Question

Consider the following population: \(\\{2,3,3,4,4\\}\). The value of \(\mu\) is \(3.2\), but suppose that this is not known to an investigator, who therefore wants to estimate \(\mu\) from sample data. Three possible statistics for estimating \(\mu\) are Statistic \(1:\) the sample mean, \(\bar{x}\) Statistic 2 : the sample median Statistic 3 : the average of the largest and the smallest values in the sample A random sample of size 3 will be selected without replacement. 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): $$\begin{array}{rlllll} 2,3,3^{*} & 2,3,4 & 2,3,4^{*} & 2,3^{*}, 4 & 2,3^{*}, 4^{*} \\ 2,4,4^{*} & 3,3^{*}, 4 & 3,3^{*}, 4^{*} & 3,4,4^{*} & 3^{*}, 4,4^{*} \end{array}$$ 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 \(\mu\) and why?

Short answer

Given the populations and samples, the calculated sampling distributions indicate that the sample mean (Statistic 1) or the sample median (Statistic 2) are the best estimators for the population mean. They both show less variance and their mean values are the closest to the population mean (\mu = 3.2).

Step-by-step solution

Calculate Statistics 1, 2, and 3 for each sample

First, enumerate all possible combinations of the 3-value sample from the population. Then calculate the sample mean (Statistic 1), the sample median (Statistic 2), and the average of the maximum and minimum values (Statistic 3) for each of these samples.

Construct Sampling Distribution of Statistics

After calculating the statistics, count the number of times every unique Statistic 1 (sample mean), Statistic 2 (sample median), and Statistic 3 (average of maximum and minimum) value occurs. These will create the sampling distributions for each statistic.

Analyze and Recommend

Examine the three sampling distributions. The statistic with a sampling distribution whose mean is the closest to the true population mean (\mu = 3.2) is the best estimator. In addition, the less dispersed (lower variance) a sampling distribution, the better the estimator.