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Appendix C Statistics used in sampling bias study
In Chapter 6, it is stated that under random sampling,
should follow an approximately normal (0,1) distribution. A justification for this is given here.
Sampling was done independently in each CU.
Therefore 
is the sum of H independent random variables,
where H is the number of CUs in Canada. There are
46,510 CUs in Canada that contain sampled households, so
H is very large. Thus, according to the central limit theorem, 
will
follow an approximately normal (0,1) distribution (see Kendall and Stuart [1963], p. 193)
as will 
if 
. 
, however, would not
have a mean of 0 if the CU level samples of households were
significantly biased for any reason.
An additional statistic will now be derived which allows us to test if the bias between
two regions or two censuses is the same. Let 
and 
be estimators (based on initial
weights) of the known population counts 
and 
for
two distinct geographic areas or for two different censuses. Let 
and
be the relative biases of and
. We wish to test if the null hypothesis
is true. This can be done using the statistic
where 
and 
are
unbiased estimators of 
and 
respectively.
Thus, if the null hypothesis 
above is true, the expectation of 
is zero. Note also
that the denominator of is the standard error of the numerator of (there is no
covariance term because estimates from separate regions or from different censuses are independent)
and hence has a variance of 1. Now if 
approximately follows a normal distribution (again based on
the central limit theorem), 
will also approximately follow a
normal distribution, as will 
and 
. Thus, follows approximately a normal (0,1)
distribution if the null hypothesis 
is true.
