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Zorluk: OrtaMeasures of Central Tendency (Mean, Median, Mode)

A regional library recorded the daily number of public computer reservations over a 7-day period: 14,25,18,31,25,12,14, 25, 18, 31, 25, 12, and 2121. During the following week, 2 additional daily reservation values, xx and yy, were added to the dataset. If the median of the resulting 9-day dataset is equal to the mode of the original 7-day dataset, which of the following values could be the sum of xx and yy?

  1. A
    36
  2. B
    42
  3. C
    46
  4. D
    48
  5. 54Cevap

Cevap

54
The mode of the original 7 values is 25. When 2 values (xx and yy) are added to form a 9-value dataset, the median is the 5th value when ordered. In the original sorted dataset (12,14,18,21,25,25,3112, 14, 18, 21, 25, 25, 31), there are only 3 values greater than or equal to 25. To make the 5th value of the combined 9-value list equal to 25, both added values must be at least 25 (x≥25x \ge 25 and y≥25y \ge 25). Consequently, their sum x+yx + y must be at least 5050. The only choice that satisfies x+y≥50x + y \ge 50 is 54.

Adım Adım Çözüm

1
Identify the mode of the original 7-day dataset.
The original values are 14,25,18,31,25,12,2114, 25, 18, 31, 25, 12, 21. The value 2525 appears twice, while all other numbers appear once. Therefore, the mode is 2525.
The question states that the target median of the 9-day dataset must equal the mode of the original dataset.
2
Sort the original 7-day dataset in ascending order.
The sorted original dataset is 12,14,18,21,25,25,3112, 14, 18, 21, 25, 25, 31.
Determining position-based metrics like median requires ordering the data.
3
Determine the condition required for the median of the 9-day dataset to be 25.
In a sorted 9-element dataset, the median is the 5th element. For the 5th element to be 2525, there must be at least 5 elements in the set that are greater than or equal to 2525. The original set contains 3 elements ≥25\ge 25 (25,25,3125, 25, 31). Thus, both new values xx and yy must be ≥25\ge 25.
If only one of xx or yy were ≥25\ge 25, there would only be 4 elements ≥25\ge 25, making the 5th element (median) less than 2525.
4
Calculate the lower bound for the sum x+yx + y and select the valid option.
Since x≥25x \geq 25 and y≥25y \geq 25, the sum must satisfy x+y≥25+25=50x + y \geq 25 + 25 = 50. Among the given options, only 5454 is greater than or equal to 5050.
Any sum less than 5050 violates the condition that both xx and yy are at least 2525.

Anahtar Kavram

Determining positional metrics (median) after dataset expansion
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Konular

Measures of Central Tendency (Mean, Median, Mode)

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