Welcome to Chapter 8 MBA 541

1. Welcome to Chapter 8MBA 541 BENEDICTINEUNIVERSITY • Sampling and Estimation • Sampling Methods and the Central Limit Theorem • Chapter 8

2. Chapter 8 Please, Read Chapter 8 in Lind before viewing this presentation. Statistical Techniques in Business & Economics Lind

3. Goals When you have completed this chapter, you will be able to: • ONE • Explain why a sample is often the only feasible way to learn about a population. • TWO • Describe methods to select a sample. • THREE • Define and construct a sampling distribution of the sample mean.

4. Goals When you have completed this chapter, you will be able to: • FOUR • Explain the Central Limit Theorem. • FIVE • Use the Central Limit Theorem to find probabilities of selecting possible sample means from a specified population.

5. Why Sample the Population? • A sample is a portion or part of the population of interest. • The reasons to sample include: • To contact the whole population would be time consuming, • The cost of studying all the items in a population may be prohibitive, • The physical impossibility of checking all items in the population, • The destructive nature of some tests, and • The sample results are adequate.

6. Probability Sample A probability sample is a sample of items or individuals selected so that each member of the population has a chance of being included in the sample.

7. Probability Sampling Methods • The methods for creating a probability sample include: • Simple random sampling, • Systematic random sampling, • Stratified random sampling, and • Cluster sampling.

8. Probability Sampling Methods • Simple Random Sample: A sample selected so that each item or person in the population has the same chance of being included. • Systematic Random Sample: A random starting point is selected, and then every kth member of the population is selected. • Stratified Random Sample: A population is divided into subgroups, called strata, and a sample is randomly selected from each stratum. • Cluster Sampling: A population is divided into clusters using naturally occurring geographic or other boundaries. Then, clusters are randomly selected and a sample is collected by random selection from each cluster.

9. Sampling Error • Sampling Error: The difference between a sample statistic and its corresponding population parameter. • The value of the sampling error is based on the random selection of a sample. • Sampling errors are random and occur by chance.

10. Sampling Distribution of the Sample Mean A Sampling Distribution of the Sample Mean is a probability distribution of all possible sample means of a given sample size.

11. Example 1 • Example 1 illustrates the concept of the sampling distribution of the sample mean. • If two partners are selected randomly, how many different samples are possible? Background: The law firm of Hoya and associates has five partners. At their weekly partner’s meeting, each reported the number of hours they billed clients for their services last week.

12. Example 1 (Continued) • Continuing, how many different samples are possible, if the hours from 2 partners are selected randomly? Mathematically, the number of combinations of 5 objects taken 2 at a time is required. A total of 10 different samples are possible.

13. Example 1 (Continued) • Continuing, the data from the previous slide can be presented in a sampling distribution.

14. Example 1 (Continued) • Continuing, compute the mean of the sample data. Compare it with the population mean. • The mean of the sample means: • The population mean: • Notice that the mean of the sample means is exactly equal to the population mean.

15. Central Limit Theorem Central Limit Theorem: If all samples of a particular size are selected from any population, the sampling distribution of the sample means is approximately a normal distribution. This approximation improves with larger samples.

16. Central Limit Theorem • The sampling distribution of the sample mean will be normal if either: • The underlying population follows the normal distribution, Or • The sample size is large enough (at least 30) even when the underlying population may be non-normal.

17. Characteristics of theSampling Distributions • Consider the sampling distribution of the sample means taken from a population with a mean, μ, and a standard deviation, σ. • The mean of the sampling distribution equals the population mean. • There is less variation in the distribution of the sample means than in the population mean. The standard error of the mean measures the variation in the sampling distribution. (Where n is the number of observations in each sample.)

18. Using theSampling Distributions • Most business decisions are made on the basis of sampling results. • By using the concepts of sampling distributions, the probability that a sample mean will fall within a certain range can be computed. • By using z values, any sampling distribution can be converted into the standard normal distribution. This conversion allows for simplified calculations of probabilities.

19. Using theSampling Distributions • To find the z value for the mean from the sampling distribution, • When the population standard deviation is known, use the following formula: • When the population standard deviation is unknown, use the following formula:

20. Example 2 • Example 2 illustrates finding probabilities using the sampling distribution of sample means. • Background: Suppose the mean selling price of a gallon of gasoline in the United States is $1.30. Further assume the distribution is positively skewed, with a standard deviation of $0.28. • What is the probability of selecting a sample of 35 gasoline stations and finding the sample mean within $0.08?

21. Example 2 (Continued) • Step One: Find the z values corresponding to $1.22 and $1.38. These are the two values within $0.08 of the population mean.

22. Example 2 (Continued) • Step Two: Determine the probability of a z value between 1.69 and 1.69. • About 91% of the sample means are expected to be within $0.08 of the population mean.