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0580/42 Oct/Nov 2022, Question 3: Estimating the Mean for Grouped Data

Cambridge IGCSE Mathematics 0580 · Paper 42 (Extended) · Calculator · 15 marks

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The question, in short

Using grouped distance data for 200 cars, estimate a mean, compare two versions of the final interval, work out histogram block heights, and find probabilities for randomly chosen cars.

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

Part (ai) · 4 marks

Estimate the mean distance from Kai's grouped frequency table.

  1. 1Mid-points: 90, 125, 175, 250, 350
  2. 2Σfx = 7×90 + 33×125 + 76×175 + 52×250 + 32×350 = 42255
  3. 3Mean = 42255 ÷ 200 = 211.275 km

Answer211.275

Part (aii) · 2 marks

Using Ann's table with a different last interval, find how much her mean estimate differs from Kai's, showing working.

  1. 1Only the last mid-point changes: 350 → 330, a change of 20
  2. 2Difference = 32 × 20 ÷ 200 = 3.2 km

Answer3.2 or −3.2

Part (aiii) · 3 marks

Given the 200 < d ≤ 300 block is 2.6 cm high, find the heights of the blocks for 80 < d ≤ 100, 150 < d ≤ 200 and 300 < d ≤ 400.

  1. 1Frequency density 200 < d ≤ 300: 52 ÷ 100 = 0.52 → 2.6 cm, so height = 5 × frequency density
  2. 280 < d ≤ 100: 7 ÷ 20 = 0.35 → 1.75 cm
  3. 3150 < d ≤ 200: 76 ÷ 50 = 1.52 → 7.6 cm
  4. 4300 < d ≤ 400: 32 ÷ 100 = 0.32 → 1.6 cm

Answer80 < d ≤ 100: 1.75 cm 150 < d ≤ 200: 7.6 cm 300 < d ≤ 400: 1.6 cm

Part (b) · 1 mark

Find the probability that a single randomly chosen car travelled more than 300 km.

  1. 1P(more than 300 km) = 32/200 = 4/25

Answer4/25

Part (ci) · 2 marks

Find the probability that two randomly chosen cars both travelled 150 km or less.

  1. 1150 km or less: 7 + 33 = 40 cars
  2. 2P = 40/200 × 39/199 = 1560/39800 = 39/995

Answer39/995

Part (cii) · 3 marks

Find the probability that one chosen car went over 200 km while the other went 100 km or less.

  1. 1More than 200 km: 52 + 32 = 84 cars, 100 km or less: 7 cars
  2. 2P = 2 × 84/200 × 7/199 = 1176/39800
  3. 3= 147/4975

Answer147/4975

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Also searched as: 0580/42/O/N/22 · 0580_w22_qp_42 · 0580 w22 42 · question 3.
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