Question

A city contains 69 precincts. A simple random sample of 10 precincts was selected, and every household in each sampled precinct was questioned.

The results are shown in the table.

a) Is this an example of a cluster or stratified sampling or something else?

b) Find a point estimate of the proportion of households in the city having cable television.

Sampled Precinct	Number of Households	Number with Cable Television
1	28	6
2	35	9
3	18	13
4	52	29
5	41	15
6	38	13
7	36	20
8	30	14
9	20	9
10	42	26

Answer 1

Before I answer the questions, let us first define cluster and stratified sampling:

1. Cluster Sampling: In cluster sampling, the researcher divides the population into separate groups called clusters. These clusters are mutually homogeneous yet internally heterogeneous groups, which are evident in population under consideration.

In cluster sampling, after the clusters have been made, a simple random sample of the groups is selected. The element in each cluster are then sampled. Cluster sampling gives each cluster equal chance of being selected.

Note: If all the elements from the selected cluster are sampled, then this is called "one-stage" clustering.

2. Stratified Sampling: Stratification is a process of dividing members of the population into homogeneous subgroups before sampling. These homogeneous subgroups are called Stratum. The strata should be mutually exclusive: every element in the population must be assigned to only one stratum and collectively exhaustive: no population element should be excluded.

After the population is divided into Strata's, then simple random sampling is applied to each stratum.

(a)From above definitions, as out of 69 precincts , we have sample 10 entire precincts,so this is an example of cluster sampling.

(b) Point estimate of population proportion(p) is given by $\hat{p}$, and $\hat{p}=x/n$ where x: no of units having attribute A in sample and n= sample size

In our case attribute A is having cable television.

Please refer the table and calculations below:

Sample Precinct	Number of Households	Number with cable television
1	28	6
2	35	9
3	18	13
4	52	29
5	41	15
6	38	13
7	36	20
8	30	14
9	20	9
10	42	26
Total:	340	154

x=Total no of household with cable television and n=Total no of household

So, $\hat{p}=x/n$=154/340=0.45294 is the point estimate of households in the city having cable television.