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Wize University 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?
Extra Practice
Drawing From a Finite Population
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?
Finite Population Sampling: Standard error and variability
The standard error of the sample mean measures the variability of the
Statistical sampling
The standard deviation of a population distribution is 6 times larger than the standard deviation of the sample mean. Find the sample size.
Population Sampling: Confidence Interval
Bob, a second-year student at Twin Pines University, is not very good at statistics. He wishes to infer about the average starting salary of recent graduates at Twin Pines University. His sample included 12 friends, all recent graduates, and it revealed a mean of $55,000. His 90% confidence interval is [39,116, 80,884].
Bob reports: "We conclude that 90% of recent graduates at Twin Pines University earn between $39,116 and $80,884."
List everything that is wrong about his statement. (Check all that applies.)