In competition, a diver must complete 8 dives. Her scores for the first 7 dives are:7.3, 7.4, 7.7, 8.4, 8.7, 8.9, 9.4Determine her score in the 8th round if the upper quartile (Q3) of all her 8 scores is 8.85.

Question

BlackTom AI · Accepted Answer

For eight data points the upper quartile Q3 is the median of the upper half; with n = 8 that is the average of the 6th and 7th smallest values in the sorted list. So Q3 = (value at position 6 + value at position 7)/2.

1. Relevant concepts
- Order the data and identify positions.
- For an even sample of size 8, split into lower half (positions 1–4) and upper half (positions 5–8); the median of the upper half is the average of the middle two of those four, i.e. positions 6 and 7.

2. Step-by-step reasoning
- Start with the given seven scores in ascending order: 7.3, 7.4, 7.7, 8.4, 8.7, 8.9, 9.4. Let x be the 8th score. After inserting x and sorting, label the ordered values 1 through 8.
- Q3 is (value at position 6 + value at position 7)/2 and we are told this equals 8.85. Therefore value6 + value7 = 2 × 8.85 = 17.7.

3. Calculations and intermediate steps
- Consider where x could fall. If x falls after 8.7 but before or equal to 8.9 (i.e. 8.7 < x ≤ 8.9), the ordered list becomes:
  7.3, 7.4, 7.7, 8.4, 8.7, x, 8.9, 9.4.
  Then position 6 = x and position 7 = 8.9.
- Plugging into the quartile equation: x + 8.9 = 17.7 → x = 17.7 − 8.9 = 8.8.
- This value 8.8 lies in the interval (8.7, 8.9], so the placement assumption is consistent.

4. Why the correct answer is correct
- With x = 8.8 the sorted scores are 7.3, 7.4, 7.7, 8.4, 8.7, 8.8, 8.9, 9.4. Positions 6 and 7 are 8.8 and 8.9, whose average is (8.8 + 8.9)/2 = 17.7/2 = 8.85, exactly the given Q3.

5. Why the other options are incorrect
- 8.5 or any x ≤ 8.4 would place x earlier so positions 6 and 7 would be 8.7 and 8.9 giving Q3 = (8.7+8.9)/2 = 8.8 (not 8.85).
- 9.0 would require x to be ≤ 8.9 or in a position that makes position 6 or 7 equal to x, but x = 9.0 does not fit the necessary interval; plugging in 9.0 would give different positions and not produce Q3 = 8.85.
- 9.2 (or any x ≥ 9.4) makes positions 6 and 7 equal to 8.9 and 9.4, giving Q3 = (8.9+9.4)/2 = 9.15, not 8.85.

6. Conclusion
The only value consistent with Q3 = 8.85 is x = 8.8. This matches the provided correct answer: 8.8.

In competition, a diver must complete 8 dives. Her scores for the first 7 dives are:7.3, 7.4, 7.7, 8.4, 8.7, 8.9, 9.4Determine her score in the 8th round if the upper quartile (Q3) of all her 8 scores is 8.85.Single choice

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