Under Neyman allocation, discussed in Section 3.4.2, the optimal sample size in stratum h is a. Show that the variance oftstr if Neyman allocation is

Under Neyman allocation, discussed in Section 3.4.2, the optimal sample size in stratum h is

a. Show that the variance ofˆtstr if Neyman allocation is used is

b. We showed in Section 3.4.1 that the variance of ṫstr if proportional allocation is used is

Prove that the theoretical variance from Neyman allocation is always less than or equal to the theoretical variance from proportional allocation by showing that

c. From (b), we see that the gain in precision from using Neyman allocation relative to using proportional allocation is higher if the stratum standard deviations Sh vary widely. When H = 2, show that