1. Introduction to Sampling Distributions
Chapter 6
Introduction to Sampling Distributions

2. Chapter 6 - Chapter Outcomes
After studying the material in this chapter, you should be able to:
• Understand the concept of sampling error.
• Determine the mean and standard deviation for the sampling distribution of the sample mean.

3. Chapter 6 - Chapter Outcomes (continued)
After studying the material in this chapter, you should be able to:
• Determine the mean and standard deviation for the sampling distribution of the sample proportion.
• Understand the importance of the Central Limit Theorem.
• Apply the sampling distributions for both the mean and proportion.

4. SAMPLING ERROR-SINGLE MEAN
The difference between a value (a statistic) computed from a sample and the corresponding value (a parameter) computed from a population.
Where:

5. Sampling Error -Parameters v. Statistics-
• A parameter is a measure computed from the entire population
• A statistic is a measure computed from a sample that has been selected from a population.

6. Sampling Error POPULATION MEAN Where:  = Population mean
x = Values in the population
N = Population size

7. Sampling Error (Example 6.1)
If  = 158,972 square feet and a sample of n = 5 shopping centers yields = 155,072 square feet, then the sampling error would be:

8. Sampling Errors
Useful Fundamental Statistical Concepts:
• The size of the sampling error depends on which sample is taken.
• The sampling error may be positive or negative.
• There is potentially a different value for each possible sample mean.

9. Sampling Error
A simple random sample is a sample selected in such a manner that each possible sample of a given size has an equal chance of being selected.

10. Sampling Error SAMPLE MEAN Where: = Sample mean
x = Sample value selected from the population
n = Sample size

11. POPULATION PROPORTION
Sampling Errors
POPULATION PROPORTION
Where:
 = Population proportion
x = Number of items having the attribute
N = Population size

12. Sampling Errors SAMPLE PROPORTION Where: p = Sample proportion
x = Number of items in the sample having the attribute
n = sample size

13. SINGLE PROPORTION SAMPLING ERROR Where:

14. Sampling Distributions
A sampling distribution is a distribution of the possible values of a statistic for a given size sample selected from a population.

15. Sampling Distribution of the Mean
THEOREM 6-1
If a population is normally distributed with a mean  and a standard deviation , the sampling distribution of the sample mean is also normally distributed with a mean equal to the population mean and a standard deviation equal to the population standard deviation divided by the square-root of the sample size

16. Sampling Distribution of the Mean
THEOREM 6-2: THE CENTRAL LIMIT THEREOM
For random samples of n observations taken from a population with mean  and standard deviation , regardless of the population’s distribution, provided the sample size is sufficiently large, the distribution of the sample mean , will be normal with a mean equal to the population mean Further, the standard deviation will equal the population standard deviation divided by the square-root of the sample size
The larger the sample size, the better the approximation to the normal distribution.

17. Sampling Distribution of the Mean
z-VALUE FOR SAMPLING DISTRIBUTION OF
where: = Sample mean
= Population mean
= Population standard deviation
n = Sample size

18. Example of Calculation z-Value for the Sample Mean (Example 6-5)
What is the probability that a sample of 100 automobile insurance claim files will yield an average claim of $4, or less if the average claim for the population is $4,560 with standard deviation of $600?

19. Sampling Distribution of a Proportion
SAMPLING DISTRIBUTION OF p
and
where: = Population proportion
= Sample proportion
n = Sample size

20. Sampling Distribution of the Mean
z-VALUE FOR PROPORTIONS
where: z = Number of standard errors p is from
p = Sample proportion
= Standard error of the sampling distribution
Mean of sample proportions

21. Example of Calculation z-Value for Proportion (Example 6-6)
What is the probability that a sample of 500 units will contain 18% or more broken items given that observation over time has shown 15% of shipments damaged?

22. Key Terms • Central Limit Theorem
• Finite Population Correction Factor
• Parameter
• Population Proportion
• Sample Proportion
• Sampling Distribution
• Sampling Error
• Simple Random Sample
• Statistic
• Theorem 6-1