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7-1 Chapter 7 Created by Bethany Stubbe and Stephan Kogitz McGraw-Hill-Ryerson © The McGraw-Hill Companies, Inc., 2004 All Rights Reserved. Chapter Seven 7-2 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 often the only feasible way to learn something 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. McGraw-Hill-Ryerson © The McGraw-Hill Companies, Inc., 2004 All Rights Reserved. Chapter Seven 7-3 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. McGraw-Hill-Ryerson © The McGraw-Hill Companies, Inc., 2004 All Rights Reserved. 7-4 Why Sample the Population? To contact the whole population would often be time consuming. The cost of studying all the items in a population is often prohibitive. The adequacy of sample results. McGraw-Hill-Ryerson © The McGraw-Hill Companies, Inc., 2004 All Rights Reserved. 7-5 Why Sample the Population The destructive nature of certain tests. The physical impossibility of checking all items in the population. McGraw-Hill-Ryerson © The McGraw-Hill Companies, Inc., 2004 All Rights Reserved. 7-6 Probability Sampling A probability sample is a sample selected such that each member of the population being studied has a known likelihood of being included in the sample. McGraw-Hill-Ryerson © The McGraw-Hill Companies, Inc., 2004 All Rights Reserved. 7-7 Methods of Probability Sampling 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: A random starting point is selected and then every kth member of the population is selected. McGraw-Hill-Ryerson © The McGraw-Hill Companies, Inc., 2004 All Rights Reserved. 7-8 Methods of Probability Sampling Stratified Random Sampling: 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 randomly selecting from each cluster. McGraw-Hill-Ryerson © The McGraw-Hill Companies, Inc., 2004 All Rights Reserved. 7-9 Methods of Probability Sampling In nonprobability sample inclusion in the sample is based on the judgment of the person selecting the sample. McGraw-Hill-Ryerson © The McGraw-Hill Companies, Inc., 2004 All Rights Reserved. 7-10 The Sampling Distribution of the Sample Mean The sampling error is the difference between a sample statistic and its corresponding population parameter. McGraw-Hill-Ryerson © The McGraw-Hill Companies, Inc., 2004 All Rights Reserved. 7-11 Sampling Distribution of the Sample Mean The sampling distribution of the sample mean is a probability distribution of all possible sample means of a given sample size. McGraw-Hill-Ryerson © The McGraw-Hill Companies, Inc., 2004 All Rights Reserved. 7-12 EXAMPLE 1 The law firm of Tybo 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 1. Dunn 22 2. Hardy 26 3. Kiers 30 4. Malinowski 26 5. Tillman 22 McGraw-Hill-Ryerson © The McGraw-Hill Companies, Inc., 2004 All Rights Reserved. 7-13 Example 1 If two partners are selected randomly, how many different samples are possible? There are 10 different samples. This is the combination of 5 objects taken 2 at a time. 5! 10 5 C2 2! (5 2)! McGraw-Hill-Ryerson © The McGraw-Hill Companies, Inc., 2004 All Rights Reserved. 7-14 Example 1 continued Partners Total Mean 1,2 48 24 1,3 52 26 1,4 48 24 1,5 44 22 2,3 56 28 2,4 52 26 2,5 48 24 3,4 56 28 3,5 52 26 4,5 48 24 McGraw-Hill-Ryerson © The McGraw-Hill Companies, Inc., 2004 All Rights Reserved. 7-15 EXAMPLE 1 continued Organize the sample means into a sampling distribution. S a m p le M ean F re q u e n c y 22 1 R e la tiv e F re q u e n c y p ro b a b ility 1 /1 0 24 4 4 /1 0 26 3 3 /1 0 28 2 2 /1 0 McGraw-Hill-Ryerson © The McGraw-Hill Companies, Inc., 2004 All Rights Reserved. 7-16 EXAMPLE 1 continued Compute the mean of the sample means. Compare it with the population mean. The mean of the sample means is 25.2 hours. X 22 (1) 24 (2) 26 (3) 28 (2) 25 .2 10 McGraw-Hill-Ryerson © The McGraw-Hill Companies, Inc., 2004 All Rights Reserved. 7-17 Example 1 continued The population mean is also 25.2 hours. 22 26 30 26 22 25 .2 5 Notice that the mean of the sample means is exactly equal to the population mean. McGraw-Hill-Ryerson © The McGraw-Hill Companies, Inc., 2004 All Rights Reserved. 7-18 Central Limit Theorem If all samples of a specified size are selected from any population, the sampling distribution of the sample mean is approximately a normal distribution. This approximation improves with larger samples. McGraw-Hill-Ryerson © The McGraw-Hill Companies, Inc., 2004 All Rights Reserved. 7-19 Mean of the Sample Means The mean of the distribution of the sample mean will be exactly equal to the population mean if we are able to select all possible samples of a particular size from the a given population. McGraw-Hill-Ryerson X © The McGraw-Hill Companies, Inc., 2004 All Rights Reserved. 7-20 Standard Error of the Mean If the standard deviation of the population is σ, the standard deviation of the distribution of the sample mean is / n x McGraw-Hill-Ryerson © The McGraw-Hill Companies, Inc., 2004 All Rights Reserved. 7-21 Sampling from a Normal Population If a population follows the normal distribution, the sampling distribution of the sample mean will also follow the normal distribution. McGraw-Hill-Ryerson © The McGraw-Hill Companies, Inc., 2004 All Rights Reserved. 7-22 Finding the z Value of X When σ is known If the population is normally distributed Assume the standard deviation, σ, is known. To determine the probability a sample mean falls within a particular region, use: z McGraw-Hill-Ryerson X n © The McGraw-Hill Companies, Inc., 2004 All Rights Reserved. 7-23 Finding the z Value of X When σ is Unknown If the population is not normally distributed, but the sample is at least 30 observations, the sampling distribution of the sample mean is approximately normal. Assume the population standard deviation is not known, use the sample standard deviation. McGraw-Hill-Ryerson © The McGraw-Hill Companies, Inc., 2004 All Rights Reserved. 7-24 Finding the z Value of X When σ is Unknown continued If the population standard deviation, σ, is unknown: To determine the probability a sample mean falls within a particular region, use: z X s McGraw-Hill-Ryerson n © The McGraw-Hill Companies, Inc., 2004 All Rights Reserved. 7-25 Example 2 The mean selling price of 100 ml. Tube of toothpaste is $1.30. The distribution is positively skewed, with a standard deviation of $0.28. What is the probability of selecting a sample of 35 stores and finding the sample mean within $.08? McGraw-Hill-Ryerson © The McGraw-Hill Companies, Inc., 2004 All Rights Reserved. 7-26 Example 2 continued The first step is to find the z-values corresponding to $1.24 and $1.36. These are the two points within $0.08 of the population mean. z X s McGraw-Hill-Ryerson n $1.38 $1.30 $0.28 1.69 35 © The McGraw-Hill Companies, Inc., 2004 All Rights Reserved. 7-27 Example 2 z X s n McGraw-Hill-Ryerson continued $1.22 $1.30 1.69 $0.28 35 © The McGraw-Hill Companies, Inc., 2004 All Rights Reserved. 7-28 Example 2 continued Next we determine the probability of a zvalue between -1.69 and 1.69. It is: 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. McGraw-Hill-Ryerson © The McGraw-Hill Companies, Inc., 2004 All Rights Reserved.