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Difficulty: MediumMeasures of Central Tendency (Mean, Median, Mode)

A quality assurance auditor reviewed the processing times, in seconds, for a batch of 7 completed tasks: 22,10,24,18,30,22,22, 10, 24, 18, 30, 22, and 1414. An 8th task with a processing time of xx seconds, where xx is a positive integer, is added to the batch. Which of the following statements must be true? Select all such statements.

  1. If x=22x = 22, the median of the 8 processing times is equal to 22.Answer
  2. If the arithmetic mean of the 8 processing times is equal to 21, then x=28x = 28.Answer
  3. C
    If x=18x = 18, the median of the 8 processing times is equal to 21.
  4. D
    If x=30x = 30, the arithmetic mean of the dataset increases by 5.
  5. E
    If x=10x = 10, the mode of the 8 processing times becomes 10.

Answer

The correct statements are those asserting that if x=22x=22, the median of the 8 processing times is 22, and if the mean of the 8 processing times is 21, then x=28x=28.
The statement regarding x=22x=22 is correct because inserting 22 places 22 at both the 4th and 5th positions of the 8-element ordered list, yielding a median of 22. The statement regarding a mean of 21 is correct because the required total sum for 8 items is 8×21=1688 \times 21 = 168, which requires x=168−140=28x = 168 - 140 = 28.

Step-by-Step Solution

1
Order the original dataset and calculate initial metrics.
Sorted original list: 10,14,18,22,22,24,3010, 14, 18, 22, 22, 24, 30. Original sum = 140, original mean = 1407=20\frac{140}{7} = 20, original median = 22 (4th term).
Establishing the initial baseline values is necessary to evaluate statements about changes in mean and median.
2
Evaluate the statement regarding median when x=22x=22.
Inserting x=22x=22 yields sorted list: 10,14,18,22,22,22,24,3010, 14, 18, 22, 22, 22, 24, 30. The 4th and 5th items are both 22, giving median 22+222=22\frac{22+22}{2} = 22.
Confirms the first statement is true.
3
Evaluate the statement regarding mean equal to 21.
Target sum for 8 items with mean 21 is 8×21=1688 \times 21 = 168. Setting 140+x=168140 + x = 168 yields x=28x = 28.
Confirms the second statement is true.
4
Evaluate remaining incorrect statements.
For x=18x=18, ordered list is 10,14,18,18,22,22,24,3010, 14, 18, 18, 22, 22, 24, 30, giving median 18+222=20≠21\frac{18+22}{2} = 20 \neq 21. For x=30x=30, new mean is 21.2521.25, an increase of 1.25≠51.25 \neq 5. For x=10x=10, 10 and 22 both appear twice, making it bimodal rather than having a single mode of 10.
Verifies that all other statements are false.

Key Concept

Measures of Central Tendency (Mean, Median, Mode) for modified datasets
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Topics

Measures of Central Tendency (Mean, Median, Mode)

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