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Average Matches

Problem

The contents of twelve boxes of matches were recorded as:

34, 31, 29, 35, 33, 30, 31, 28, 27, 35, 32, 31

On the box it stated, "Average contents 32 matches." Is this correct?


Solution

The statement does not mention which average was used so we shall calculate the mean, median, and mode.

The mean is 376/12 approximately 31.3.
By arranging in order: 27, 28, 29, 30, 31, 31, 31, 32, 33, 33, 34, 35, we can see that the middle two numbers are 31, so the median is 31.
As 31 occurs most often the mode is also 31.

As all three averages are below 32 it is tempting to dismiss the claim. However, this is a classic error of sampling. If a coin were spun twelve times and exactly six heads and six tails were not obtained, would we suspect the coin of being biased? Of course not!

How far below the stated "average" must the sample mean be in order to invalidate the claim? (see Mean Claim)

Problem ID: 303 (12 Jan 2007)     Difficulty: 2 Star

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