A Sampling Experiment: Evaluating Coal Cars for Tampa Electric Company
Mission
Tampa Electric Company (TECO) has hired the Saint Leo Institute for Chemical and
Environmental Studies (SLICES) to sample coal cars that arrive by train to their Apollo Beach
facility.
TECO would like to establish a method by which the entire population (a train-load of
coal) could be effectively sampled to determine the % clean burning environmentally friendly
coal (represented as white beans) versus the % unclean coal (in this study black beans).
The institute is tasked with determining the mean percentage of clean vs. unclean coal
that enter the electric power plant.
Discussion
In most analytical methods, only a small fraction of the entire population is analyzed. The results
from the determination of an analyte in a laboratory sample are assumed to be similar to the
concentration of the analyte in the whole population. Consequently, a laboratory sample taken
from the entire batch must be representative of the population.
Reference:
J. E. Vitt and R. C. Engstrom, J. Chem. Educ., 1999, 76, 99.
In this experiment, you will investigate how the sample size influences the uncertainty associated
with the sampling step. Generally, the required sample size must increase as the sample
heterogeneity increases, as the fraction of the analyte decreases, or as the desired uncertainty
decreases. The model system used in this experiment consists of a collection of beans that are
identical in size, shape, and density but that are different in color. If p represents the fraction of
the particles of the analyte (beans of the first color), then 1- p is the fraction of the second type of
particles (beans of the second color). If a sample of N particles is drawn from the population, then
the number of particles of the analyte in the sample should be Np. It can be shown that the
standard deviation of the number of particles of analyte Np obtained from a sample of the twocomponent mixture is SQRT [Np(1- p) ]. The relative standard deviation (σr) is SQRT [(1-p)/NP]
This equation suggests that as the number of particles sampled increases, the relative uncertainty
decreases. Using a mixture of beans of two colors, you will determine the uncertainty of sampling
as a function of sample size.
PROCEDURE
CALCULATIONS
1. Using the compiled class data, calculate the mean percentage of beans of the specified color
and the relative standard deviation of that percentage for each sample size.
2. Using the equation given previously, based on sampling theory, calculate the theoretical
relative standard deviation using the values of p and the mean number of particles for each of the
three sample sizes.
3. Compare your class data with the theoretical result. Does the relative standard deviation
decrease as the sample size increases, as predicted by sampling theory?
4. Use the equation for the relative standard deviation to find the number of beans that would
have to be sampled to achieve a relative standard deviation of 0.002.
5. Suggest two reasons why this theory might not be adequate to describe the sampling of many
materials for chemical analysis.