8- 1 Chapter Eight McGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved.

8- 2 Chapter Eight Sampling Methods and the Central Limit Theorem GOALS When you have completed this chapter, you will be able to: ONE Explain why a sample is 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. FOUR Explain the central limit theorem. Goals

8- 3 Chapter Eight continued Sampling Methods and the Central Limit Theorem GOALS When you have completed this chapter, you will be able to: FIVE Use the Central Limit Theorem to find probabilities of selecting possible sample means from a specified population. Goals

8- 4 Why Sample the Population? Why sample? The destructive nature of certain tests. The physical impossibility of checking all items in the population. The cost of studying all the items in a population. The adequacy of sample results in most cases. The time-consuming aspect of contacting the whole population.

8- 5 Probability Sampling/Methods Systematic Random Sampling The items or individuals of the population are arranged in some order. A random starting point is selected and then every k th member of the population is selected for the sample. (i.e. 1 in every 10 items) Simple Random Sample A sample selected so that each item or person in the population has the same chance of being included.

8- 6 Methods of Probability Sampling Stratified Random Sampling: A population is first divided into subgroups, called strata, and a sample is selected from each stratum.

8- 7 Cluster Sampling Cluster Sampling: A population is first divided into primary units then samples are selected from the primary units.

8- 8 Methods of Probability Sampling The sampling error is the difference between a sample statistic and its corresponding population parameter. The sampling distribution of the sample mean is a probability distribution consisting of all possible sample means of a given sample size selected from a population.

8- 9 Example 1 The law firm of Hoya and Associates has five partners. At their weekly partners meeting each reported the number of hours they billed clients for their services last week. If two partners are selected randomly, how many different samples are possible?

8- 10 Example 1 5 objects taken 2 at a time. A total of 10 different samples

8- 11 Example 1 continued As a sampling distribution

8- 12 Example 1 continued Compute the mean of the sample means. 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.

8- 13 Central Limit Theorem  x x =   n If all samples of a particular size are selected from any population, the sampling distribution of the sample mean is approximately a normal distribution. This approximation improves with larger samples. The mean of the sampling distribution equal to m and the variance equal to  2 /n. The standard error of the mean is the standard deviation of the population means divided by the square root of n given as: Central Limit Theorem

8- 14  States that any distribution of sample means from a large population approaches the normal distribution as n increases to infinity  The mean of the population of means is always equal to the mean of the parent population.  The standard deviation of the population of means is always equal to the standard deviation of the parent population divided by the square root of the sample size (N).  If you chart the values, the values will have less variation than the individual measurements  This is true if the sample size is sufficiently large.  What does this mean? VII-5 Central Limit Theorem Central Limit Theorem

8- 15 Central Limit Theorem For almost all populations, the sampling distribution of the mean can be closely approximated by a normal distribution, provided the sample is sufficiently large. Collect many x children, (assumption is infinite number of samples), create histograms. VII-6 Central Limit Theorem

8- 16 Sample Means the sample size is large enough even when the underlying population may be nonnormal Sample means follow the normal probability distribution under two conditions: the underlying population follows the normal distribution OR

8- 17 Use  in place of s if the population standard deviation is known. Sample Means To determine the probability that a sample mean falls within a particular region, use

8- 18 Example 2 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 $.08?

8- 19 Example 2 continued Step One : Find the z-values corresponding to $1.22 and $1.38. These are the two points within $0.08 of the population mean.

8- 20 Example 2 continued Step Two: determine the probability of a z-value between and We would expect about 91 percent of the sample means to be within $0.08 of the population mean.