A population consists of 4 strata containing respectively 5% 25% 30% and 40% of the total population of 1600 individuals. The means for strata are 61.2, 162.0, 141.2 and 100.0 respectively & the s.d. are 8.3, 9.3, 9.2 & 10.1 respectively. For a stratified sample of 150 individuals, find the sizes of the sample for each stratum under proportional allocation and optimum allocation, out of both which is of more effecient.
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Answer:
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Step-by-step explanation:
Stratified Sampling
An important objective in any estimation problem is to obtain an estimator of a population parameter
which can take care of the salient features of the population. If the population is homogeneous with
respect to the characteristic under study, then the method of simple random sampling will yield a
homogeneous sample, and in turn, the sample mean will serve as a good estimator of the population
mean. Thus, if the population is homogeneous with respect to the characteristic under study, then the
sample drawn through simple random sampling is expected to provide a representative sample.
Moreover, the variance of the sample mean not only depends on the sample size and sampling fraction
but also on the population variance. In order to increase the precision of an estimator, we need to use a
sampling scheme which can reduce the heterogeneity in the population. If the population is
heterogeneous with respect to the characteristic under study, then one such sampling procedure is a
stratified sampling.
The basic idea behind the stratified sampling is to
divide the whole heterogeneous population into smaller groups or subpopulations, such that the
sampling units are homogeneous with respect to the characteristic under study within the
subpopulation and
heterogeneous with respect to the characteristic under study between/among the
subpopulations. Such subpopulations are termed as strata.
Treat each subpopulation as a separate population and draw a sample by SRS from each
stratum.
[Note: ‘Stratum’ is singular and ‘strata’ is plural].
Example: In order to find the average height of the students in a school of class 1 to class 12, the
height varies a lot as the students in class 1 are of age around 6 years, and students in class 10 are of
age around 16 years. So one can divide all the students into different subpopulations or strata such as
Students of class 1, 2 and 3: Stratum 1
Students of class 4, 5 and 6: Stratum 2
Students of class 7, 8 and 9: Stratum 3
Students of class 10, 11 and 12: Stratum 4
Now draw the samples by SRS from each of the strata 1, 2, 3 and 4. All the drawn samples combined
together will constitute the final stratified sample for further analysis.