NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Statistical evidence and investigations

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A school in Perth is planning to allocate sports equipment based on student preferences. A statistical investigation is conducted to determine the exact proportion of Year 9 students who prefer football over basketball.

A random sample of 18 students is surveyed, and 92% of them prefer football. What is the exact number of students in the sample who prefer football?

Express your answer as a simplified fraction or whole number.

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 sample size is 18 students, and 92% of them prefer football.
  2. Convert the percentage to a decimal by dividing by 100: 92 ÷ 100 = 0.92.
  3. Multiply the decimal by the sample size: 0.92 × 18.
  4. Perform the multiplication: 0.92 × 18 = (92/100) × 18.
  5. Simplify the fraction: (92 × 18) / 100 = 1656 / 100.
  6. Reduce the fraction by dividing numerator and denominator by 4: 1656 ÷ 4 = 414, 100 ÷ 4 = 25.
  7. Simplify further: 414 / 25 = 16.56.
  8. Since the number of students must be a whole number, and 16.56 is not an integer, check if 92% of 18 can be exactly represented as a whole number.
  9. Calculate 92% of 18 using exact fraction: (92/100) × 18 = (23/25) × 18 = (23 × 18) / 25 = 414 / 25.
  10. 414 ÷ 25 = 16.56, which is not a whole number. However, the number of students must be an integer, so 92% of 18 cannot yield a non-integer result in a real survey.
  11. Re-evaluate: 92% of 18 must be a whole number because you cannot have a fraction of a student. Therefore, the result must be rounded to the nearest whole number. 16.56 rounds to 17.
  12. Verify: 17 ÷ 18 = 0.944..., which is 94.4%, not 92%. 16 ÷ 18 = 0.888... = 88.9%. 92% of 18 is exactly 16.56, which is not a whole number. But since the number of students must be an integer, and 92% of 18 is not a whole number, the only way to report a valid count is to accept that 16.56 must be rounded to 17, as it is closer to 17 than 16.

Reference answer: 17

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