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

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Required skills (click to reveal)

A statistical investigation is conducted to compare the average weekly spending on technology among Year 9 students in Melbourne and Sydney. A random sample of 28 students in Melbourne reported a total weekly spending of $1,400 on technology.

A separate random sample of 72 students in Sydney reported a total weekly spending of $3,600 on technology. What is the exact combined mean weekly spending on technology for the two samples?

Express your answer 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 total weekly spending for Melbourne is $1,400 from 28 students. Calculate the mean spending for Melbourne by dividing total spending by number of students.
  2. Mean spending in Melbourne = 1400 / 28. Perform division: 1400 ÷ 28 = 50. So, mean = $50 per student.
  3. The total weekly spending for Sydney is $3,600 from 72 students. Calculate the mean spending for Sydney by dividing total spending by number of students.
  4. Mean spending in Sydney = 3600 / 72. Perform division: 3600 ÷ 72 = 50. So, mean = $50 per student.
  5. The combined total spending is the sum of Melbourne and Sydney totals: 1400 + 3600 = $5,000.
  6. The combined total number of students is 28 + 72 = 100.
  7. The combined mean weekly spending is total spending divided by total number of students: 5000 / 100.
  8. Perform division: 5000 ÷ 100 = 50. So, combined mean = $50 per student.
  9. Express $50 as a simplified fraction: 50/1.
  10. Check that 50/1 is in simplest form: 50 and 1 have no common factors other than 1.
  11. Verify the combined mean: (28 × 50) + (72 × 50) = 1400 + 3600 = 5000. Total students = 100. 5000 / 100 = 50. Correct.
  12. The exact combined mean weekly spending on technology for the two samples is $50, which as a simplified fraction is 50/1.

Reference answer: 50/1

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