(Requires probability.) For two-phase sampling with ratio estimation (Section 12.3.1), suppose the phase I sample is an SRS of size n(1), and the phase II sample is an SRS of fixed size n(2).
a. Show that P (Zi = 1) = n(1) /N, and P(Di = 1 | Z) = Zin(2)/n(1).
b. Show that (12.10) gives the approximate variance of ṫ (2) yr .
c. Let ei = yi ˆ’ Ḃ(2)xi and let s2y and s2e be the sample variances of the yi€™s and the ei€™s from the phase II sample,

Show that (12.11) is an approximately unbiased estimator of V (ṫ(2)yr ).

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ieS2) n2)1 ieS2)