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Wize AP Statistics Textbook > Sampling Distribution

Finite Population Sampling

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Finite Population Sampling

A population is considered finite if it is:
  • small
  • possible to count its individuals without difficulty
  • your sample is a relatively large fraction of the population (see: 10% Condition)
A finite population can also be called a countable population.

Examples
  • The number of twins born at a hospital.
  • The number of employees that were fired in a small company.
  • The number of people with a rare disease.


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Standard Deviation for a Finite Population

What happens if you are drawing from finite population such that your sample is not small compared to the population? In this case, when you sample – without replacement – the remaining dynamics of the population may be changed and the sampling variability is large.

Example

Finite Population (N=5)\left(N=5\right): {10,20,25,42, 71}\left\{10,20,25,42,\ 71\right\}

Suppose we draw a sample of n=2n=2 to find the sample mean.
  • Drawing a sample of 2 out of a population of 5 means the sample is 40% of the population.
  • The sampling variability is large because the sample mean is expected to differ greater each time you draw another sample.
  • e.g. Sample mean of {10, 42}\left\{10,\ 42\right\} is x‾=26\overline{x}=26
  • e.g. Sample mean of {25, 71}\left\{25,\ 71\right\} is x‾=48\overline{x}=48
  • e.g. Sample mean of {20, 42}\left\{20,\ 42\right\} is x‾=31\overline{x}=31
  • The population mean is μ=33.6\mu=33.6
  • The population standard deviation is σ=23.9\sigma=23.9
  • The standard deviation of the sample mean is σn=23.92=16.9\dfrac{\sigma}{\sqrt{n}}=\dfrac{23.9}{\sqrt{2}}=16.9

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Finite Population Correction

For drawing from a finite population, NN, we multiply the factor N−nN−1\boxed{\sqrt{\frac{N-n}{N-1}}} (finite population correction) to the standard deviation.


Note: Proportions are taught later in the course.

Wize Tip
When the population size is infinite or large relative to the sample size, the factor N−nN−1≈1\frac{N-n}{N-1}\approx 1; and the standard error is therefore approximated by σn\frac{\sigma}{\sqrt{n}} for sample means and pqn\sqrt{\frac{pq}{n}} for sample proportions.

From the example earlier:

σn⋅N−nN−1=23.92⋅5−25−1=14.6\displaystyle{\frac{\sigma}{\sqrt{n}}\cdot\sqrt{\frac{N-n}{N-1}}=\frac{23.9}{\sqrt{2}}\cdot\sqrt{\frac{5-2}{5-1}}=14.6}

Notice that the standard deviation of the sample mean with the "finite population correction" (14.6) is smaller than the standard deviation of the sample mean without the correction (16.9).



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Example: Drawing Samples From a Finite Population

Note: Proportions are taught later in the course.

There are only 120 houses in Twin Pines, an affluent neighborhood. The average number of bedrooms is 5.5 with a standard deviation of 1.7. Furthermore, 66 houses in Twin Pines have panic rooms. Based on a sample of 40 houses, we have the following statistics:

  • x‾=6.2\overline{x}=6.2 (sample mean: based on a sample of 40 houses, the average number of bedrooms is 6.2)
  • p^=2840=0.7\hat p =\frac{28}{40}=0.7 (sample proportion: based on a sample of 40 houses, 28 (or 70%) of them have panic rooms)

(a) What is the standard deviation of the sample mean?

For drawing from a finite population, NN, we multiply N−nN−1\boxed{\sqrt{\frac{N-n}{N-1}}} to the standard deviation of the sample mean:

σn⋅N−nN−1=1.740⋅120−40120−1=0.2204\displaystyle{\frac{\sigma}{\sqrt{n}}\cdot\sqrt{\frac{N-n}{N-1}}=\frac{1.7}{\sqrt{40}}\cdot\sqrt{\frac{120-40}{120-1}}=0.2204}



(b) What is the standard deviation of the sample proportion?

For drawing from a finite population, NN, we multiply N−nN−1\boxed{\sqrt{\frac{N-n}{N-1}}} to the standard deviation of the sample proportion:

pqn⋅N−nN−1=(0.55)(0.45)40⋅120−40120−1=0.0645\displaystyle{\sqrt{\frac{pq}{n}}\cdot\sqrt{\frac{N-n}{N-1}}=\sqrt{\frac{(0.55)(0.45)}{40}}\cdot\sqrt{\frac{120-40}{120-1}}=0.0645}




The Pogs Fan Club only has 150 members. The number of pogs a member owns is normally distributed with a mean of 265 and standard deviation of 35. What is the probability of selecting a random sample of 10 members with a sample mean of 280 or more?