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Question

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

A healthcare clinic logged the waiting times, in minutes, for 7 patients on Monday: 12,41,24,19,33,28,12, 41, 24, 19, 33, 28, and 4949. On Tuesday, 3 additional patients were logged. The arithmetic mean waiting time for all 10 patients combined was 3030 minutes, and the median waiting time of the 3 patients logged on Tuesday was 3434 minutes. If one of the patients logged on Tuesday had a waiting time of 2222 minutes and another had a waiting time of 3434 minutes, what was the waiting time, in minutes, of the third patient logged on Tuesday?

  1. A
    28 minutes
  2. B
    32 minutes
  3. C
    34 minutes
  4. 38 minutesAnswer
  5. E
    48 minutes

Answer

38 minutes
The option stating 38 minutes is correct because the total sum for all 10 patients combined is 10×30=30010 \times 30 = 300 minutes. The sum of Monday's 7 patient waiting times is 12+41+24+19+33+28+49=20612 + 41 + 24 + 19 + 33 + 28 + 49 = 206 minutes. Therefore, Tuesday's 3 patients must sum to 300−206=94300 - 206 = 94 minutes. Since two of Tuesday's patients waited 22 minutes and 34 minutes, the third patient's waiting time is 94−(22+34)=3894 - (22 + 34) = 38 minutes. Ordering the three Tuesday times (22,34,38)(22, 34, 38) confirms the median is indeed 34 minutes.

Step-by-Step Solution

1
Calculate the total sum of waiting times for all 10 patients.
Total sum = 10×30=30010 \times 30 = 300 minutes.
The arithmetic mean of nn values is Sum/n\text{Sum} / n, so Sum=n×Mean\text{Sum} = n \times \text{Mean}.
2
Calculate the sum of waiting times for the 7 patients logged on Monday.
Monday sum = 12+41+24+19+33+28+49=20612 + 41 + 24 + 19 + 33 + 28 + 49 = 206 minutes.
Adding the individual values gives the total Monday waiting time.
3
Find the total sum of waiting times for the 3 patients logged on Tuesday.
Tuesday sum = 300−206=94300 - 206 = 94 minutes.
Subtracting Monday's total from the combined total yields Tuesday's total.
4
Determine the third patient's waiting time on Tuesday using the given values.
Third patient's waiting time = 94−(22+34)=94−56=3894 - (22 + 34) = 94 - 56 = 38 minutes.
The sum of the 3 Tuesday patients is 94 minutes, and two known patients waited 22 and 34 minutes. Note that the sorted set (22,34,38)(22, 34, 38) has a median of 34 minutes, consistent with the problem statement.

Key Concept

Combined Mean and Median Properties
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Topics

Measures of Central Tendency (Mean, Median, Mode)

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