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Welcome to Chapter 8 MBA 541. B ENEDICTINE U NIVERSITY Sampling and Estimation Sampling Methods and the Central Limit Theorem Chapter 8. Chapter 8. Please, Read Chapter 8 in Lind before viewing this presentation. Statistical Techniques in Business & Economics Lind. Goals.
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Welcome to Chapter 8MBA 541 BENEDICTINEUNIVERSITY • Sampling and Estimation • Sampling Methods and the Central Limit Theorem • Chapter 8
Chapter 8 Please, Read Chapter 8 in Lind before viewing this presentation. Statistical Techniques in Business & Economics Lind
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.
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.
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.
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.
Probability Sampling Methods • The methods for creating a probability sample include: • Simple random sampling, • Systematic random sampling, • Stratified random sampling, and • Cluster sampling.
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.
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.
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.
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.
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.
Example 1 (Continued) • Continuing, the data from the previous slide can be presented in a sampling distribution.
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.
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.
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.
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.)
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.
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:
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?
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.
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.