1. Chapter 7 Sampling and Sampling Distributions
Yandell – Econ 216

2. Chapter Goals After completing this chapter, you should be able to: _
Determine the mean and standard deviation for the sampling distribution of the sample mean, X
Determine the mean and standard deviation for the sampling distribution of the sample proportion, p
Describe the Central Limit Theorem and its importance
Apply sampling distributions for both X and p _ _
Yandell – Econ 216

3. Sampling Distribution
A sampling distribution is a distribution of the possible values of a statistic for a given size sample selected from a population
Yandell – Econ 216

4. Developing a Sampling Distribution
Assume there is a population …
Population size N=4
Random variable, x, is age of individuals
Values of x: 18, 20, 22, 24 (years) D A C B
Yandell – Econ 216

5. Developing a Sampling Distribution
(continued)
Summary Measures for the Population Distribution:
P(x) .3 .2 .1 x
A B C D
Equally likely
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6. Now consider all possible samples of size n=2
Developing a Sampling Distribution
(continued)
Now consider all possible samples of size n=2
16 Sample Means
16 possible samples (sampling with replacement)
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7. Sampling Distribution of All Sample Means
Developing a
Sampling Distribution
(continued)
Sampling Distribution of All Sample Means
Sample Means Distribution
16 Sample Means _
P(X) .3 .2 .1 _ X
(no longer uniform)
Yandell – Econ 216
listen

8. Summary Measures of this Sampling Distribution:
Developing a
Sampling Distribution
(continued)
Summary Measures of this Sampling Distribution:
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9. Comparing the Population with its Sampling Distribution
Sample Means Distribution
n = 2 _
P(X)
P(X) .3 .3 .2 .2 .1 .1 _ X
A B C D X
Yandell – Econ 216

10. If the Population is Normal
If a population is normal with mean μ and standard deviation σ, the sampling distribution
of is also normally distributed with
and
Yandell – Econ 216

11. Z-value for Sampling Distribution of X
Z-value for the sampling distribution of :
where: = sample mean
= population mean
= population standard deviation
n = sample size
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12. Finite Population Correction
Apply the Finite Population Correction if:
the sample is large relative to the population
(n is greater than 5% of N)
and…
Sampling is without replacement
Then
Yandell – Econ 216

13. Sampling Distribution Properties
Normal Population Distribution
(i.e is unbiased )
Normal Sampling Distribution
(has the same mean)
listen
Yandell – Econ 216

14. Sampling Distribution Properties
(continued)
For sampling with replacement:
As n increases,
decreases
Larger sample size
Smaller sample size
Yandell – Econ 216

15. If the Population is not Normal
We can apply the Central Limit Theorem:
Even if the population is not normal,
…sample means from the population will be approximately normal as long as the sample size is large enough
…and the sampling distribution will have
and
Yandell – Econ 216

16. Central Limit Theorem n↑
the sampling distribution becomes almost normal regardless of shape of population n↑
As the sample size gets large enough…
listen
Yandell – Econ 216

17. If the Population is not Normal
(continued)
Population Distribution
Sampling distribution properties:
Central Tendency
Sampling Distribution
(becomes normal as n increases)
Variation
Larger sample size
Smaller sample size
(Sampling with replacement)
Yandell – Econ 216

18. How Large is Large Enough?
For most distributions, n > 30 will give a sampling distribution that is nearly normal
For fairly symmetric distributions, n > 15
For normal population distributions, the sampling distribution of the mean is always normally distributed
Yandell – Econ 216

19. Example
Suppose a population has mean μ = 8 and standard deviation σ = 3. Suppose a random sample of size n = 36 is selected.
What is the probability that the sample mean is between 7.8 and 8.2?
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20. Example
(continued)
Solution:
Even if the population is not normally distributed, the central limit theorem can be used (n > 30)
… so the sampling distribution of is approximately normal
… with mean = 8
…and standard deviation
Yandell – Econ 216

21. Example Solution (continued): (continued) Z X Population Distribution
Sampling Distribution
Standard Normal Distribution ? ? ? ? ? ? ? ? ? ?
Sample
Standardize ? ? Z X
Yandell – Econ 216

22. Population Proportions, π
π = the proportion of population having
some characteristic
Sample proportion ( p ) provides an estimate
of π :
If two outcomes, p has a binomial distribution
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23. Sampling Distribution of p
Approximated by a normal distribution if:
where
and
Sampling Distribution
P( p ) .3 .2 .1 p
listen
(where π = population proportion)
Yandell – Econ 216

24. Z-Value for Proportions
Standardize p to a Z value with the formula:
If sampling is without replacement and n is greater than 5% of the population size, then must use the finite population correction factor:
Yandell – Econ 216

25. Example
If the true proportion of voters who support Proposition A is π = .4, what is the probability that a sample of size 200 yields a sample proportion between .40 and .45?
i.e.: if π = .4 and n = 200, what is
P(.40 ≤ p ≤ .45) ?
Yandell – Econ 216

26. Example if π = .4 and n = 200, what is P(.40 ≤ p ≤ .45) ? Find :
(continued)
if π = .4 and n = 200, what is
P(.40 ≤ p ≤ .45) ?
Find :
Convert to standard normal:
Yandell – Econ 216

27. Standardized Normal Distribution
Example
(continued)
if π = .4 and n = 200, what is
P(.40 ≤ p ≤ .45) ?
Use standard normal table: P(0 ≤ Z ≤ 1.44) = .4251
Standardized Normal Distribution
Sampling Distribution
.4251
Standardize
.40
.45
1.44 p Z
Yandell – Econ 216

28. Chapter Summary Introduced sampling distributions
Described the sampling distribution of the mean
For normal populations
Using the Central Limit Theorem
Described the sampling distribution of a proportion
Calculated probabilities using sampling distributions
Discussed sampling from finite populations
Yandell – Econ 216

29. Demonstration of Central Limit Theorem
Visit this website to see a Normal Distribution Demonstration
(watch a live electronic quincunx demo)
Yandell – Econ 216