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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 physical impossibility of checking all items in the population. The cost of studying all the items in a population. The destructive nature of certain tests. The time-consuming aspect of contacting the whole population. The adequacy of sample results in most cases. Why Sample the Population? 8- 5 Simple Random Sample A sample selected so that each item or person in the population has the same chance of being included. Systematic Random Sampling The items or individuals of the population are arranged in some order. A random starting point is selected and then every kth member of the population is selected for the sample. (i.e. 1 in every 10 items) Probability Sampling/Methods 8- 6 Stratified Random Sampling: A population is first divided into subgroups, called strata, and a sample is selected from each stratum. Methods of Probability Sampling 8- 7 Cluster Sampling: A population is first divided into primary units then samples are selected from the primary units. Cluster Sampling 8- 8 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. Methods of Probability Sampling 8- 9 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. Partner Hours Dunn 22 Hardy 26 Kiers 30 Malory 26 Tillman 22 If two partners are selected randomly, how many different samples are possible? Example 1 8- 10 5 objects taken 2 at a time. 5! 10 5 C2 2! (5 2)! Partners 1,2 1,3 1,4 1,5 2,3 2,4 2,5 3,4 3,5 4,5 Total 48 52 48 44 56 52 48 56 52 48 A total of 10 different samples Mean 24 26 24 22 28 26 24 28 26 24 Example 1 8- 11 As a sampling distribution Sample Mean Frequency Relative Frequency probability 22 1 1/10 24 4 4/10 26 3 3/10 28 2 2/10 Example 1 continued 8- 12 Compute the mean of the sample means. Compare it with the population mean. The mean of the sample means X 22 (1) 24 (2) 26 (3) 28 (2) 25 .2 10 The population mean 22 26 30 26 22 25 .2 5 Notice that the mean of the sample means is exactly equal to the population mean. Example 1 continued 8- 13 Central Limit Theorem If all samples of a particular size are selected from any population, the sampling distribution of the sample mean is approximately a normal distribution. The standard error of the mean is the standard deviation of the population means divided by the square root of n given as: sx = s n This approximation improves with larger samples. The mean of the sampling distribution equal to m and the variance equal to s2/n. Central Limit Theorem 8- 14 Central Limit Theorem States that any distribution of sample means x 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 x 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? Central Limit Theorem http://www.chem.uoa.gr/applets/appletcentrallimit/appl_centrallimit2.html VII-5 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. Central Limit Theorem VII-6 8- 16 Sample means follow the normal probability distribution under two conditions: the underlying population follows the normal distribution OR the sample size is large enough even when the underlying population may be nonnormal Sample Means 8- 17 To determine the probability that a sample mean falls within a particular region, use z X s n Use s in place of s if the population standard deviation is known. Sample Means 8- 18 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? Example 2 8- 19 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. z X s z n X s n $1.38 $1.30 $0.28 35 $1.22 $1.30 $0.28 1.69 1.69 35 Example 2 continued 8- 20 Step Two: determine the probability of a z-value between -1.69 and 1.69. P(1.69 z 1.69) 2(.4545 ) .9090 We would expect about 91 percent of the sample means to be within $0.08 of the population mean. Example 2 continued