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Chapter 7 Sampling Methods and the Central Limit Theorem Prepared by: Jean-Paul Olivier Red River College © 2009 McGraw-Hill Ryerson Limited 1 of 39 Learning Objectives 1. Explain why a sample is often the only feasible way to learn something about a population 2. Describe methods to select a sample 3. Define and construct a sampling distribution of the sample mean 4. Explain the central limit theorem 5. Use the central limit theorem to find probabilities of selecting possible sample means from a specified population 6. Define and construct a sampling distribution of a proportion © 2009 McGraw-Hill Ryerson Limited 2 of 39 Reasons To Sample 1. To contact the whole population would be time consuming 2. The cost of studying all the items in a population may be prohibitive 3. The physical impossibility of checking all items in the population 4. The destructive nature of certain tests 5. The sample results are adequate © 2009 McGraw-Hill Ryerson Limited 3 of 39 Methods of Sampling 1. 2. 3. 4. Simple random sampling Systematic random sampling Stratified random sampling Cluster sampling © 2009 McGraw-Hill Ryerson Limited 4 of 39 Simple Random Sampling • A sample selected so that each item or person in the population has the same chance of being included • Can use tables of random numbers • Suppose a population consists of 845 employees of Nitra Industries – A sample of 20 employees is to be selected from that population © 2009 McGraw-Hill Ryerson Limited 5 of 39 Simple Random Sampling In Excel © 2009 McGraw-Hill Ryerson Limited 6 of 39 You Try It Out! The following class roster lists the students enrolling in an introductory course in business statistics. Three students are to be randomly selected and asked various questions regarding course content and method of instruction. a) The numbers 00 through 45 are handwritten on slips of paper and placed in a bowl. The three numbers selected are 31, 7, and 25. Which students would be included in the sample? b) Now use the table of random digits, Appendix E, to select your own sample c) What would you do if you encountered the number 59 in the table of random digits? © 2009 McGraw-Hill Ryerson Limited 7 of 39 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 • k is calculated as the population size divided by the sample size • When the physical order is related to the population characteristic, then systematic random sampling should not be used © 2009 McGraw-Hill Ryerson Limited 8 of 39 Stratified Random Sampling • A population is divided into subgroups, called strata, and a sample is randomly selected from each stratum • Once the strata are defined, we can apply simple random sampling within each group or strata to collect the sample © 2009 McGraw-Hill Ryerson Limited 9 of 39 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 © 2009 McGraw-Hill Ryerson Limited 10 of 39 You Try It Out! The following class roster lists the students enrolling in an introductory course in business statistics. Three students are to be randomly selected and asked various questions regarding course content and method of instruction. a) Suppose a systematic random sample will select every ninth student enrolled in the class. Initially, the fourth student on the list was selected at random. That student is numbered 03. Remembering that the random numbers start with 00, which students will be chosen to be members of the sample? © 2009 McGraw-Hill Ryerson Limited 11 of 39 Sampling Error • The difference between a sample statistic and its corresponding population parameter • Since the sample is a part or portion of the population, it is unlikely that the sample mean would be exactly equal to the population mean • Similarly, it is unlikely that the sample standard deviation would be exactly equal to the population standard deviation • We can therefore expect a difference between a sample statistic and its corresponding population parameter © 2009 McGraw-Hill Ryerson Limited 12 of 39 EXAMPLE – Sampling Error • Jane and Joe Miley operate a bed and breakfast called the Foxtrot Inn. There are eight rooms available for rent • Rentals for June are displayed • During the month of June there were 94 rentals so the mean number of units rented per night is 3.13 • The population mean, μ=3.13 © 2009 McGraw-Hill Ryerson Limited 13 of 39 EXAMPLE – Sampling Error • We take three random samples of five nights rented – The first random sample of five nights resulted in the following number of rooms rented: 4, 7, 4, 3, and 1 • The mean of this sample is 3.8 rooms, so the sampling error is 3.8-3.13=+0.67 – The second random sample of five nights resulted in the following number of rooms rented: 3, 3, 2, 3, and 6 • The mean of this sample is 3.4 rooms, so the sampling error is 3.4-3.13=+0.27 – In the third sample, the sampling error was found to be -1.33 © 2009 McGraw-Hill Ryerson Limited 14 of 39 Sampling Distribution of the Sample Mean • The sample means in the previous example varied from one sample to the next. The mean of the first sample of 5 days was 3.8 rooms, and the second sample was 3.4 rooms. The population mean was 3.13 rooms. • If we organize the means of all possible samples of five days into a probability distribution, the result is called the sampling distribution of the sample mean – a probability distribution of all possible sample means of a given sample size © 2009 McGraw-Hill Ryerson Limited 15 of 39 EXAMPLE – Sampling Distribution of the Sample Mean Tartus Industries has seven production employees (considered the population). The hourly earnings of each employee are given in the table. 1. 2. 3. 4. What is the population mean? What is the sampling distribution of the sample mean for samples of size 2? What is the mean of the sampling distribution? What observations can be made about the population and the sampling distribution? © 2009 McGraw-Hill Ryerson Limited 16 of 39 EXAMPLE – Sampling Distribution of the Sample Mean © 2009 McGraw-Hill Ryerson Limited 17 of 39 EXAMPLE – Sampling Distribution of the Sample Mean © 2009 McGraw-Hill Ryerson Limited 18 of 39 EXAMPLE – Sampling Distribution of the Sample Mean © 2009 McGraw-Hill Ryerson Limited 19 of 39 Relationship Between Population Distribution and Sampling Distribution 1. The mean of the sample means is exactly equal to the population mean 2. The variance of the sample means is equal to the population variance divided by n. 3. The sampling distribution of the sample means tends to become bell-shaped and to approximate the normal probability distribution – This approximation improves with larger samples © 2009 McGraw-Hill Ryerson Limited 20 of 39 You Try It Out! The lengths of service of all the executives employed by Standard Chemicals are: a) Using the combination formula, how many samples of size 2 are possible? b) List all possible samples of two executives from the population and compute their means. c) Organize the means into a sampling distribution. d) Compare the population mean and the mean of the sample means. e) Compare the dispersion in the population with that in the distribution of the sample mean. f) A chart portraying the population values follows. Is the distribution of population values normally distributed (bell-shaped)? g) Is the distribution of the sample mean computed in part (c) starting to show some tendency toward being bell-shaped? © 2009 McGraw-Hill Ryerson Limited 21 of 39 The 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 • This approximation improves with larger samples © 2009 McGraw-Hill Ryerson Limited 22 of 39 Illustration © 2009 McGraw-Hill Ryerson Limited 23 of 39 EXAMPLE – The Central Limit Theorem Spence Sprockets Inc. faces some major decisions regarding health care for these employees. Ed Spence decides to form a committee of five representative employees to study the health care issue carefully and make a recommendation. Ed feels the views of newer employees toward health care may differ from those of more experienced employees. If Ed randomly selects this committee, what can he expect in terms of the mean years with Spence Sprockets for those on the committee? How does the shape of the distribution of years of experience of all employees (the population) compare with the shape of the sampling distribution of the mean? The lengths of service of the 40 employees currently on the Spence Sprockets Inc. payroll are: © 2009 McGraw-Hill Ryerson Limited 24 of 39 EXAMPLE – The Central Limit Theorem This sample mean is 3.80 years. © 2009 McGraw-Hill Ryerson Limited 25 of 39 EXAMPLE – The Central Limit Theorem © 2009 McGraw-Hill Ryerson Limited 26 of 39 Standard Error of the Mean • The standard deviation of the sampling distribution of the sample mean • n is the number of observations in each sample © 2009 McGraw-Hill Ryerson Limited 27 of 39 Important Conclusions 1. 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 a given population. That is: X Even if we do not select all samples, we can expect the mean of the distribution of the sample mean to be close to the population mean. 2. There will be less dispersion in the sampling distribution of the sample mean than in the population. If the standard deviation of the population is σ, the standard deviation of the distribution of the sample mean is σ/√n. Note that when we increase the size of the sample, the standard error of the mean decreases. © 2009 McGraw-Hill Ryerson Limited 28 of 39 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 • To determine the probability a sample mean falls within a particular region, use: © 2009 McGraw-Hill Ryerson Limited 29 of 39 EXAMPLE -- Using the Sampling Distribution of the Sample Mean (σ Known) The quality control department for Cola Inc. maintains records regarding the amount of cola in its “Jumbo” bottle. The actual amount of cola in each bottle is critical, but varies a small amount from one bottle to the next. Cola Inc. does not wish to underfill the bottles, because it will have a problem with truth in labelling. On the other hand, it cannot overfill each bottle, because it would be giving cola away, hence reducing its profits. Its records indicate that the amount of cola follows a normal probability distribution. The mean amount per bottle is 1 L and the population standard deviation is 12.8 ml. At 8 a.m. today the quality control technician randomly selected 16 bottles from the filling line. The mean amount of cola contained in the bottles is 1.006 L. Is this an unlikely result? Is it likely the process is putting too much cola in the bottles? © 2009 McGraw-Hill Ryerson Limited 30 of 39 EXAMPLE -- Using the Sampling Distribution of the Sample Mean (σ Known) Step 1: Find the z-value corresponding to the sample mean of 1006ml given µ=1000ml and σ=12.8ml © 2009 McGraw-Hill Ryerson Limited 31 of 39 EXAMPLE -- Using the Sampling Distribution of the Sample Mean (σ Known) Step 2: Find the probability of observing a Z equal to or greater than 1.875 © 2009 McGraw-Hill Ryerson Limited 32 of 39 EXAMPLE -- Using the Sampling Distribution of the Sample Mean (σ Known) What do we conclude? It is unlikely, about a 3 percent chance, we could select a sample of 16 observations from a normal population with a mean of 1 L and a population standard deviation of 12.8 ml and find the sample mean equal to or greater than 1.006 L. We conclude the process is putting too much cola in the bottles. The quality control technician should see the production supervisor about reducing the amount of cola in each bottle. © 2009 McGraw-Hill Ryerson Limited 33 of 39 You Try It Out! Refer to the Cola Inc. information. Suppose the quality technician selected a sample of 16 Jumbo Cola bottles that averaged 0.996L. What can you conclude about the filling process? © 2009 McGraw-Hill Ryerson Limited 34 of 39 Sampling Distribution of The Proportion • For the nominal scale of measurement • A proportion is the fraction, ratio, or percent indicating the part of the sample or the population having a particular trait of interest © 2009 McGraw-Hill Ryerson Limited 35 of 39 EXAMPLE – Using the Sampling Distribution of a Proportion Alpha Corporation receives a shipment of flour every morning from its supplier. The flour is in 40 kg bags and Alpha will reject any shipment that is more than 5 percent underweight. The foreman samples 50 bags with each shipment and if the bags average more than 5 percent underweight, the whole shipment is returned to the supplier. What is the probability that in a sample of 150 bags, the foreman will find that less than 3 percent are underweight? © 2009 McGraw-Hill Ryerson Limited 36 of 39 EXAMPLE – Using the Sampling Distribution of a Proportion © 2009 McGraw-Hill Ryerson Limited 37 of 39 You Try It Out! Refer to the Alpha Corporation information. Compute the probability that in a sample of 200, the foreman will find more than 4 percent of the bags underweight. © 2009 McGraw-Hill Ryerson Limited 38 of 39 Chapter Summary • There are many reasons for sampling a population • In an unbiased sample, all members of the population have a chance of being selected for the sample. • There are several probability sampling methods including simple random sample, systematic sample, stratified sample, and cluster sampling • The sampling error is the difference between a population parameter and a sample statistic • The sampling distribution of the sample mean is a probability distribution of all possible sample means of a given size from a population © 2009 McGraw-Hill Ryerson Limited 39 of 39