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Sample Size and Statistical theory - Summary This chapter has introduced some useful concepts and applied them to the problem of determining the sample size. A population  characteristic, such as the attitude of symphony season ticketholders, is to be estimatecTby the sample. A sample statistic such as the average attitude of a sample of season ticketholders, is used to estimate the population characteristic. The sample statistic will have a variance (it will not be the same each time a sample is drawn), and this will be a measure of its reliability. The estimate, based on the sample statistic, has an interval associated with it that reflects

Sequential Sampling Sometimes a researcher may want to take a modest sample, look at the results, and then decide if more information, in the form of a larger sample, is needed. Such a procedure is termed sequential sampling. If a new industrial product were being evaluated, a small probability sample of potential users might be contacted. Suppose it were found that their average annual usage level at a 95 percent confidence level was between 10 and 30 units, and it was known that for the product to be economically viable the average would have to be 50 units. This is sufficient information for a decision to drop the product. If, however, the interval estimate from the original sample were from 45 to 65, then the information would be inadequate for making that decision and an additional sample might be obtained. The combined samples then would provide a smaller interval estimate. If the resulting interval were still inadequate, the sample size could be increased a third time. Of cou...

Sample Size and Statistical theory - Stratified Sampling In stratified sampling, the population is divided into subgroups or strata and a sample is taken from each. Stratified sampling is worthwhile when one or both of the following are true: 1. The population standard deviation differs by strata. 2. The interview cost differs by strata. Suppose we desired to estimate the usage of electricity to heat swimming pools. The population of swimming pools might be stratified into commercial pools at hotels and clubs and individual home swimming pools. The latter may have a small variation and thus would require a smaller sample. If, however, the home-pool owners were less costly to interview, that would allow more of them to be interviewed than if the two groups involved the same interview cost. How does one determine the best allocation of the sampling budget to the various strata? This classic problem of sampling was solved in 1935 by Jerzy Neyman.  His solution is represent...

Sample Size and Statistical theory - Sample-Size Question Now, we are finally ready to use these concepts to help determine sample size. To proceed, the analyst must specify: 1. Size of the sampling error that is desired. 2. Confidence level, for example, the 95 percent confidence level. This specification will depend on a trade-off between the value of more accurate information and the cost of an increased sample size. For a given confidence level, a smaller sampling error will "cost" in terms of a larger sample size. Similarly, for a given sampling error, a higher confidence level will "cost" in terms of a larger sample size. These statements will become more tangible in the context of some examples. Using the general formula for the interval estimate (recall that a and ax are the same) We know that Sampling error = VnT Dividing through by the sampling error and multiplying by Vn (sampling error) and squaring both sides, we get an expression f...

Interval Estimation The sample mean, X, is used to estimate the unknown population mean (p.). Because X varies from sample to sample, it is not, of course, equal to the population mean (u-). There is a sampling error. It is useful to provide an interval estimate around X that reflects our judgment of the extent of this sampling error: X ± sampling error = the interval estimate of u. The size of the interval will depend on how confident we want to be tha: the interval contains the true unknown population mean. If it were necessary to be 95 percent confident that the interval estimate contained the true population mean, the interval estimate would be 2o-„ X ± 28^ = X ± —f= = 95 percent interval estimate of p. (recall that ox = a^/Vri). The interval size is based on 28* because, as Figure 12-3 shows, the probability that X will be within 28* of the population mean is 0.95. In our example, the interval would be X ± 28x- = 0.5 ± 2 x .47 = 0.5 ± .94 a„ sinceS - = —7= = .47. ...

Sample Size and Statistical theory - Sample Reliability Of course, all samples will not generate the same value of X (or s). If another simple random sample of size 10 were taken from the population, X might be 0.3 or 1.2 or 0.4 or whatever. The point is that X will vary from sample to sample. Intuitively, it is reasonable to believe that the variation in X will be larger as the variance in the population, cr2, is larger. At one extreme, if there is no variation in the population, there will be no variation inX. It also is reasonable to believe that, as the size of the sample increases, the variation in X will decrease. When the sample is small, it takes only one or two extreme scores to substantially affect the sample mean, thus generating a relatively large or small X. As the sample size increases, these extreme values will have less impact when they do appear, because they will be averaged with more values. The variation in X is measured by its standard error,  which is a...

Sample Size and Statistical theory - Sample Characteristics The problem is that the population mean is not known but must be estimated from a sample. Assume that a simple random sample of size 10 is taken from the population. The 10 people selected and their respective attitudes are shown in Figure 12-2. 10 X = -jrrlXt = 0.5 iu i=l s2 = -A_2(Xt-X)2 = ^=1.61 s = Vs2" = 1.27 Just as the population has a set of characteristics, each sample also has a set of characteristics. One sample characteristic is the sample average or mean: n X=-EXi = 0.5 Two means now have been introduced, and it is important to keep them separate. One is the population mean (p.), a population characteristic. The second is the sample mean (X), a sample characteristic. Because the X is a sample characteristic, it would change if a new sample were obtained. The sample mean (X) is used to estimate the unknown population mean (p Another sample characteristic or statistic is the sample variance (s2)...

Sample Size and Statistical theory - Population Characteristics Let us assume that we are interested in the attitudes of symphony season ticketholders toward changing the starting time of weekday performances from 8:00 P.M. to 7:30 P.M. The population is comprised of the 10,00C symphony season ticketholders. Their response to the proposal is shown in Figure 12-1. Of these ticket holders, 3000 responded "definitely yes" (which is coded as +2). Another 2000 would "prefer yes" (coded as +1), and so on The needed information is the average or mean response of the population (the 10,000 season ticketholders), which is termed |x = the population mean = 0.3 This population mean is one population characteristic of interest. It normally is unknown, and our goal is to determine its value as closely as possible by taking a sample from the population. Another population characteristic of interest is the population variance, a2, and its square root, the population standa...

Sample Size and Statistical Theory A practical question in much of marketing research involves the determination of sample size. A survey cannot be planned or implemented without knowing the sample size. Further, the sample-size decision is related directly to research cost and therefore must be justified. In the previous chapter, several practical approaches to obtaining sample size were presented. These approaches are extremely sensible, will lead to reasonable sample-size decisions, and in fact are used often in marketing research. There is, however, a formal approach to determining sample size using statistical theory. It is useful to understand this formal approach— the subject of this chapter—for several reasons. First, in some contexts it can be applied directly to make more precise sample-size decisions. Second, it can provide worthwhile guidance even when it may not be easy to apply the statistical theory. Finally, the discussion serves to introduce some important concept...

Sampling Fundamentals - Summary There are four main considerations in developing a probability sample. First, the target population must be defined. In doing so, the researcher should look to the research objectives for guidance and consider alternative definitions. Second, the mechanism for selecting the sample needs to be determined The simple random sampling, cluster sampling, stratified sampling, and multistage designs are among the available choices. It is important to consider the differences that may exist between the population list or sampling frame from which the sample is drawn and the target population. Potential biases should be Identified. For example, in telephone interviewing, the telephone directory will not Include those with unlisted numbers. The third consideration is sample size. Several ad hoc methods are available, such as ensuring that there are at least 100 sample members for each group within the population that is of interest. In the next chapter we will...

Shopping Center Sampling Shopping center studies in which shoppers are intercepted present some difficult sampling problems. As noted in Chapter 7, well over 20 percent of all questionnaires completed or interviews granted were store-intercept interview.13 One limitation with shopping center surveys is the bias introduced by the methods used to select the sample. In particular, biases that are potentially damaging to a study can be caused by the selection of the shopping center, the part of the shopping center from which the respondents are drawn, the time of day, and the fact that more frequent shoppers will be more likely to be selected. Sudman suggests approaches to minimize these problems and, in doing so, clarifies the nature of these biases.14 Shopping Center Selection A shopping center sample usually will reflect primarily those famiJle&wfhc —uY£jn_ihe ju^a. Obviously, there can be great differences between people 13"Shoppers Grant 91 Million Interviews Yearly,...

Sampling Fundamentals - Nonprobability Sampling In probability sampling, the theory of probability allows the researcher to calculate the nature and extent of any biases in the estimate and to determine what variation in the estimate is due to the sampling procedure. It requires a sampling frame—a list of sampling units or a procedure to reach respondents with a known probability. In nonprobability sampling, the costs and trouble of developing a sampling frame are eliminated, but so is the precision with which the resulting information can be presented. In fact, the results can contain hidden biases and uncertainties that make them worse than no information at all. These problems, it should be noted, are not alleviated by increasing the sample size. For this reason, statisticians prefer to avoid nonprobability sampling designs; however, they often are used legitimately and effectively. It is worthwhile to distinguish among four types of nonprobability sampling procedures: judgment...

Sampling Fundamentals - Nonresponse Problems The object of sampling is to obtain a body of data that is representative of the population. Unfortunately, some sample members become nonrespondents because they (1) refuse to respond, (2) lack the ability to respond, (3) are not at home, or (4) are inaccessible. Nonresponse can be a serious problem. It means, of course, that the sample size has to be large enough to allow for nonresponse. If a sample size of 1000 is needed and only a 50-percent response rate is expected, then 2000 people will need to be identified as possible sample members. Second, and more serious, is the possibility that those who respond differ from nonrespondents in a meaningful way, thereby creating biases. The seriousness of nonresponse bias depends on the extent of the nonresponse. If the percentage involved is small, the bias is small. Unfortunately, however, as the discussion in Chapter 7 made clear, the percentage can be significant. For example, a review o...

Sampling Fundamentals - Determining the Sample Size: Ad Hoc Methods How large should the sample be? This question is simple and straightforward, but to answer it with precision is not so easy. Statistical theory does provide some tools and a structure with which to address the question which will be described in more detail in Chapter 12. In this chapter several ad hoc but practical approaches are discussed. Rules of Thumb One approach is to use some rulesof thumb. Sudman suggests that the sample should be large enough so that when it is divided into groups, each group will have a minimum sample size of 100 or more.6 Suppose the opinions of citizens regarding municipal parks was desired. In particular, an estimation was to be made of the percentage who fell that tennis courts were needed. Suppose, further, that a comparison was desired among those who (1) used parks frequently, (2) used parks occasionally, and (3) never used parks. Thus, the sample size should be such that each...

Sampling Fundamentals - Selecting the Probability Sample There are a variety of methods that can be used to select a probability sample. The simplest, conceptually, is termed "simple random sampling." It not only has practical value, but it is a good vehicle for gaining intuitive understanding of the logic and power of random sampling. Simple Random Sampling Simple random sampling is an approach in which each population member, and thus each po^ibie^an-UDke^Jias^ an equal probability of being se-lected. The implementation is straightforward. Put the name of each person in the population on a tag and place the tags in a large bowl. Mix the contents of the bowl thoroughly and then draw out the desired number for the sample. Such a method was, in fact, used to select the order in which men would be drafted for military service during the Vietnam War, using birth dates. Despite the fact that the bowl was well mixed, the early drawing revealed a much higher number of Decemb...

Sampling Fundamentals - Target Population Sampling is intended to gain information about a population. Thus, it is critical at the outset to identify the population properly and accurately. If the population is defined in a fuzzy way, the results also will be fuzzy. If the population is defined improperly, the research probably will answer the wrong question as a result. For example, if some research questions involve prospective car buyers and the population contains all adults with driver's licenses, the research output will be unlikely to provide the relevant information. Although the definition of the target population is important, it often is neglected because it seems obvious and noncontroversial. But considerable effort in identifying the target population usually will pay off. The following guidelines should be considered. Look to the Research Objectives If the research objectives are well thought out, the target population definition will be clear as well. Recall...

Sampling Fundamentals - Sampling Fundamentals Marketing research often involves the estimation of a characteristic of some population. For instance, the average usage level of a park by community residents might be of interest, or information on the attitudes of a student body toward a proposed intramural facility could be needed. In either case, it would be unlikely that all members of the population would be surveyed. Contacting the entire population, that is, the entire census list, simply would not be worthwhile from a cost—benefit viewpoint. It would be both costly and, in nearly all cases, unnecessary, since adequate reliability usually can be obtained from a sample. Further, it often would be less accurate since nonsampling errors like nonresponse, cheating, and data-coding errors are more difficult to control. There are many ways of obtaining a sample. Some are informal and even casual. Passers-by may be queried as to their opinions of a new product. If the response of eve...

Experimentation - Appendix Latin Square Design The Latin square design is a method to reduce the number of groups involved when interactions between the treatment levels and the control variables can be considered relatively unimportant. We will use a laboratory nutritional labeling experiment to describe and illustrate the Latin square design. The goal of the experiment was to contribute to the judgment of those proposing and evaluating several public policy nutritional labeling alternatives. In particular, the research goal was to determine the impact upon shopper perceptions and preferences of variations in nutritional information on labels of canned peas. Four levels of information were tested. The first provided only a simple quality statement. The second listed some major nutrient components and whether the product was high or low on them. The third provided the amounts of each nutrient. The fourth listed all nutritional components and was the most complete. There were two c...

Experimentation - Summary Experiments are conducted to identify and quantify causal relationships Laboratory experiments are often relatively inexpensive and provide the opportunity to exercise tight control. In a laboratory, for example, an exposure to a concept can be controlled whereas in the more realistic field context there are many factors that can distort an exposure, such as weather, competitive reactions, and family activities. However, the laboratory experiment suffers from the testing effect and from the artificiality of the situation. Thus, the external validity (the ability to generalize from the experiment) is limited. Field experiments have greater external validity but are more costly to run (in expense, time, and security), are difficult to implement, and lack the tight control possible in the laboratory. As a result their internal validity is often a problem. There are several experimental design alternatives to consider. The use of a control group can serve to h...

Experimentation - Limitations of Experiments Experimentation is a powerful tool in the search for unambiguous relationships that we hope may be used to make valid predictions about the effects of marketing decisions and to develop basic theories. The laboratory experiment is the preferred method because of its internal validity; however because of acute external validity problems in the laboratory setting, managers are reluctant to rely upon it. Unfortunately, the field experiment is beset by a number of problems whose net effect has been to limit the vast majority of marketing experiments to short-run comparisons across store; home placements of product variations, and so forth. Relatively few large-scale experiments with social programs, marketing programs, or advertising campaigns are conducted in any given year. What are the reasons? Cost Cost and time pressures are the first hurdle. Even "simple" in-store testa require additional efforts to gain cooperation; to pr...