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Sampling Methods and the Central Limit Theorem Chapter 8 McGraw-Hill/Irwin ©The McGraw-Hill Companies, Inc. 2008 GOALS 2 Understand why using sample is the only feasible way to learn about a population. Describe methods to select a sample. Define and construct a sampling distribution of the sample mean. Explain the central limit theorem. Use the central limit theorem to find probabilities of selecting possible sample means from a specified population. Why Sample? 3 The physical impossibility of checking all items in the population. The cost of studying all the items in a population. Contacting the whole population would often be time-consuming. The destructive nature of certain tests. Sample results are usually adequate. Methods of Probability Sampling Simple Random Sample: A sample formulated 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. – 4 k = population size/sample size Methods of Probability Sampling Stratified Random Sampling: A population is first divided into subgroups, called strata, and a sample is selected from each stratum. – Cluster Sampling: A population is first divided into clusters using a natural criterion, like geographic areas. Then some clusters are randomly selected, and a sample is selected from the selected clusters (primary units). – 5 To represent diverse groups. To reduce the cost of sampling. Sampling Error The sampling error is the difference between a sample statistic and the corresponding population parameter. – Sample is used to estimate population parameter. – – – – 6 Use sample mean (s.d) for population mean (s.d.). It is unlikely that sample mean (s.d.) is equal to population mean (s.d.). (Eg.) = 3 (N=200) for a population. The first sample (n=20) has mean= 3.5 (sampling error .5) The second sample (n=20) has mean= 3.2 ( ----- .2) The third sample (n=20) has mean= 2.6 (sampling error -.4) Sampling Distribution of the Sample Means The sample mean itself is a random variable having a certain probability distribution. – The sampling distribution of the sample mean is – 7 Consider all possible samples of the given size, getting the mean for each sample. a probability distribution of all possible sample means of a given sample size. Sampling Distribution of the Sample Means - Example A company has seven employees (the population). The hourly earnings of each employee are given in the table below. 1. What is the population mean? 2. What is the sampling distribution of the sample mean of samples of size 2? 3. What is the mean of the sampling distribution? 4. What observations can be made about the population and the sampling distribution? 8 Sampling Distribution of the Sample Means - Example 9 Sampling Distribution of the Sample Means - Example 10 Sampling Distribution of the Sample Means - Example 11 Sampling Distribution of the Sample Means - Example 12 From the above comparison, we can make following observations. 1) The mean of the sample means is equal to the mean of the population. 2) The dispersion of the sampling distribution of sample means is narrower than the population distribution. 3) The sampling distribution of sample means tends to become bell-shaped, and to approximate a normal distribution. Central Limit Theorem For a population with a mean μ and a variance σ2, the sampling distribution of the means of all possible samples of size n will be approximately normally distributed. – The mean of the sampling distribution equal to μ and the variance (of the sampling distrubition of the sample mean) equal to σ2/n. – 13 This approximation improves with larger samples. The standard error of the sample means equals to σ/n1/2 14 Central Limit Theorem (example) The example in the box (pp. 276-279) illustrates the central limit theorem – – sample means from a population that is not normally distributed will converge to normal distribution. Chart 8-3 the population distribution (non-normal). – – 15 The years of employment for a total of 40 employees. Chart 8-4 histogram of the means for 25 samples of 5 employees. Chart 8-5 histogram of the means for 25 samples of 20 employees Using the Sampling Distribution of the Sample Mean (σ Known) If a population follows the normal distribution, the sampling distribution of the sample mean will also follow the normal distribution. – 16 Regardless of the sample size. If the distribution of a population is not known (or it follows the non-normal distribution), the sampling distribution of the sample means from at least 30 observations will follow the normal distribution. (by the CLT). Using the Sampling Distribution of the Sample Mean (σ Known) We can convert any normal distribution to standard normal distribution. To determine the probability a sample mean falls within a particular region, we use: z 17 X n Using the Sampling Distribution of the Sample Mean (σ Unknown) Suppose the variance (the standard deviation) of the population is not known. To determine the probability a sample mean falls within a particular region, use: – Topic of the next chapter. X t s n 18 Using the Sampling Distribution of the Sample Mean (σ Known) - Example A Cola company maintains records of the amount of cola in a bottle. The actual amount of cola in each bottle is critical, but varies a little from one bottle to the next. The company does not wish to underfill or overfill the bottles. Its records indicate that the amount of cola follows the normal probability distribution. The mean amount per bottle is 31.2 ounces and the population standard deviation is 0.4 ounces. This morning a technician randomly selected 16 bottles from the filling line. The (sample) mean amount of cola contained in the bottles was 31.38 ounces. Is this an unlikely result? To put it another way, is the sampling error of 0.18 ounces unusual? 19 Using the Sampling Distribution of the Sample Mean (σ Known) - Example Step 1: Find the z-values corresponding to the sample mean of 31.38 X 31.38 32.20 z 1.80 n $0.2 16 20 Using the Sampling Distribution of the Sample Mean (σ Known) - Example Step 2: Find the probability of observing a Z equal to or greater than 1.80 21 Using the Sampling Distribution of the Sample Mean (σ Known) - Example What do we conclude? It is unlikely, less than a 4 percent chance, we could select a sample of 16 observations from a normal population with a mean of 31.2 ounces and a population standard deviation of 0.4 ounces and find the sample mean equal to or greater than 31.38 ounces. We conclude the process is putting too much cola in the bottles (They have to check the bottling process).. 22 End of Chapter 8 23