NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Evaluating statistical claims

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A school in Brisbane ordered 70 new science lab kits and 33 new sets of protective goggles for its Year 10 students. The order was placed based on a survey of 120 students, which found that 70 students preferred lab kits and 33 preferred goggles.

The school administration claims that the survey results are representative of the entire student body of 1100 students.

What is the exact proportion of students in the survey who preferred lab kits, expressed as a simplified fraction?

Concept illustrations

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Median, mode and range in sorted data. The ordered values are 2,3,3,8,9. The middle value is 3, the most frequent value is 3, and the range is 9 − 2 = 7. The row uses equal spacing, not a number-line scale.23389↑ medianmode 3 range 7
The ordered values are 2,3,3,8,9. The middle value is 3, the most frequent value is 3, and the range is 9 − 2 = 7. The row uses equal spacing, not a number-line scale.
Comparing three category counts. Illustrative category counts: A = 2, B = 4, C = 9. Separate bars compare categories; the vertical scale starts at zero.05102A4B9C5
Illustrative category counts: A = 2, B = 4, C = 9. Separate bars compare categories; the vertical scale starts at zero.

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Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. The survey included 120 students, and 70 of them preferred lab kits.
  2. The proportion of students who preferred lab kits is calculated as the number who preferred lab kits divided by the total number surveyed.
  3. Set up the fraction: 70 / 120.
  4. Simplify the fraction by dividing both numerator and denominator by their greatest common divisor (GCD).
  5. Find the GCD of 70 and 120. The prime factors of 70 are 2 × 5 × 7. The prime factors of 120 are 2 × 2 × 2 × 3 × 5. The common factors are 2 and 5, so GCD = 2 × 5 = 10.
  6. Divide both numerator and denominator by 10: 70 ÷ 10 = 7, 120 ÷ 10 = 12.
  7. The simplified fraction is 7/12.
  8. Verify the simplification: 7 × 10 = 70, 12 × 10 = 120, so 7/12 = 70/120.
  9. Check that 7 and 12 have no common factors other than 1. 7 is prime, and 12 is not divisible by 7.
  10. The proportion of students who preferred lab kits is exactly 7/12.
  11. The final answer is 7/12, which is in simplest form.
  12. The question asks for the exact proportion as a simplified fraction, which is 7/12.

Reference answer: 7/12

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