To determine which value would change the most when replacing 91 with 99 in the set of numbers \(46, 27, 46, 91, 10, 28, 16, 96\), we will calculate the mean, median, and mode before and after the replacement.
Original Set: \(46, 27, 46, 91, 10, 28, 16, 96\)
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Mean:
- Sum of original set: \(46 + 27 + 46 + 91 + 10 + 28 + 16 + 96 = 364\)
- Number of values: \(8\)
- Mean: \(\frac{364}{8} = 45.5\)
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Median:
- Sorted order: \(10, 16, 27, 28, 46, 46, 91, 96\)
- Median (average of two middle numbers): \(\frac{28 + 46}{2} = 37\)
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Mode:
- The number that appears most frequently: \(46\) (appears twice)
New Set: \(46, 27, 46, 99, 10, 28, 16, 96\)
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Mean:
- Sum of new set: \(46 + 27 + 46 + 99 + 10 + 28 + 16 + 96 = 368\)
- Mean: \(\frac{368}{8} = 46\)
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Median:
- Sorted order: \(10, 16, 27, 28, 46, 46, 96, 99\)
- Median: \(\frac{28 + 46}{2} = 37\)
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Mode:
- The number that appears most frequently: \(46\) (still appears twice)
Changes:
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Mean:
- Original Mean: \(45.5\)
- New Mean: \(46\)
- Change: \(46 - 45.5 = 0.5\)
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Median:
- Original Median: \(37\)
- New Median: \(37\)
- Change: \(0\)
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Mode:
- Original Mode: \(46\)
- New Mode: \(46\)
- Change: \(0\)
Conclusion:
- The mean changes by \(0.5\), while the median and mode remain unchanged.
- Therefore, the mean changes the most as a result of replacing \(91\) with \(99\).
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