1990 POPULATION CENSUS Detailed data based on a 2 per cent representative sample (1992)
IV. SAMPLING ERROR OF THE ESTIMATES
Nj — the number of ultimate sampling units (persons, households, dwellings) in the jth primary sampling unit in the stratum, j = 1, 2, ..., m, nw-1, ..., M; nj — the number of ultimate units selected in the sample from the jth PSU, j = 1, 2, ..., m, (nj~Nj/10); y^j — value of the characteristic for unit i in PSU j, i = 1, 2, ..., nj; nj yj = Yij - sample totál for the jth PSU; - Nj Yj - — Yj — sample estimate of the totál of the jth sample PSU; nj 1 5 Y = -jjj- . ^j — average of the estimated totals over the sample PSU's; - M "! Nj Y = MY = — L — Yj — sample estimate of the totál for the stratum. m j=l nj In terms of these notations the variance of Y can be estimated as follows: + 5l Z n? l-g) ) E UyüY, Y Y1 Y2 [ ti] ( m_i) m j=i \ J ) m y-i 3 \ Njj nj(nj-l) i= 1 \ U nj j where s2 and s2 represent variability among PSU's and within PSU's, respectively. On the basis of this formula Y1 Y2 2 variance of somé national aggregate or county data is given in the form £ s*^ where r refers to somé stratum and the sum is taken over all strata belonging to the geographical unit in consideration. As a consequence of the adjustment described in the preceding chapter we have a sampling fraction of 2 per cent and therefore MNj/(nrtj) = 50, i.e. the common inflation factor; this will be denoted by w in the following formulas. Making use of the fact that in the case of the variables selected y^j may assume only the values 0 and 1 it is obvious that the term (nj-1) i=l within the expression s 2 may be replaced by the well-known variance expression for estimated proportions. This leads to the equality Y 2 - ii m m iT i h) 2 , . . ,2 , . * 1-Sl WYil + l Íl " r I * 1/5 \ fm 15 M] m-l|j = 1 J iM wI j = 1 nj-1 * where yj = E yjj and y^j is an auxiliary variable defined as y^j = 1 - y^j . Estimating the variance of an estimated ratio reduces to that of an estimated totál, since the ratios considered are of the form R = Y / X where Y and X are estimated totals. By means of a simple approximation we obtain 2 E(Y-RX) 2 s \ S » - 55* - • R X 2 X 2 ék A where Z denotes estimated totál of the new variable z^j = y^j - Rx-jj. Applying the above variance expression for the estimated totals Y and X we have s2 - k s 2- 2 R cov (X,Y) + R 2s 2 R x2 1 Y A A 2 where the covariance term cov(X,Y) has the same structure as s* with the slight difference that the squared terms * a A A are replaced by corresponding cross-product terms of x and y. Estimated totals X, Y and Z in the expressions of R 242
