[Math] Stratified random sampling

statistics

Which of the following is NOT a characteristic of stratified random sampling?

(A) Random sampling is part of the sampling procedure.

(B) The population is divided into groups of units that are similar on some characteristic.

(D) Each individual unit in the population belongs to one and only one of the strata.

(E) Every possible subset of the population, of the desired sample size, has an equal chance of being selected.

The correct answer is (e), but why (e)? isn't every subset of the population has an equal chance of being selected. the idea of stratified sampling?

Best Answer

The correct answer is $E$.

For stratified random sampling, we get to choose the sample size for each stratum. By picking larger/smaller numbers for one group, we're changing their probability of being selected without changing anyone else's. That means that (unless by coincidence) the chance of different samples being selected is not the same.

It seems like you're thinking of 'simple random sampling'. That has the property of independence, which means that every possible subset of the sample size has the same chance of occurring. While simple random sampling is used in stratified random sampling, the extra leeway in picking the sample sizes for each strata changes the independence property, and means choice $E$ is not true.

Related Solutions

[Math] Random Sample vs Simple Random Sample

Okay, let me try to address your question. I am sorry, I did not have time to read other responses. Here is the deal: In the definition of the random sample Triola is talking about a probability of selecting an object (individual). In the definition of a simple random sample, he talks about the probability of selecting varied samples of size n (groups of objects). That is the difference between the two (individual vs. a group)

Here is an example: You have a class of 50 students, 30 males and 20 females. You are randomly selecting 3 males and 2 females. This is an example of a random sample because each person in the class has the same probability of being selected 3/30=2/20. However, all your samples are going to be the same: 3 males and 2 females. It is not possible to get 2 males and 3 females, or 4 males and 1 female. Therefore, it is not a simple random sample.

I hope this helps )

Unifrom sampling

If you look at Equation 2.3 of that paper, I believe they are describing uniform subsampling. You just draw samples from a data set with uniform probability. This will non-parametrically approximate the underlying distribution, and is often called boostrapping. In this case (Eq.2.3), they're using bootstrapping to approximate the expected value.

See the wiki entry: https://en.wikipedia.org/wiki/Bootstrapping_(statistics)

In python, you can get a uniform sample easily using libraries like numpy:

from numpy.random import uniform, randint

num_data = 100

//
// Get 10 samples in the range 0 to num_data
p = uniform(size=(10,)) * num_data
print(p.astype(int))

//
// Alternatively, you can just uniformly sample 
// random integers which is probably easier 
p = randint(low=0, high=num_data, size=(10,))
print(p)

//
// Now if you have data... you just get it like this
pretend_this_is_data = arange(100)
uniform_data_sample = pretend_this_is_data[p]

Best Answer

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[Math] Random Sample vs Simple Random Sample

Unifrom sampling

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