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0580/42 Feb/Mar 2022, Question 5: Estimating the Mean for Grouped Data

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

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The question, in short

From a grouped frequency table of 120 letter masses, estimate the mean, scale histogram bar heights, build and plot cumulative frequency, then read off median, quartiles, percentile and a count.

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

Part (a) · 4 marks

Estimate the mean mass using the grouped frequency table with 120 letters.

  1. 1Midpoints: 25, 75, 150, 350
  2. 2Σfx = 43×25 + 31×75 + 25×150 + 21×350 = 1075 + 2325 + 3750 + 7350 = 14500
  3. 3Mean = 14500 ÷ 120 = 120.83… ≈ 121 g

Answer121 or 120.8… or 120 5/6

Part (b) · 3 marks

Given the first bar height is 17.2 cm, calculate the heights of the other three histogram bars.

  1. 1Frequency densities: 43/50 = 0.86, 31/50 = 0.62, 25/100 = 0.25, 21/300 = 0.07
  2. 2Scale: 17.2 ÷ 0.86 = 20 cm per unit
  3. 3Heights: 0.62 × 20 = 12.4 cm, 0.25 × 20 = 5 cm, 0.07 × 20 = 1.4 cm

Answer50 < m ≤ 100: 12.4 cm; 100 < m ≤ 200: 5 cm; 200 < m ≤ 500: 1.4 cm

Part (c) · 2 marks

Fill in the cumulative frequency values for m ≤ 50, 100, 200 and 500.

  1. 1Cumulative frequency is a running total: keep adding the next frequency.
  2. 2m ≤ 50: 43
  3. 3m ≤ 100: 43 + 31 = 74
  4. 4m ≤ 200: 74 + 25 = 99
  5. 5m ≤ 500: 99 + 21 = 120

Answer| m ≤ 50 | m ≤ 100 | m ≤ 200 | m ≤ 500 | | 43 | 74 | 99 | 120 |

Part (d) · 3 marks

Plot the cumulative frequency diagram on the given axes of mass against cumulative frequency.

  1. 1Plot (0, 0), (50, 43), (100, 74), (200, 99), (500, 120) at the upper class boundaries
  2. 2Join with a smooth increasing curve (or straight lines)

AnswerCorrect diagram

Part (ei) · 1 mark

Use your cumulative frequency diagram to estimate the median mass.

  1. 1Median position = 120 ÷ 2 = 60
  2. 2Read across from cumulative frequency 60 to the curve and down to the mass axis

Answer76 (accept 72 to 80)

Part (eii) · 1 mark

Use your cumulative frequency diagram to estimate the upper quartile mass.

  1. 1Upper quartile position = 3/4 × 120 = 90
  2. 2Read across from cumulative frequency 90 to the curve and down to the mass axis

Answer158 (accept 150 to 170)

Part (eiii) · 2 marks

Use your cumulative frequency diagram to estimate the 40th percentile mass.

  1. 140% of 120 = 48
  2. 2Read across from cumulative frequency 48 to the curve and down to the mass axis

Answer58 (accept 55 to 61)

Part (eiv) · 2 marks

Use your cumulative frequency diagram to estimate how many letters have mass 250 < m ≤ 400.

  1. 1Read up from m = 400 and from m = 250 to the curve and across to the cumulative frequency axis
  2. 2Number = (reading at 400) − (reading at 250)

Answer11 (accept 9 to 13)

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