1. Chapter 7 Introduction to Sampling Distributions
Business Statistics:
A Decision-Making Approach
7th Edition
Chapter 7 Introduction to Sampling Distributions
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

2. Chapter Goals After completing this chapter, you should be able to:
Define the concept of sampling error
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 _ _ _ _
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

3. Sampling Error Problems:
Sample Statistics are used to estimate Population Parameters
ex: X is an estimate of the population mean, μ
Problems:
Different samples provide different estimates of the population parameter
Sample results have potential variability, thus sampling error exits
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

4. Calculating Sampling Error
The difference between a value (a statistic) computed from a sample and the corresponding value (a parameter) computed from a population
Example: (for the mean)
where:
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

5. Review Population mean: Sample Mean: where: μ = Population mean
x = sample mean
xi = Values in the population or sample
N = Population size
n = sample size
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

6. Example
If the population mean is μ = 98.6 degrees and a sample of n = 5 temperatures yields a sample mean of = degrees, then the sampling error is
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

7. Sampling Errors Different samples will yield different sampling errors
The sampling error may be positive or negative
( may be greater than or less than μ)
The expected sampling error decreases as the sample size increases
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

8. Sampling Distribution
A sampling distribution is a distribution of the possible values of a statistic for a given size sample selected from a population
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

9. 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
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

10. Developing a Sampling Distribution
(continued)
Summary Measures for the Population Distribution:
P(x) .3 .2 .1 x
A B C D
Uniform Distribution
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

11. 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)
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

12. 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)
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

13. Summary Measures of this Sampling Distribution:
Developing a
Sampling Distribution
(continued)
Summary Measures of this Sampling Distribution:
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

14. 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
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

15. Properties of a Sampling Distribution
For any population,
the average value of all possible sample means computed from all possible random samples of a given size from the population is equal to the population mean:
The standard deviation of the possible sample means computed from all random samples of size n is equal to the population standard deviation divided by the square root of the sample size:
Theorem 1
Theorem 2
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

16. 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
Theorem 3
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

17. 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
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

18. 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
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

19. Sampling Distribution Properties
The sample mean is an unbiased estimator
Normal Population Distribution
Normal Sampling Distribution
(has the same mean)
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

20. Sampling Distribution Properties
(continued)
The sample mean is a consistent estimator
(the value of x becomes closer to μ as n increases):
Population x
Small sample size
As n increases,
decreases
Larger sample size
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

21. 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
Theorem 4
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

22. Central Limit Theorem
the sampling distribution becomes almost normal regardless of shape of population
As the sample size gets large enough… n↑
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

23. 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)
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

24. 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 is sufficient
For normal population distributions, the sampling distribution of the mean is always normally distributed
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

25. 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?
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

26. 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 = μ = 8
…and standard deviation
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

27. Example Solution (continued) -- find z-scores: x (continued) z
Population Distribution
Sampling Distribution
Standard Normal Distribution ? ? ? ? ? ? ? ? ? ?
Sample
Standardize ? ? x z
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

28. Population Proportions, π
π = the proportion of the population having
some characteristic
Sample proportion ( p ) provides an estimate
of π :
If two outcomes, p has a binomial distribution
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

29. Sampling Distribution of p
Approximated by a normal distribution if:
where
and
Sampling Distribution
P( p ) .3 .2 .1 p
(where π = population proportion)
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

30. 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:
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

31. 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) ?
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

32. 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:
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

33. 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
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

34. Chapter Summary Discussed sampling error
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
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.