Sample Size and Sample Allocation - Sampling Spatial Units for Agricultural Surveys - page 197

8.2 Sample Size Estimation for Simple Random Sampling

When planning a sample survey of spatial units, it is important to appropriately
determine the sample size. If it is too large , a huge amount of resources are required;
if it is too small , the results may become inefficient and as a consequence not useful.
The first thing we should consider when determining the sample size is what the
researcher expects from the sample. This argument can be expressed in terms of the
desired limits of error; that is, the amount of error that will be acceptable in the
sample estimates. Or it can be expressed in terms of some criterion to be decided
when the sample results are known. The amount is dependent on the intended use of
the sample results. Sometimes, it is difficult to decide how much error is acceptable,
particularly when the results have several different uses.
We must also define equations that connect n with the desired precision of the
sample. The equation will vary as a function of the desired precision, and according
to the sampling design under consideration. Suppose that the aim is to estimate a
population parameter using an estimator θ . We want the estimate to be close to the
true value with a high probability. Specifying a maximum acceptable difference ( ʷ )
between the estimator and the true value, and allowing for a small probability (

)
that the error may exceed this difference, the objective is to choose a sample size
n such that

ʱ

ʷ ʱ:

θ θ

Pr ð 8 : 1 Þ

The desired precision may be also expressed in relative terms as

!

θ θ
θ

Pr c ʱ: ð 8 : 2 Þ

The confidence interval yields an equation that links the precision and the sample
size. If θ

is an unbiased, normally distributed estimator of θ , then the statistic

θ θ
Var

q

ð 8 : 3 Þ

θ

has a standard normal distribution. If we let z denote the upper ʱ /2 points of the

standard normal distribution, then