This is a technique to break down the variation of a random variable into
useful components (called stratum) in order to decrease experimental variation
and increase accuracy of results. It has been found that a more accurate
estimate of population mean \(\mu\) can often be obtained by taking measurements
from naturally occurring subpopulations and combining the results using
weighted averages. For example, suppose an accurate estimate of the mean
weight of sixth grade students is desired for a large school system. Suppose
(for cost reasons) we can only take a random sample of \(m=100\) students,
Instead of taking a simple random sample of 100 students from the entire
population of all sixth grade students, we use stratified sampling as follows.
The school system under study consists three large schools. School A has
\(N_{1}=310\) sixth grade students, School B has \(N_{2}=420\) sixth grade
students, and School C has \(N_{3}=516\) sixth grade students. This is a total
population of 1246 sixth grade students in our study and we have strata
consisting of the 3 schools. A preliminary study in each school with
relatively small sample size has given estimates for the sample standard
deviation \(s\) of sixth grade student weights in each school. These are shown
in the following table.
$$\begin{array}{lll}
\text { School A } & \text { School B } & \text { School C } \\
N_{1}=310 & N_{2}=420 & N_{3}=516 \\
s_{1}=3 \mathrm{lb} & s_{2}=12 \mathrm{lb} & s_{3}=6 \mathrm{lb}
\end{array}$$
How many students should we randomly choose from each school for a best
estimate \(\mu\) for the population mean weight? A lot of mathematics goes into
the answer. Fortunately, Bill Williams of Bell Laboratories wrote a book
called
A Sampler on Sampling (John Wiley and Sons, publisher), which provides an
answer. Let \(n_{1}\) be the number of students randomly chosen from School
\(\mathrm{A}\),
\(n_{2}\) be the number chosen from School \(\mathrm{B},\) and \(n_{3}\) be the
number chosen from School C. This means our total sample size will be
\(m=n_{1}+n_{2}+n_{3} .\) What is the formula for \(n_{i} ?\) A popular and widely
used technique is the following.
$$n_{i}=\left[\frac{N_{i} s_{i}}{N_{1} s_{1}+N_{2} s_{2}+N_{3} s_{3}}\right]
m$$
The \(n_{i}\) are usually not whole numbers, so we need to round to the nearest
whole number. This formula allocates more students to schools that have a
larger population of sixth graders and/or have larger sample standard
deviations. Remember, this is a popular and widely used technique for
stratified sampling. It is not an absolute rule. There are other methods of
stratified sampling also in use. In general practice, according to Bill
Williams, the use of naturally occurring strata seems to reduce overall
variability in measurements by about \(20 \%\) compared to simple random samples
taken from the entire (unstratified) population.
Now suppose you have taken a random sample size \(n_{i}\) from each appropriate
school and you got a sample mean weight \(\bar{x}_{i}\) from each school. How do
you get the best estimate for population mean weight \(\mu\) of the all 1246
students? The answer is, we use a weighted average.
$$\mu \approx \frac{n_{1}}{m} \overline{x_{1}}+\frac{n_{2}}{m}
\overline{x_{2}}+\frac{n_{3}}{m} \overline{x_{3}}$$
This is an example with three strata. Applications with any number of strata
can be solved in a similar way with obvious extensions of formulas.
(a) Compute the size of the random samples \(n_{1}, n_{2}, n_{3}\) to be taken
from each school. Round each sample size to the nearest whole number and make
sure they add up to \(m=100\).
(b) Suppose you took the appropriate random sample from each school and you
got the following average student weights: \(\overline{x_{1}}=82 l b,
\overline{x_{2}}=115 l b, \overline{x_{3}}=90 l b\). Compute your best estimate
for the population mean weight \(\mu\).
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