Question

Math 1342 Project: Unbiased vs Biased Estimators In our class, we worked with a population of size " 3 "' and showed that the n−1 computation for variance gave us a figure such that the mean of the variances of all the samples was equal to the variance of the population - sample variance is an unbiased estimator for population variance. Here you will be examining a population of size " 5 " and examining which of mean, median, range, standard deviation, and variance are unbiased. a) Consider the population of the size of the households of my immediate family members: {1,2,4,4,6}; compute the mean, median, range, standard deviation, and variance of this POPULATION. b) Now consider the possible random samples from this population with n=2 (you should identify 25 such samples). For each of these SAMPLES, compute the mean, median, range, standard deviation, and variance (hint: with n=2, the median is the same as the mean). c) At this point, you should have produced five data sets (four distinct) with 25 data values each; find the mean (only the mean) of each of these sets. d) Compare the mean of the sample statistic data from c) with the population parameters computed in a) - which of these are unbiased estimators? e) Write a brief (1-2 paragraph) summary of your procedure and findings.

Answer 1

To assess unbiased estimators, population parameters such as mean and variance were calculated for a family size population, then samples of n=2 were taken to compute sample statistics. By comparing the means of these sample statistics to the population values, the unbiased estimators were identified.

Step-by-step explanation:

To determine which statistics are unbiased estimators for a population, a population of size 5 with household sizes {1,2,4,4,6} is used. First, compute the population parameters: mean, median, range, standard deviation, and variance. Then, consider all possible samples of size n=2 (25 unique combinations), and calculate sample means, medians, ranges, standard deviations, and variances. Notably, for samples of n=2, the median and mean are identical. Next, find the mean of each sample statistic across the 25 samples. Finally, compare these means to the population parameters to identify which statistics are unbiased, meaning the mean of the sample statistic is equal to the corresponding population parameter.

A statistic is unbiased if the expected value of the sample statistic is equal to the population parameter it estimates. The calculation of sample variance uses n-1 instead of n to ensure it is an unbiased estimator of the population variance. The Central Limit Theorem is also relevant as it states that the distribution of the sample means will be approximately normal, with the mean of the sample means equal to the population mean for sufficiently large samples.