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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.