8- Chapter Eight McGraw- © 2005 The McGraw-Hill Companies, Inc., All 8- Chapter Eight Sampling Methods and the Central Limit Theorem GOALS When you have completed this chapter, you will be ONEable to: 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- 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- 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- A probability sample is a sample selected such that each item or person in the population being studied has a known likelihood of being included in the sample 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 Probability Sampling/Methods selected for the sample Simple Random Sample A sample formulated so that each item or person in the population has the same chance of being included 8- 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- Cluster Sampling: A population is first divided into primary units then samples are selected from the primary units Cluster Sampling 8- In nonprobability sample inclusion in the sample is based on the judgment of the person selecting the sample 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- 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? Example 8- 10 objects taken at a time 5! 10 C2 2! (5 2)! A total of 10 different samples Example 8- 11 As a sampling distribution Example 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 Notice that the mean of the sample means is exactly equal to the population mean Example continued 8- 13 Central Limit Theorem For a population with a mean and a variance 2 the sampling distribution of the means of all possible samples of size n generated from the population will be approximately normally distributed The standard error of the mean is the standard deviation of the standard deviation of the sample means given as: x = n This approximation improves with larger samples The mean of the sampling distribution equal to m and the variance equal to 2/n Central Limit Theorem 8- 14 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- 15 To determine the probability that a sample mean falls within a particular region, use z X s n Use in place of s if the population standard deviation is known Sample Means 8- 16 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 8- 17 Step One : Find the z-values corresponding to $1.24 and $1.36 These are the two points within $0.08 of the population mean z z X s n X s n $1.38 $1.30 $0.28 35 $1.22 $1.30 $0.28 35 1.69 1.69 Example continued 8- 18 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 continued ... or person in the population being studied has a known likelihood of being included in the sample Systematic Random Sampling The items or individuals of the population are arranged in some order... sample of 35 gasoline stations and finding the sample mean within $.08? Example 8- 17 Step One : Find the z-values corresponding to $1.24 and $1.36 These are the two points within $0.08 of the... Probability Sampling 8- Cluster Sampling: A population is first divided into primary units then samples are selected from the primary units Cluster Sampling 8- In nonprobability sample inclusion in the