A \"population\" is formed by placing five balls in a bag. The balls are labeled

Question

A "population" is formed by placing five balls in a bag. The balls are labeled 1, 2, 3, 4, and 5, respectively. The mean of this population is mu = 3. Someone who does not know the contents of the bag will estimate the value of mu by randomly taking a sample of three of the balls (without replacement) and finding either the sample mean pr the sample median. In the meantime, a statistician has listed all the possible samples of size three (sampling without replacement) and has calculated the sample mean and the sample median for each possible sample. The statistician finds that: All the possible sample means form a distribution whose mean is 3 and whose standard deviation is 0.577. All the possible sample medians form a distribution w hose mean is 3 and whose standard deviation is 0.775. Regarding the choice between using the sample mean and using the sample median for estimating mu, which of the following is true? (A) Both the sample mean and the sample median are unbiased, but the sample median is preferable as it has the larger standard deviation. (B) Both the sample mean and the sample median are unbiased, but the sample mean is preferable as it has the smaller standard deviation. (C) The sample mean is unbiased and the sample median is biased, so the sample mean is preferable. (D) The sample median is unbiased and the sample mean is biased, so the sample median is preferable. (E) Both the sample mean and the sample median are biased.

Explanation / Answer

Answer B

Here as both mean and median are same, the distribution is symmetrical. As mean is unbiased, median will also be. The one with less variance is more preferrable. Thus, both are unbiased but mean is preferrable as its variation is less.

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