NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Survey means and medians

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A survey of 87 students in Canberra was conducted to estimate the median number of minutes spent daily on homework. The data, arranged in ascending order, shows that the 44th value is 45 minutes and the 45th value is 52 minutes.

What is the exact median number of minutes spent on homework per day for this sample?

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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

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Worked solution

  1. The survey includes 87 students, which is an odd number, so the median is the value at the middle position when the data is ordered.
  2. To find the middle position in a data set of size n, use the formula: (n + 1) / 2.
  3. Substitute n = 87 into the formula: (87 + 1) / 2 = 88 / 2 = 44.
  4. The 44th value in the ordered data set is 45 minutes.
  5. Since the median is the 44th value in this odd-sized data set, and the 44th value is 45 minutes, the median is 45 minutes.
  6. The 45th value is 52 minutes, but this is not needed because the median is determined by the 44th value in an odd-sized sample.
  7. No interpolation is required because the median position corresponds exactly to a data point in the list.
  8. The median is therefore exactly 45 minutes.
  9. Check: for n = 87, the middle index is 44, and the 44th value is given as 45 minutes.
  10. The result is consistent with the definition of median for an odd number of observations.
  11. The median number of minutes spent on homework per day is 45 minutes.
  12. Final verification: 87 students, ordered, 44th value is 45 minutes — this is the median.

Reference answer: 45 minutes

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