Consider the following population: {2, 3, 3, 4, 4}. The value of m is 3.2, but suppose that this is not known to an investigator, who therefore wants to estimate m from sample data. Three possible statistics for estimating m are
Statistic 1: the sample mean, x
Statistic 2: the sample median
Statistic 3: the average of the largest and the small- est values in the sample
A random sample of size 3 will be selected without re- placement. Provided that we disregard the order in which the observations are selected, there are 10 possible samples that might result (writing 3 and 3*, 4 and 4* to distinguish the two 3's and the two 4's in the population):
2, 3, 3* 2, 3, 4 2, 3, 4* 2, 3*, 4 2, 3*, 4*
2, 4, 4* 3, 3*, 4 3, 3*, 4* 3, 4, 4* 3*, 4, 4*
For each of these 10 samples, compute Statistics 1, 2, and 3. Construct the sampling distribution of each of these statistics. Which statistic would you recommend for estimating m and why?