##
Overview
Section* *

In Section 9.1, we introduce multi-stage design and give a few practical examples. We then provide the notations for the two-stage design. The unbiased estimators for the two-stage design with simple random sampling at each stage are discussed. Then we discuss the ratio estimator for a two-stage design with simple random sampling at each stage.

In Section 9.2, we discuss the two-stage design with primary units selected with probability proportional to the size and secondary units selected with simple random sampling. The Hansen-Hurwitz estimator is computed for this situation. We also show how to compute the estimated variance for the H-H estimator.

Lesson 9: Ch. 13.1, 13.2 of *Sampling *by Steven Thompson, 3rd edition.

## Objectives

- Identify the appropriate reasons and situations to use multi-stage sampling ,
- Compute the unbiased estimator and its estimated variance for the two-stage design when srs is used at each stage,
- Compute ratio estimator and its estimated variance for the two-stage design when srs is used at each stage, and
- Compute the Hansen-Hurwitz estimator and its estimated variance when primary units are selected with probability proportional to the size and secondary units selected with srs.