part (aii) · 2 marks

Given 2/5 of the 80 trees are Class I, find the height y above which trees are Class I.

  1. 1Class I trees = 2/5 × 80 = 32, so 80 − 32 = 48 trees are y m or less
  2. 2Read across from 48 to the curve and down: y ≈ 1.36

Answer1.355 to 1.36

Part (aiii) · 2 marks

Complete the frequency table for the five height intervals using the curve.

  1. 1Read cumulative frequencies: 30 at 1.3, 58 at 1.4, 76 at 1.5, 80 at 1.6
  2. 2Frequencies: 58 − 30 = 28, 76 − 58 = 18, 80 − 76 = 4

Answer28, 18, 4

Part (b) · 6 marks

With midpoints and an estimated mean of 1.81 m for 50 trees, calculate p and q.

  1. 1p + 15 + 17 + q = 50, so p + q = 18
  2. 2Midpoints: 1.6, 1.75, 1.85, 2.1
  3. 31.6p + 1.75 × 15 + 1.85 × 17 + 2.1q = 1.81 × 50 = 90.5
  4. 41.6p + 26.25 + 31.45 + 2.1q = 90.5, so 1.6p + 2.1q = 32.8
  5. 51.6(18 − q) + 2.1q = 32.8 → 28.8 + 0.5q = 32.8 → q = 8
  6. 6p = 10, q = 8

Answerp = 10, q = 8

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