# statistics binomial distribution

This strategy for examining is utilized as a part of circumstance where the population can be effortlessly partitioned into gatherings or strata which are particularly not quite the same as one another, yet the components inside of a gathering are homogeneous regarding a few attributes e. g. understudies of school can be separated into strata on the premise of sexual orientation, courses offered, age and so forth. In this the population is initially partitioned into strata and afterward a basic irregular specimen is taken from every stratum. Stratified testing is of two sorts: proportionate stratified inspecting and disproportionate stratified examining.

• Proportionate Stratified Sampling – In this the number of units selected from each stratum is proportionate to the share of stratum in the population e.g. in a college there are total 2500 students out of which 1500 students are enrolled in graduate courses and 1000 are enrolled in post graduate courses. If a sample of 100 is to be chosen using proportionate stratified sampling then the number of undergraduate students in sample would be 60 and 40 would be post graduate students. Thus the two strata are represented in the same proportion in the sample as is their representation in the population.

This method is most suitable when the purpose of sampling is to estimate the population value of some characteristic and there is no difference in within- stratum variances.

• Disproportionate Stratified Sampling – When the purpose of study is to compare the differences among strata then it become necessary to draw equal units from all strata irrespective of their share in population. Sometimes some strata are more variable with respect to some characteristic than other strata, in such a case a larger number of units may be drawn from the more variable strata. In both the situations the sample drawn is a disproportionate stratified sample.

The difference in stratum size and stratum variability can be optimally allocated using the following formula for determining the sample size from different strata

## Formula

${n_i = \frac{n.n_i\sigma_i}{n_1\sigma_1+n_2\sigma_2+…+n_k\sigma_k}\ for\ i = 1,2 …k}$

Where −

• ${n_i}$ = the sample size of i strata.
• ${n}$ = the size of strata.
• ${\sigma_1}$ = the standard deviation of i strata.

In addition to it, there might be a situation where cost of collecting a sample might be more in one strata than in other. The optimal disproportionate sampling should be done in a manner that

${\frac{n_1}{n_1\sigma_1\sqrt{c_1}} = \frac{n_2}{n_2\sigma_1\sqrt{c_2}} = … = \frac{n_k}{n_k\sigma_k\sqrt{c_k}}}$

Where ${c_1, c_2, … ,c_k}$ refer to the cost of sampling in k strata. The sample size from different strata can be determined using the following formula:

${n_i = \frac{\frac{n.n_i\sigma_i}{\sqrt{c_i}}}{\frac{n_1\sigma_1}{\sqrt{c_i}}+\frac{n_2\sigma_2}{\sqrt{c_2}}+…+\frac{n_k\sigma_k}{\sqrt{c_k}}}\ for\ i = 1,2 …k}$

### Example

Problem Statement:

An organisation has 5000 employees who have been stratified into three levels.

• Stratum A: 50 executives with standard deviation = 9
• Stratum B: 1250 non-manual workers with standard deviation = 4
• Stratum C: 3700 manual workers with standard deviation = 1

How will a sample of 300 employees are drawn on a disproportionate basis having optimum allocation?

Solution:

Using the formula of disproportionate sampling for optimum allocation.

${n_i = \frac{n.n_i\sigma_i}{n_1\sigma_1+n_2\sigma_2+n_3\sigma_3}} \\[7pt] \, For Stream A, {n_1 = \frac{300(50)(9)}{(50)(9)+(1250)(4)+(3700)(1)}} \\[7pt] \, = {\frac{135000}{1950} = {14.75}\ or\ say\ {15}} \\[7pt] \, For Stream B, {n_1 = \frac{300(1250)(4)}{(50)(9)+(1250)(4)+(3700)(1)}} \\[7pt] \, = {\frac{150000}{1950} = {163.93}\ or\ say\ {167}} \\[7pt] \, For Stream C, {n_1 = \frac{300(3700)(1)}{(50)(9)+(1250)(4)+(3700)(1)}} \\[7pt] \, = {\frac{110000}{1950} = {121.3}\ or\ say\ {121}}$

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