Ebook Statistics for managers using - Microsoft excel (5th edition): Part 2

CHAPTER 7 Sampling and Sampling Distributions USING STATISTICS @ Oxford Cereals 7.1 TYPES OF SAMPLING METHODS Simple Random Samples Sampling from Normally Distributed Populations Sampling from Non-Normally Distributed Systematic Samples Populations The Central Limit Theorem Stratified Samples Cluster Samples 7.5 SAMPLING DISTRIBUTION OF THE PROPORTION 7.2 EVALUATING SURVEY WORTHINESS Survey Errors Ethical Issues 7.3 SAMPLING DISTRIBUTIONS 7.4 SAMPLING DISTRIBUTION OF THE MEAN The Unbiased Property of the Sample Mean Standard Error of the Mean 7.6 (CD-ROM TOPIC) SAMPLING FROM FINITE POPULATIONS EXCEL COMPANION TO CHAPTER 7 E7.1 Creating Simple Random Samples (Without Replacement) E7.2 Creating Simulated Sampling Distributions LEARNING OBJECTIVES In this chapter, you learn: * To distinguish between different sampling methods * The concept of the sampling distribution * To compute probabilities related to the sample mean and the sample proportion * The importance of the Central LimitTheorem Statistics for Managers Using Microsoft Excel, Fifth Edition, by David M. Levine, Mark L. Berenson, and Timothy C. Krehbiel. Published by Prentice Hall. Copyright 2008 by Pearson Education, Inc. 252 CHAPTER SEVEN Sampling and Sampling Distributions Using Statistics @ Oxford Cereals Oxford Cereals fills thousands of boxes of cereal during an eight-hour shift. As the plant operations manager, you are responsible for monitoring the amount of cereal placed in each box. To be consistent with package labeling, boxes should contain a mean of 368 grams of cereal. Because of the speed of the process, the cereal weight varies from box to box, causing some boxes to be underfilled and others overfilled. If the process is not working properly, the mean weight in the boxes could vary too much from the label weight of 368 grams to be acceptable. Because weighing every single box is too time-consuming, costly, and inefficient, you must take a sample of boxes. For each sample you select, you plan to weigh the individual boxes and calculate a sample mean. You need to determine the probability that such a sample mean could have been randomly selected from a population whose mean is 368 grams. Based on your analysis, you will have to decide whether to maintain, alter, or shut down the process. n Chapter 6, you used the normal distribution to study the distribution of download times for the OurCampus! Web site. In this chapter, you need to make a decision about the cereal-filling process, based on a sample of cereal boxes. You will learn different methods of sampling and about sampling distributions and how to use them to solve business problems. 7.1 TYPES OF SAMPLING METHODS In Section 1.3, a sample was defined as the portion of a population that has been selected for analysis. Rather than selecting every item in the population, statistical sampling procedures focus on collecting a small representative group of the larger population. The results of the sample are then used to estimate characteristics of the entire population. There are three main reasons for selecting a sample: * Selecting a sample is less time-consuming than selecting every item in the population. * Selecting a sample is less costly than selecting every item in the population. * An analysis of a sample is less cumbersome and more practical than an analysis of the entire population. The sampling process begins by defining the frame. The frame is a listing of items that make up the population. Frames are data sources such as population lists, directories, or maps. Samples are drawn from frames. Inaccurate or biased results can result if a frame excludes cer-tain portions of the population. Using different frames to generate data can lead to opposite conclusions. After you select a frame, you draw a sample from the frame. As illustrated in Figure 7.1, there are two kinds of samples: the nonprobability sample and the probability sample. 7.1: Types of Sampling Methods 253 FIGURE 7.1 Types of samples Types of Samples Used Nonprobability Samples Probability Samples Judgment Quota Sample Sample Chunk Convenience Simple Sample Sample Random Sample Systematic Stratified Sample Sample Cluster Sample In a nonprobability sample, you select the items or individuals without knowing their probabilities of selection. Thus, the theory that has been developed for probability sampling cannot be applied to nonprobability samples. A common type of nonprobability sampling is convenience sampling. In convenience sampling, items are selected based only on the fact that they are easy, inexpensive, or convenient to sample. In many cases, participants are self-selected. For example, many companies conduct surveys by giving visitors to theirWeb site the opportunity to complete survey forms and submit them electronically. The responses to these surveys can provide large amounts of data quickly and inexpensively, but the sample consists of self-selected Web users (see p. 8). For many studies, only a nonprobability sample such as a judgment sample is available. In a judgment sample, you get the opinions of preselected experts in the subject matter. Some other common procedures of nonprobability sampling are quota sampling and chunk sampling.These are discussed in detail in specialized books on sam-pling methods (see reference 1). Nonprobability samples can have certain advantages, such as convenience, speed, and low cost. However, their lack of accuracy due to selection bias and the fact that the results cannot be generalized more than offset these advantages. Therefore, you should use nonprobability sam-pling methods only for small-scale studies that precede larger investigations. In a probability sample, you select the items based on known probabilities. Whenever possible, you should use probability sampling methods. Probability samples allow you to make unbiased inferences about the population of interest. In practice, it is often difficult or impossible to take a probability sample. However, you should work toward achieving a probability sample and acknowledge any potential biases that might exist. The four types of probability samples most commonly used are simple random, systematic, stratified, and cluster samples. These sampling methods vary in their cost, accuracy, and complexity. Simple Random Samples In a simple random sample, every item from a frame has the same chance of selection as every other item. In addition, every sample of a fixed size has the same chance of selection as every other sample of that size. Simple random sampling is the most elementary random sam-pling technique. It forms the basis for the other random sampling techniques. With simple random sampling, you use n to represent the sample size and N to represent the frame size.You number every item in the frame from 1 to N.The chance that you will select any particular member of the frame on the first selection is 1/N. You select samples with replacement or without replacement. Sampling with replace-ment means that after you select an item, you return it to the frame, where it has the same prob-ability of being selected again. Imagine that you have a fishbowl containing N business cards. Statistics for Managers Using Microsoft Excel, Fifth Edition, by David M. Levine, Mark L. Berenson, and Timothy C. Krehbiel. Published by Prentice Hall. Copyright 2008 by Pearson Education, Inc. 254 CHAPTER SEVEN Sampling and Sampling Distributions On the first selection, you select the card for Judy Craven. You record pertinent information and replace the business card in the bowl.You then mix up the cards in the bowl and select the second card. On the second selection, Judy Craven has the same probability of being selected again, 1/N.You repeat this process until you have selected the desired sample size, n. However, usually you do not want the same item to be selected again. Sampling without replacement means that once you select an item, you cannot select it again.The chance that you will select any particular item in the frame for example, the busi-ness card for Judy Craven on the first draw is 1/N. The chance that you will select any card not previously selected on the second draw is now 1 out of N ∗ 1. This process continues until you have selected the desired sample of size n. Regardless of whether you have sampled with or without replacement, fishbowl methods of sample selection have a major drawback the ability to thoroughly mix the cards and ran-domly select the sample.As a result, fishbowl methods are not very useful.You need to use less cumbersome and more scientific methods of selection. One such method uses a table of random numbers (see Table E.1) for selecting the sam-ple. A table of random numbers consists of a series of digits listed in a randomly generated sequence (see reference 8). Because the numeric system uses 10 digits (0, 1, 2, . . . , 9), the chance that you will randomly generate any particular digit is equal to the probability of gener-ating any other digit.This probability is1 outof10.Hence,ifyou generate asequence of800 dig-its, you would expect about 80 to be the digit 0, 80 to be the digit 1, and so on. In fact, those who use tables of random numbers usually test the generated digits for randomness prior to using them. Table E.1 has met all such criteria for randomness. Because every digit or sequence of digits in the table is random, the table can be read either horizontally or vertically. The mar-gins of the table designate row numbers and column numbers. The digits themselves are grouped into sequences of five in order to make reading the table easier. To use such a table instead of a fishbowl for selecting the sample, you first need to assign code numbers to the individual members of the frame. Then you generate the random sample by reading the table of random numbers and selecting those individuals from the frame whose assigned code numbers match the digits found in the table. You can better understand the process of sample selection by examining Example 7.1. EXAMPLE 7.1 SELECTING A SIMPLE RANDOM SAMPLE BY USING A TABLE OF RANDOM NUMBERS A company wants to select a sample of 32 full-time workers from a population of 800 full-time employees in order to collect information on expenditures concerning a company-sponsored dental plan. How do you select a simple random sample? SOLUTION The company decides to conduct an email survey.Assuming that not everyone will respond to the survey, you need to send more than 32 surveys to get the necessary 32 responses. Assuming that 8 out of 10 full-time workers will respond to such a survey (that is, a response rate of 80%), you decide to send 40 surveys. The frame consists of a listing of the names and email addresses of all N = 800 full-time employees taken from the company personnel files. Thus, the frame is an accurate and com-plete listing of the population. To select the random sample of 40 employees from this frame, you use a table of random numbers. Because the population size (800) is a three-digit number, each assigned code number must also be three digits so that every full-time worker has an equal chance of selection.You assign a code of 001 to the first full-time employee in the population listing, a code of 002 to the second full-time employee in the population listing, and so on, until a code of 800 is assigned to the Nth full-time worker in the listing. Because N = 800 is the largest possible coded value, you discard all three-digit code sequences greater than 800 (that is, 801 through 999 and 000). Statistics for Managers Using Microsoft Excel, Fifth Edition, by David M. Levine, Mark L. Berenson, and Timothy C. Krehbiel. Published by Prentice Hall. Copyright 2008 by Pearson Education, Inc. 7.1: Types of Sampling Methods 255 To select the simple random sample, you choose an arbitrary starting point from the table of random numbers. One method you can use is to close your eyes and strike the table of ran-dom numbers with a pencil. Suppose you used this procedure and you selected row 06, column 05, of Table 7.1 (which is extracted from Table E.1) as the starting point. Although you can go in any direction, in this example, you read the table from left to right, in sequences of three dig-its, without skipping. TABLE 7.1 Column Using a Table of Random Numbers Row 01 02 03 04 05 Begin 06 selection 07 (row 06, 08 column 5) 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 00000 00001 12345 67890 49280 88924 61870 41657 43898 65923 62993 93912 33850 58555 97340 03364 70543 29776 89382 93809 37818 72142 60430 22834 82975 66158 39087 71938 55700 24586 14756 23997 32166 53251 23236 73751 45794 26926 09893 20505 54382 74598 94750 89923 70297 34135 85157 47954 11100 02340 36871 50775 23913 48357 11111 11112 22222 12345 67890 12345 35779 00283 81163 07468 08612 98083 25078 86129 78496 30454 84598 56095 51438 85507 71865 88472 04334 63919 10087 10072 55980 00796 95945 34101 67140 50785 22380 14130 96593 23298 84731 19436 55790 40355 54324 08401 93247 32596 11865 78643 75912 83832 70654 92827 63491 31888 81718 06546 15130 82455 78305 14225 68514 46427 91499 14523 68479 37089 20048 80336 53140 33340 42050 32979 26575 57600 12860 74697 96644 30592 57143 17381 63308 16090 51690 22223 33333 33334 67890 12345 67890 07275 89863 02348 97349 20775 45091 97653 91550 08078 20664 12872 64647 79488 76783 31708 36394 11095 92470 64688 68239 20461 81277 66090 88872 16703 53362 44940 56203 92671 15925 69229 28661 13675 26299 49420 59208 63397 44251 43189 32768 18928 57070 04233 33825 69662 83246 47651 04877 55058 52551 47182 56788 96297 78822 27686 46162 83554 94598 26940 36858 82341 44104 82949 40881 12250 73742 89439 28707 25815 68856 25853 35041 54607 72407 55538 Source: Partially extracted from The Rand Corporation,A Million Random Digits with 100,000 Normal Deviates (Glencoe, IL:The Free Press, 1955) and displayed in Table E.1 in Appendix E. The individual with code number 003 is the first full-time employee in the sample (row 06 and columns 05 07), the second individual has code number 364 (row 06 and columns 08 10), and the third individual has code number 884. Because the highest code for any employee is 800, you discard the number 884. Individuals with code numbers 720, 433, 463, 363, 109, 592, 470, and 705 are selected third through tenth, respectively. You continue the selection process until you get the required sample size of 40 full-time employees. During the selection process, if any three-digit coded sequence repeats, you include the employee corresponding to that coded sequence again as part of the sample if you are sam-pling with replacement.You discard the repeating coded sequence if you are sampling without replacement. Statistics for Managers Using Microsoft Excel, Fifth Edition, by David M. Levine, Mark L. Berenson, and Timothy C. Krehbiel. Published by Prentice Hall. 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