###### The Cumulative Distribution Function: ...

A cumulative distribution offers a convenient tool for determining probabilities for a given... **Read More**

Simple random and stratified random sampling are both sampling techniques used by analysts during statistical analyses.

Simple random sampling involves the selection of a sample from an entire population such that each member or element of the population has an equal probability of being picked. The method attempts to come up with a sample that represents the population in an unbiased manner.

However, it is not appropriate when there are glaring differences within the population such that statisticians can divide the members into different, distinctive categories. That’s where stratified random sampling comes in.

In stratified random sampling, analysts subdivide the population into separate groups known as strata (singular – stratum). Each stratum is composed of elements that have a common characteristic (attribute) that distinguishes them from all the others. The method is most appropriate for large populations that are **heterogeneous** in nature.

A simple random sample is then taken from within each stratum and combined to form the overall, final sample that takes heterogeneity into account. The number of members chosen from any one stratum depends on its size relative to the population as a whole.

An advertising firm wants to determine the extent to which they should emphasize television ads in a district. They decide to carry out a survey aimed at estimating the mean number of hours spent by households watching TV per week. The district has three distinct towns – A, B, which are urbanized, and C, located in a rural area. Town A is adjacent a major factory where most residents work, with most having school-aged kids. Town B mainly harbors retirees while most people in town C practice agriculture.

There are 160 households in town A, 60 in town B, and 80 in C. Given the differences in the composition of each region, the firm decides to draw a sample of 50 households, taking the total number of families in each into account.

Determine the number of homes that have been sampled in each region.

**Solution**

We have 3 strata: A, B, and C. We use the following formula to determine the number of households to be included in the sample from each region:

$$ \text{Number of households in sample} = \left( \cfrac {\text{number of households in region}}{ \text {total number of households} }\right) * \text {the required sample size} $$

Therefore, the number of households to be sampled in A = \(\frac {160}{300} * 50 = 27\) (approximately)

Similarly, the number of households to be sampled in B = \(\frac {60}{300} * 50 = 10\)

Finally, the firm would need \( \left(\frac {80}{300} * 50 \right) = 13\) households in region C.

- Stratification is associated with a smaller error of estimation compared to simple random sampling, especially when each stratum is homogeneous.
- Stratification enables analysts to estimate the population parameter, say, the mean for all the subgroups of the entire population.