NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Sampling methods and bias

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A school in Brisbane is allocating 46 new basketballs and 38 new footballs to 10 sports classrooms. The allocation is based on a survey of student preferences conducted across 5 randomly selected classrooms, which represent 1/2 of the total classrooms.

The survey found that 46 students preferred basketballs and 38 preferred footballs. What is the exact proportion of surveyed students who preferred basketballs?

Express your answer as a simplified fraction.

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A sample is part of a population. The large outline represents a population of 24 members. Six filled members illustrate a sample. This picture does not establish that a selection process was random.6 selected from 24
The large outline represents a population of 24 members. Six filled members illustrate a sample. This picture does not establish that a selection process was random.
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.

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Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. The survey was conducted in 5 classrooms, and the number of students who preferred basketballs is given as 46.
  2. The number of students who preferred footballs is 38.
  3. The total number of surveyed students is the sum of those who preferred basketballs and footballs: 46 + 38 = 84 students.
  4. The proportion of surveyed students who preferred basketballs is calculated as the number who preferred basketballs divided by the total number of surveyed students: 46 / 84.
  5. Simplify the fraction 46/84 by finding the greatest common divisor (GCD) of 46 and 84. The GCD of 46 and 84 is 2.
  6. Divide both numerator and denominator by 2: 46 ÷ 2 = 23, 84 ÷ 2 = 42.
  7. The simplified fraction is 23/42.
  8. Check that 23 and 42 have no common factors other than 1. 23 is a prime number, and 42 is not divisible by 23.
  9. Therefore, 23/42 is in its simplest form.
  10. The exact proportion of surveyed students who preferred basketballs is 23/42.
  11. Verify: 23 ÷ 42 ≈ 0.5476, and 46 ÷ 84 ≈ 0.5476, confirming the fraction is correct.
  12. The final answer is 23/42, as a simplified fraction.

Reference answer: 23/42

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