Day 31

MATH 313: Survey Design and Sampling

Recap: Key Notations

\(M\): Total clusters in the population.
\(m\): Clusters in the sample.
\(n_i\): Elements in cluster \(i\).
\(a_i\): Elements in cluster \(i\) with the characteristic of interest.
\(N\): Total population elements.
\(\bar{N} = \frac{N}{M}\): Mean cluster size in the population.
\(\bar{n} = \frac{1}{m} \sum_{i=1}^{m} n_i\): Sample mean cluster size.

Estimation of Population Proportion (\(p\))

Estimator: \[ \hat{p} = \frac{\sum_{i=1}^{m} a_i}{\sum_{i=1}^{m} n_i} \]
- Ratio estimation method.

Variance Estimation of \(\hat{p}\)

Variance Formula: \[ \hat{V}(\hat{p}) = \left(1 - \frac{m}{M}\right) \frac{S_p^2}{m \bar{n}^2} \]
- \(S_p^2 = \frac{\sum_{i=1}^{m}(a_i - \hat{p} n_i)^2}{m-1}\): Sample variance of proportions.

Error Bound and Confidence Interval

Error Bound: \[ B = \lambda \sqrt{\hat{V}(\hat{p})}, \quad \lambda = \sqrt{\frac{4}{9\alpha}} \]
Confidence Interval: \[ \hat{p} \pm B \]
- Avoids \(t\)-distribution due to ratio nature of \(\hat{p}\).

Sample Size Calculation

Formula:

\[ m=\frac{M S_p^2}{M \cdot\left(\frac{B \bar{n}}{\lambda}\right)^2+S_p^2} \] - For estimating \(p\) with a pre-defined error margin (\(B\)).

Example 1: Recall Example 1 of Day 28, in addition to being asked about their income, the residents are asked whether they rent or own their homes. The results are given in the following table.

Cluster	Residents, \(n_i\)	Renters, \(a_i\)
1	8	4
2	12	7
3	4	1
4	5	3
5	6	3
6	6	4
7	7	4
8	5	2
9	8	3
10	3	2
11	2	1
12	6	3
13	5	2

Cluster	Residents, \(n_i\)	Renters, \(a_i\)
14	10	5
15	9	4
16	3	1
17	6	4
18	5	2
19	5	3
20	4	1
21	6	3
22	8	3
23	7	4
24	3	0
25	8	3

Use the data to estimate the proportion of residents who live in rented housing. Use \(95 \%\) bound.

If the data is out of date, a new study will be conducted in the same city for the purpose of estimating the proportion \(p\) of residents who rent their homes. How large a cluster sample should be taken to estimate \(p\) with a bound of 0.04 on the error of estimation?