Question

A population of size 500 is divided into 4 strata. The following table gives the (5)
data on size and standard deviation of each stratum.
STRATA
I II III IV
Size 100 150 150 100
S.D. 5 8 7 10
A stratified random sample of size 100 is to be drawn from the population.
Determine the size of samples from each of these strata for:
i) proportional allocation
ii) Neyman’s optimal allocation

Answer 1

Answer on Question #46501 – Math - Statistics and Probability

Problem.

A population of size 500 is divided into 4 strata. The following table gives the (5) data on size and standard deviation of each stratum.

STRATA

A stratified random sample of size 100 is to be drawn from the population.

Determine the size of samples from each of these strata for:

i) proportional allocation

ii) Neyman’s optimal allocation

Solution:

(i) Strata sample sizes are determined by the following equation for proportional allocation:

$$n_i = \frac{N_i}{N} \cdot n$$

where $n_i$ is the sample size for stratum $i$, $N_i$ is the population size for stratum $i$ ($N_1 = 100$, $N_2 = 150$, $N_3 = 150$, $N_4 = 100$), $N = 500$ is total population size, and $n = 100$ is total sample size.

Hence for proportional allocation and random sample of size 100 we will obtain the following table:

(ii) Strata sample sizes are determined by the following equation for Neyman’s optimal allocation:

$$n_i = \frac{N_i \sigma_i}{\sum_i N_i \sigma_i} \cdot n$$

where $n_i$ is the sample size for stratum $i$, $N_i$ is the population size for stratum $i$ ($N_1 = 100$, $N_2 = 150$, $N_3 = 150$, $N_4 = 100$), $\sigma_i$ is the standard deviation of stratum $i$ ($\sigma_1 = 5$, $\sigma_2 = 8$, $\sigma_3 = 7$, $\sigma_4 = 10$) and $n = 100$ is total sample size.

Hence for Neyman’s optimal allocation and random sample of size 100 we will obtain the following table:

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