A technology startup recorded the monthly software subscription costs, in dollars, for 6 d… — GRE General Test

A technology startup recorded the monthly software subscription costs, in dollars, for 6 department teams: $450, 120, 310, 120, 280,$ and $500$ . A seventh department team with a monthly subscription cost of $x$ dollars is added to the dataset, where $x$ is a positive integer. Which of the following statements regarding the measures of central tendency for the updated 7-team dataset must be true? Select all that apply.

The median of the subscription costs for the 7 teams cannot be greater than 310.Answer
If $x = 600$ , the arithmetic mean of the subscription costs for the 7 teams is greater than the median.Answer
C
If $x = 120$ , the median of the subscription costs for the 7 teams is 120.
D
The arithmetic mean of the subscription costs for the 7 teams will equal the median if $x = 290$ .
E
The mode of the subscription costs for the 7 teams is uniquely 120 for any positive integer value of $x$ .

Answer

The statements that must be true are that the median cannot exceed 310, and that if x = 600, the arithmetic mean is greater than the median.

Sorting the six known values gives 120, 120, 280, 310, 450, 500. In a dataset of 7 numbers, the median is the 4th value when sorted. If x is added, the 4th value will be 280 if x ≤ 280, x if 280 < x < 310, or 310 if x ≥ 310. Thus, the median is capped at 310. Furthermore, setting x = 600 yields a median of 310 and a mean of (1780 + 600) / 7 = 340, which is greater than the median.

Step-by-Step Solution

Sort the existing 6 dataset elements in ascending order and compute their sum.

Sorted list: 120, 120, 280, 310, 450, 500. Sum = 1780.

Establishing the baseline position of values helps determine how inserting x shifts the median, mean, and mode.

Analyze the position of the median for a 7-element dataset.

The median is the 4th element when sorted. If x ≤ 280, the 4th element is 280. If 280 < x < 310, the 4th element is x. If x ≥ 310, the 4th element is 310. Thus, 280 ≤ median ≤ 310.

This establishes upper and lower bounds for the median regardless of x.

Evaluate the statement regarding median upper bound (median ≤ 310).

Since the maximum possible 4th element in the sorted 7-element list is 310, the median can never be greater than 310.

Confirms the validity of the statement concerning the maximum median.

Evaluate the scenario where x = 600.

Sum = 1780 + 600 = 2380. Mean = 2380 / 7 = 340. Median = 310 (since 600 ≥ 310). 340 > 310.

Confirms that when x = 600, mean (340) is strictly greater than median (310).

Key Concept

Measures of Central Tendency (Mean, Median, Mode)

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