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4024/21 May/Jun 2021, Question 9: Estimating the Mean for Grouped Data

Cambridge O Level Mathematics 4024 · Paper 21 (Calculator) · Calculator · 10 marks

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The question, in short

From a cumulative frequency curve for 80 students' Maths marks, estimate the median, find the pass mark for a 60% pass rate, complete a grouped frequency table, and use a Science test table with mean 67.625 to find p and q.

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

Part (ai) · 1 mark

Use the cumulative frequency diagram to estimate the median mark.

  1. 1Median at cumulative frequency 80 ÷ 2 = 40
  2. 2Read across and down: median ≈ 48

Answer48

Part (aii) · 2 marks

Use the diagram to find the mark needed for a pass, given 60% of students passed.

  1. 140% failed: 40/100 × 80 = 32 students
  2. 2Read mark at cumulative frequency 32
  3. 3Pass mark ≈ 44

Answer43 to 45

Part (aiii) · 2 marks

Use the diagram to complete the frequency table for the five mark intervals, given the first frequency is 8.

  1. 1Cumulative frequencies at 20, 40, 60, 80, 100: 8, 26, 58, 74, 80
  2. 2Frequencies: 26 − 8 = 18, 58 − 26 = 32, 74 − 58 = 16, 80 − 74 = 6

Answer| Mark (m) | 20 ≤ m < 40 | 40 ≤ m < 60 | 60 ≤ m < 80 | 80 ≤ m < 100 | | Frequency | 18 | 32 | 16 | 6 |

Part (b) · 5 marks

Using mid-interval values and the mean 67.625 minutes for 80 students, calculate p and q in the Science test frequency table.

  1. 18 + 13 + p + 20 + q = 80 → p + q = 39
  2. 2Midpoints 45, 55, 65, 75, 85: 8 × 45 + 13 × 55 + 65p + 20 × 75 + 85q = 67.625 × 80 = 5410
  3. 3360 + 715 + 1500 + 65p + 85q = 5410 → 65p + 85q = 2835
  4. 465p + 85(39 − p) = 2835 → 3315 − 20p = 2835 → p = 24
  5. 5p = 24, q = 15

Answerp = 24, q = 15

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