Problem:
A sample consisting of five observations has an arithmetic mean of and a median of . The smallest value that the range (largest observation minus smallest) can assume for such a sample is
Answer Choices:
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Solution:
Since the mean is , the sum of the observations must be . The median of forces one observation to be , two more to be no more than , and the remaining two to be at least . If a maximal observation is increased by , the sum of those no larger than must be reduced by in order to keep the mean at . However, this expands the range. Thus the minimum range will occur when three observations are and the remaining two observations are equal and sum to . Hence the sample minimizes the range, and the smallest value that the range can assume is .