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0580/43 May/Jun 2022, Question 5: Estimating the Mean for Grouped Data

Cambridge IGCSE Mathematics 0580 · Paper 43 (Extended) · Calculator · 19 marks

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The question, in short

From a grouped frequency table of 80 travel times, find the median class, estimate the mean, work out probabilities, complete and graph cumulative frequency, and estimate a percentile and a percentage.

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

Part (ai) · 1 mark

State which time class interval holds the median travel time.

  1. 1Median is the 40th−41st value; cumulative frequencies 3, 10, 28, 56
  2. 2Median class: 20 < t ≤ 35

Answer20 < t ≤ 35

Part (aii) · 4 marks

Use class midpoints and frequencies to calculate an estimated mean travel time.

  1. 1Mid-values: 2.5, 7.5, 15, 27.5, 47.5
  2. 2Σfm = 3×2.5 + 7×7.5 + 18×15 + 28×27.5 + 24×47.5
  3. 3= 7.5 + 52.5 + 270 + 770 + 1140 = 2240
  4. 4Mean = 2240 ÷ 80 = 28 min

Answer28

Part (bi) · 2 marks

Find, as a simplified fraction, the probability a randomly chosen person travelled more than 10 minutes.

  1. 1Longer than 10 min: 18 + 28 + 24 = 70
  2. 2P = 70/80 = 7/8

Answer7/8

Part (bii) · 3 marks

Choosing two people from those taking 20 minutes or less, find the probability one took 5 minutes or less and the other more than 5.

  1. 120 minutes or less: 3 + 7 + 18 = 28 people; 3 took 5 min or less, 25 took more
  2. 2P = 3/28 × 25/27 + 25/28 × 3/27
  3. 3= 2 × 75/756 = 150/756 = 25/126

Answer25/126

Part (ci) · 1 mark

Fill in the missing cumulative frequencies for t ≤ 20 and t ≤ 35.

  1. 110 + 18 = 28, 28 + 28 = 56
  2. 2t ≤ 20: 28, t ≤ 35: 56

Answer28 and 56

Part (cii) · 3 marks

Plot the cumulative frequency diagram on the grid using the completed table.

  1. 1Plot (5, 3), (10, 10), (20, 28), (35, 56), (60, 80) at the upper class boundaries
  2. 2Start at (0, 0) and join with a smooth increasing curve (or straight lines)

AnswerCorrect diagram

Part (ciii) · 2 marks

Use the cumulative frequency diagram to estimate the 80th percentile time.

  1. 180th percentile: 80% of 80 = 64, read across from cumulative frequency 64
  2. 2With straight lines: 35 + (64 − 56)/24 × 25 = 43.3…
  3. 380th percentile ≈ 43 minutes

Answer43 (accept 41 to 45)

Part (civ) · 3 marks

Estimate the percentage of people whose travel time exceeded 45 minutes, showing working.

  1. 1Read up from t = 45: cumulative frequency ≈ 56 + 10/25 × 24 = 65.6 (with straight lines)
  2. 2Longer than 45 min: 80 − 65.6 = 14.4 people
  3. 314.4 ÷ 80 × 100 ≈ 18%

Answer17.5 (accept 14 to 20)

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Also searched as: 0580/43/M/J/22 · 0580_s22_qp_43 · 0580 s22 43 · question 5.
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