1. Chapter 7 Introduction to Sampling Distributions
Business Statistics:
A Decision-Making Approach
8th Edition
Chapter 7 Introduction to Sampling Distributions

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 _ _ _ _

3. Sampling Error Population Parameters
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

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:
Always present just because you sample!

5. 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

6. Sampling Errors The sampling error may be positive or negative
( may be greater than or less than μ)
The size of the error depends on the sample selected
i.e., a larger sample does not necessarily produce a smaller error if it is not a representative sample
Download “Sampling Distributions using Excel”
First sheet: “Sampling Error”

7. Sampling Distribution
A sampling distribution is a distribution of all possible values of a statistic for a given sample size that has been randomly selected from a population.
Download “Sampling Distributions using Excel”
Second sheet: “Sampling Dist”

8. Sampling Distribution Properties
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 (call “Unbiased Estimator”):
See “Sampling Distributions using Excel”
Considered an “unbiased” estimator
Theorem 1

9. Sampling Distribution Properties
(continued)
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 (call Standard Error):
Try using “Sampling Distributions using Excel”  Not Equal!
Also called the standard error
Theorem 2

10. Finite Population Correction
Virtually all survey research, sampling is conducted without replacement from populations that are of a finite size N.
In these cases, particularly when the sample size n is not small in comparison with the population size N (i.e., more than 5% of the population is sampled)
so that n/N > 0.05, a finite population correction factor (fpc) is used to define both the standard error of the mean and the standard error of the proportion.

11. Impact of fpc by Sample Size

12. Finite Population Correction
Apply the Finite Population Correction (fpc) if:
The sample size is greater than 5% of population size.
Only with sampling without replacement
Then
See fpc by “Sampling Distributions using Excel”

13. Sampling Distribution Properties
(continued)
As the sample size is increased, the StdDev of the sampling distribution is reduced…..
That is, the potential for extreme sampling error is reduced when larger sample size are used.
Graphical illustration: next slide

14. Sampling Distribution Properties
(continued)
The value of becomes closer to μ as n increases):
Population x
Small sample size
As n increases,
decreases
Larger sample size

15. 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
As n increases the data behaves more like a normal distribution

16. 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

17. Example
Suppose that a population is known to be normally distributed with μ = 2,000 and σ = 230 and random sample of size n = 8 is selected.
Because the population is normally distributed, the sampling distribution for the mean will also be normally distributed.
What is the probability that the sample mean will exceed 2,100?
Convert to Z value
Using Excel: 1-NORMSDIST(1.23)
Answer on the website

18. 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

19. If necessary, watch the Video
Central Limit Theorem
If necessary, watch the Video
the sampling distribution becomes almost normal regardless of shape of population
As the sample size gets large enough… n↑

20. 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

21. Example
Suppose a population (not normally distributed) has mean μ = 8 and standard deviation σ = 3 and random sample of size n = 36 (greater than 30) is selected.
What is the probability that the sample mean is between 7.8 and 8.2?

22. Example Solution (continued) -- find z-scores: x (continued) z
Population Distribution
Sampling Distribution
Standard Normal Distribution ? ? ? ? ? ? ? ? ? ?
Sample
Standardize ? ? x z

23. Sampling Distribution of a Proportion
Try this by yourself!
In many instances, the objective of sample is to estimate a population proportion.
An accountant may be interested in determining the proportion of accounts payable balances that are correct.
A production supervisor may wish to determine the percentage of product that is defect free.
A marketing research department might want to know the proportion of potential customers who will purchase a prticular product.

24. Population Proportions Example
If the true proportion of voters who support Proposition A is π (population proportion) = 0.4, what is the probability that a sample of size 200 yields a sample proportion between 0.40 and 0.45?
i.e.: if π = 0.4 and n = 200, what is
P(0.40 ≤ p ≤ 0.45) ?

25. Example if π = .4 and n = 200, what is P(0.40 ≤ p ≤ 0.45) ? Find :
(continued)
if π = .4 and n = 200, what is
P(0.40 ≤ p ≤ 0.45) ?
Find :
Convert to standard normal (z-values):

26. Standardized Normal Distribution
Example
(continued)
if π = 0.4 and n = 200, what is
P(0.40 ≤ p ≤ 0.45) ?
Use standard normal table: P(0 ≤ z ≤ 1.44) =
Standardized Normal Distribution
Sampling Distribution
0.4251
Standardize
0.40
0.45
1.44 p z

27. Population Proportions, π
π = the proportion of the population having
some characteristic
Sample proportion ( p ) provides an estimate
of π :
If two outcomes, p is a binomial distribution

28. Sampling Distribution of p
Approximated by a normal distribution if:
where
and
Sampling Distribution
P( p ) .3 .2 .1 p
Theorem 5
(where π = population proportion)

29. 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:

30. Using the Sample Distribution for Proportions
Determine the population proportion, p
Calculate the sample proportion, p
Derive the mean and standard deviation of the sampling distribution
Define the event of interest
If np and n(1-p) are both > 5, then covert p to z-value
Use standard normal table (Appendix D) to determine the probability

31. Chapter Summary Discussed sampling error
Introduced sampling distributions
Described the sampling distribution of the mean
For normal populations
Using the Central Limit Theorem (normality unknown)
Described the sampling distribution of a proportion
Calculated probabilities using sampling distributions
Discussed sampling from finite populations

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Printed in the United States of America.