To analyze the changes in the median and mean when the number 178 is changed to 259 in the given list of student enrollments, we start with the original dataset:
Original data:
178, 184, 267, 268, 279, 280, 316, 361, 387
New data after change:
184, 259, 267, 268, 279, 280, 316, 361, 387
Part (A): Changes to the Median
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Original Median Calculation:
- The original dataset has 9 entries (odd number), so the median is the middle number.
- The median is the 5th number: 279.
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New Median Calculation:
- The new dataset also has 9 entries (odd number).
- The median is still the 5th number: 279.
Conclusion for Part (A):
- Since the median remains 279, there is no change to the median.
Part (B): Changes to the Mean
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Original Mean Calculation:
- First, we sum the original data: \[ 178 + 184 + 267 + 268 + 279 + 280 + 316 + 361 + 387 = 2220 \]
- Now, we calculate the mean by dividing the total by the number of entries (9): \[ \text{Original Mean} = \frac{2220}{9} \approx 246.67 \]
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New Mean Calculation:
- Next, we sum the new data: \[ 184 + 259 + 267 + 268 + 279 + 280 + 316 + 361 + 387 = 2281 \]
- Now, we calculate the mean: \[ \text{New Mean} = \frac{2281}{9} \approx 253.44 \]
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Change in Mean Calculation:
- The change in the mean is: \[ \text{Change} = 253.44 - 246.67 \approx 6.77 \]
Conclusion for Part (B):
- The mean increases by approximately 6.77.
Summary:
- (A) The median remains 279 (no change).
- (B) The mean increases by approximately 6.77.