CHAPTER 20. Sample Survey CONTENTS

Transcription

1 CHAPTER 20 Sample Survey CONTENTS STATISTICS IN PRACTICE: CINERGY 20.1 TERMINOLOGY USED IN SAMPLE SURVEYS 20.2 TYPES OF SURVEYS AND SAMPLING METHODS 20.3 SURVEY ERRORS Nonsampling Error Sampling Error 20.4 SIMPLE RANDOM SAMPLING Population Mean Population Total Population Proportion Using Excel for Simple Random Sampling Determining the Sample Size 20.5 STRATIFIED SIMPLE RANDOM SAMPLING Population Mean Using Excel: Population Mean Population Total Using Excel: Population Total Population Proportion Using Excel: Population Proportion Determining the Sample Size 20.6 CLUSTER SAMPLING Population Mean Population Total Population Proportion Using Excel for Cluster Sampling Determining the Sample Size 20.7 SYSTEMATIC SAMPLING

2 20-2 Chapter 20 Sample Survey STATISTICS in PRACTICE CINERGY* Cincinnati, Ohio *The authors are indebted to Jim Riddle of Cinergy for providing this Statistics in Practice. Statistical surveys enable Cinergy Company to determine the energy needs of its customers. PhotoDisc Red/Getty Images Cinergy, formerly Cincinnati Gas & Electric Company (CG&E), is a public utility that provides gas and electric power to customers in the Greater Cincinnati area. To improve service to its customers, Cinergy continually strives to stay up-to-date with its customers needs. Cinergy undertook a sample survey, the Building Characteristics Survey, to learn about the energy requirements of commercial buildings in its service area. Avariety of information concerning commercial buildings was sought, such as the floor space, number of employees, energy end-use, age of the building, type of building materials, and energy conservation measures. During preparations for the survey, Cinergy analysts determined that approximately 27,000 commercial buildings were in operation in the Cinergy service area. Based on available funds and the precision desired in the results, they recommended that a sample of 616 commercial buildings be surveyed. The sample design chosen was stratified simple random sampling. Total electrical usage over the past year for each commercial building in the service area was available from company records, and because many of the building characteristics of interest (size, number of employees, etc.) were related to usage, it was the criterion used to divide the population of buildings into six strata. The first stratum contained the commercial buildings for the 100 largest energy users; each building in this stratum was included in the sample. Although these buildings constituted only.2% of the population, they accounted for 14.4% of the total electrical usage. For the other strata, the number of buildings sampled was determined on the basis of obtaining the greatest precision possible per unit cost. A questionnaire was carefully developed and pretested before the actual survey was conducted. Data were collected through personal interviews. Completed surveys totaled 526 out of the sample of 616 commercial buildings. This response rate of 85.4% was considered to be excellent. Currently, Cinergy is using the survey results to improve the forecasts of energy demand and to improve service to its commercial customers. In this chapter you will learn about the issues that statisticians consider in the design and execution of a sample survey such as the one conducted by Cinergy. Sample surveys are often used to develop profiles of a company s customers; they are also used by the government and other agencies to learn about various segments of the population Terminology Used in Sample Surveys In Chapter 1 we gave the following definitions of an element, a population, and a sample. An element is the entity on which data are collected. A population is the collection of all the elements of interest. A sample is a subset of the population. To illustrate these concepts, consider the following situation. Dunning Microsystems, Inc. (DMI), a manufacturer of personal computers and peripherals, would like to collect data

3 20.2 Types of Surveys and Sampling Methods 20-3 If inferences from a sample are to be valid, the sampled population must be representative of the target population. about the characteristics of individuals who have purchased a DMI personal computer. To obtain such data, a sample survey of DMI personal computer owners could be conducted. The elements in this sample survey would be individuals who have purchased a DMI personal computer. The population would be the collection of all people who have purchased a DMI personal computer, and the sample would be the subset of DMI personal computer owners who are surveyed. In sample surveys it is necessary to distinguish between the target population and the sampled population. The target population is the population we want to make inferences about, while the sampled population is the population from which the sample is actually selected. It is important to understand that these two populations are not always the same. In the DMI example, the target population consists of all people who have purchased a DMI personal computer. The sampled population, however, might be all owners who have sent warranty registration cards back to DMI. Not every person who buys a DMI personal computer sends in the warranty card, so the sampled population would differ from the target population. Conclusions drawn from a sample survey apply only to the sampled population. Whether these conclusions can be extended to the target population depends on the judgment of the analyst. The key issue is whether the correspondence between the sampled population and the target population on the elements of interest is close enough to allow this extension. Before sampling, the population must be divided into sampling units. In some cases, the sampling units are simply the elements. In other cases, the sampling units are groups of the elements. For example, suppose we want to survey certified professional engineers who are involved in the design of heating and air conditioning systems for commercial buildings. If a list of all professional engineers involved in such work were available, the sampling units would be the professional engineers we want to survey. If such a list is not available, we must find an alternative approach. A business telephone directory might provide a list of all engineering firms involved in the design of heating and air conditioning systems. Given this list, we could select a sample of the engineering firms to survey; then, for each firm surveyed, we might interview all the professional engineers. In this case, the engineering firms would be the sampling units and the engineers interviewed would be the elements. A list of the sampling units for a particular study is called a frame. In the sample survey of professional engineers, the frame is defined as all engineering firms listed in the business telephone directory; the frame is not a list of all professional engineers because no such list is available. The choice of a particular frame and hence the definition of the sampling units are often determined by the availability and reliability of a list. In practice, the development of the frame can be one of the most difficult and important steps in conducting a sample survey Types of Surveys and Sampling Methods The three most common types of surveys are mail surveys, telephone surveys, and personal interview surveys; each of these types involves the design and administration of a questionnaire. Other types of surveys used to collect data do not involve questionnaires. For example, accounting firms are often hired to sample a company s inventory of goods to estimate the value of inventory on the company s balance sheet. In such surveys, someone simply counts the items and records the results. In surveys that use questionnaires, the design of the questionnaire is critical. The designer must resist the temptation to include questions that might be of interest, because every question adds to the length of the questionnaire. Long questionnaires lead not only to

4 20-4 Chapter 20 Sample Survey Survey costs are lower for mail and telephone surveys. But with well-trained interviewers, higher response rates and longer questionnaires are possible with personal interviews. Given the amount of data collected on each element, personal interviews were used to collect the data for the Cinergy study in the Statistics in Practice feature in this chapter. With a nonprobabilistic sampling method, if one can ensure that a representative sample is obtained, point estimates based on the sample can be useful. However, even then the precision of the results is unknown. respondent fatigue, but also to interviewer fatigue, especially in telephone surveys. However, if personal interviews are to be used, a longer and more complex questionnaire is feasible. A large body of knowledge exists concerning the phrasing, sequencing, and grouping of questions for a questionnaire. These issues are discussed in more comprehensive books on survey sampling; several sources for this type of information are listed in the bibliography. Sample surveys can also be classified in terms of the sampling method used. With probabilistic sampling, the probability of obtaining each possible sample can be computed; with a nonprobabilistic sampling, this probability is unknown. Nonprobabilistic sampling methods should not be used if the researcher wants to make statements about the precision of the estimates. In contrast, probabilistic sampling methods can be used to develop confidence intervals that provide bounds on the sampling error. In the following sections, four of the most popular probabilistic sampling methods are discussed: simple random sampling, stratified simple random sampling, cluster sampling, and systematic sampling. Although statisticians prefer to use a probabilistic sampling method, nonprobabilistic sampling methods often are necessary. The advantages of nonprobabilistic sampling methods are their low expense and ease of implementation. The disadvantage is that statistically valid statements cannot be made about the precision of the estimates. Two of the more common nonprobabilistic methods are convenience sampling and judgment sampling. With convenience sampling, the units included in the sample are chosen because of accessibility. For example, a professor conducting a research study at a university may ask student volunteers to participate in the study simply because they are in the professor s class; in this case, the sample of students is referred to as a convenience sample. In some situations, convenience sampling is the only practical approach. For example, to sample a shipment of oranges, an inspector might select oranges haphazardly from several crates since labeling each orange in the entire shipment to create a frame and employing a probabilistic sampling method would be impractical. Wildlife captures and volunteer panels for consumer research are other examples of convenience samples. Although convenience sampling is a relatively easy approach to sample selection and data gathering, it is impossible to evaluate the goodness of the sample statistics obtained in terms of their ability to estimate the population parameters of interest. A convenience sample may provide good results or it may not; no statistically justified procedures exist for making statistical inferences from the sample results. Nevertheless, at times some researchers apply a statistical method designed for a probability sample to the data gathered from a convenience sample. In doing so, the researcher may argue that the convenience sample can be treated as though it were a random sample in the sense that it is representative of the population. However, this argument should be questioned; one should be cautious in using a convenience sample to make statistical inferences about population parameters. In using the nonprobability sampling technique referred to as judgment sampling, a person knowledgeable on the subject of the study selects sampling units that he or she feels are most representative of the population. Although judgment sampling is often a relatively easy way to select samples, users of the survey results must recognize that the quality of the results is dependent on the judgment of the person selecting the sample. Consequently, caution must be exercised in using judgment samples to make statistical inferences about a population parameter. In general, no statistical statements should be made about the precision of the results from a judgment sample. Both probabilistic and nonprobabilistic sampling methods can be used to select a sample. The advantages of nonprobabilistic methods are that they are generally inexpen-

5 20.3 Survey Errors 20-5 sive and easy to use. However, if it is necessary to provide statements about the precision of the estimates, a probabilistic sampling method must be used. Almost all large sample surveys employ probabilistic sampling methods Survey Errors A census is a survey of the entire population. Nonsampling error can be minimized by proper interviewer training, good questionnaire design, pretesting, and careful management of the process of coding and transferring the data to the computer. In the 1990 U.S. Census, 25.9% of households did not respond. Census 2000 employed a sample survey of nonrespondents to estimate the characteristics of this portion of the population. Two types of errors can occur in conducting a survey. One type, sampling error, is defined as the magnitude of the difference between the point estimator developed from the sample and the population parameter. In other words, sampling error is the error that occurs because not every element in the population is surveyed. The second type, nonsampling error, refers to all other types of errors that can occur when a survey is conducted, such as measurement error, interviewer error, and processing error. Although sampling error can occur only in a sample survey, nonsampling errors can occur in both a census and a sample survey. Nonsampling Error One of the most common types of nonsampling error occurs whenever we incorrectly measure the characteristic of interest. Measurement error can occur in a census or a sample survey. For either type of survey, the researcher must exercise care to ensure that any measuring instruments (e.g., the questionnaire) are properly calibrated and that the people who take the measurements are properly trained. Attention to detail is the best precaution in most situations. Errors due to nonresponse are a concern to both the statistician responsible for designing the survey and the manager using the results. This type of nonsampling error occurs when data cannot be obtained for some of the elements surveyed or when only partial data are obtained. The problem is most serious when a bias is created. For example, if interviews were conducted to assess women s attitudes toward working outside the home, making calls to homes only during the daytime would create an obvious bias because women who work outside the home would be excluded from the sample. Nonsampling errors due to lack of respondent knowledge are common in technical surveys. For example, suppose building managers were surveyed to obtain detailed information about the types of ventilation systems used in office buildings. Managers of large office buildings may be especially knowledgeable about such systems because they may attend training seminars and have support staff to help keep them current. In contrast, managers of smaller office buildings may be less knowledgeable about such systems because of the wide variety of duties they must perform. This difference in knowledge can significantly affect the survey results. Two other types of nonsampling error are selection error and processing error. Selection errors occur when an inappropriate item is included in the survey. Suppose a sample survey was designed to develop a profile of men with beards; if some interviewers interpreted the statement men with beards to include men with mustaches while other interviewers did not, the resulting data would be flawed. Processing errors occur whenever data are incorrectly recorded or incorrectly transferred from recording forms, such as from questionnaires to computer files. Although some nonsampling errors will occur in most surveys, they can be minimized by careful planning. Care should be taken to ensure that the sampled population corresponds closely to the target population, good questionnaire design principles are followed, interviewers are well trained, and so on. The final report on a survey should include some discussion of the likely impact of nonsampling errors on the results.

6 20-6 Chapter 20 Sample Survey Sampling error is minimized by proper choice of a sample design. Sampling Error Recall the Dunning Microsystems (DMI) example introduced in Section Suppose DMI wants to estimate the mean age of people who have purchased a DMI personal computer. If the entire population of DMI personal computer owners could be surveyed (a census) and nonsampling errors were not present, we could determine the mean age exactly. But what if less than 100% of the population of DMI owners can be surveyed? In this case, there will most likely be a difference between the sample mean and the population mean; the absolute value of this difference is the sampling error. In practice, it is not possible to know what the sampling error will be for any one particular sample because the population mean is unknown; however, it is possible to provide probability statements about the size of the sampling error. As stated, sampling error occurs because a sample, and not the entire population, is surveyed. Even though sampling error cannot be avoided, it can be controlled. Selecting an appropriate sampling method or design is one important way to control this type of error. In the following sections we will discuss four probabilistic sampling methods: simple random sampling, stratified simple random sampling, cluster sampling, and systematic sampling Simple Random Sampling Recall the definition of simple random sampling from Chapter 7: A simple random sample of size n from a finite population of size N is a sample selected such that every possible sample of size n has the same probability of being selected. In Chapter 7, we showed how Excel could be used to select a simple random sample. To conduct a sample survey using simple random sampling, we begin by developing a frame or list of all elements in the sampled population. Then a selection procedure, based on the use of random numbers, is used to ensure that each element in the sampled population has the same probability of being selected. In this section we show how estimates of a population mean, total, and proportion are made for sample surveys that use simple random sampling. Population Mean In Chapter 8 we showed that the sample mean x is an estimate of the population mean µ, and the sample standard deviation s is an estimate the population standard deviation σ. For a sample of size n, the t distribution can be used to provide the following interval estimate of µ. s x t α/2 n (20.1) In expression (20.1), s n is the estimate of σ x, the standard error of the mean. When a simple random sample of size n is selected from a finite population of size N, an estimate of the standard error of the mean is s x N n N n s (20.2)

7 20.4 Simple Random Sampling 20-7 Using as an estimate of, the interval estimate of the population mean becomes s x σ x x t α/2 s x (20.3) In a sample survey it is common practice to use a value of t 2 when developing interval estimates. Thus, when simple random sampling is used, an approximate 95% confidence interval estimate of the population mean is given by the following expression. APPROXIMATE 95% CONFIDENCE INTERVAL ESTIMATE OF THE POPULATION MEAN x 2 s x (20.4) As an example, consider the situation of the publisher of Great Lakes Recreation, a regional magazine specializing in articles on boating and fishing. The magazine currently has N 8000 subscribers. A simple random sample of n 484 subscribers shows mean annual income to be $30,500 with a standard deviation of $7040. An unbiased estimate of the mean annual income of all subscribers is given by x $30,500. Using the sample results and equation (20.2), we obtain the following estimate of the standard error of the mean s x Therefore, using equation (20.4), we find that an approximate 95% confidence interval estimate of the mean annual income for the magazine subscribers is 30,500 2(310) 30, or $29,880 to $31,120. The preceding procedure can be used to compute an interval estimate for other population parameters such as the population total or the population proportion. In these cases, the approximate 95% confidence interval can be written as Point Estimator 2(Estimate of the Standard Error of the Point Estimator) In sample survey terminology, the value is often referred to as the bound on the sampling error. It is the same as the margin of error discussed in Chapter 8. For example, in the Great Lakes Recreation sample survey, an estimate of the standard error of the point estimator is s x $310, and the bound on the sampling error, denoted B, is B 2($310) $620. In general, when estimating a population mean the bound on the sampling error is 2. s x Population Total Consider the problem facing Northeast Electric and Gas (NEG). As part of an energy usage study, NEG needs to estimate the total square footage for the 500 public schools in its service area. We will denote the total square footage for the 500 schools as X; in other words,

8 20-8 Chapter 20 Sample Survey X denotes the population total. Note that if µ, the mean square footage for the 500 public schools, were known, the value of X could be computed by multiplying N times µ. However, because µ is unknown, a point estimate of X is obtained by multiplying N times x. We will denote the point estimator of X as Xˆ. POINT ESTIMATOR OF A POPULATION TOTAL Xˆ Nx (20.5) The point estimator of the standard error of Xˆ is given by s Xˆ Ns x (20.6) where s x N n N n s (20.7) Note that equation (20.7) is the formula for the estimated standard error of the mean. With this standard error and equation (20.6), an approximate 95% confidence interval for the population total is given by the following expression. When estimating a population total, the bound on the sampling error is 2 s Xˆ. APPROXIMATE 95% CONFIDENCE INTERVAL ESTIMATE OF THE POPULATION TOTAL Nx 2 s Xˆ (20.8) CD file NEG Suppose that in the NEG example a simple random sample of n 50 public schools is selected from the population of N 500 schools. The data for the NEG study are shown in the data set named NEG; the sample mean is x 22,000 square feet and the sample standard deviation is s 4000 square feet. Using equation (20.5), we find that the point estimator of the population total is Equation (20.7) can be used to compute an estimate of the standard error of the mean s x Then, using equation (20.6), we can obtain an estimate of the standard error of Xˆ. Therefore, using equation (20.8), we find that an approximate 95% confidence interval estimate of the total square footage for the 500 public schools in NEG s service area is 11,000,000 2(268,330) 11,000, ,660 or 10,463,340 to 11,536,660 square feet. Xˆ (500)(22,000) 11,000,000 s Xˆ (500)(536.66) 268,330

9 20.4 Simple Random Sampling 20-9 Population Proportion The population proportion p is the fraction of the elements in the population with some characteristic of interest. In a market research study, for example, one might be interested in the proportion of consumers preferring a certain brand of product. The sample proportion p is an unbiased point estimator of the population proportion. An estimate of the standard error of the sample proportion p is given by N n p (1 p ) s p N n 1 (20.9) An approximate 95% confidence interval estimate of the population proportion is given by the following expression. When estimating a population proportion, the bound on the sampling error is 2 s p. APPROXIMATE 95% CONFIDENCE INTERVAL ESTIMATE OF THE POPULATION PROPORTION p 2 s p (20.10) As an illustration, suppose that in the Northeast Electric and Gas example, NEG would also like to estimate the proportion of the 500 public schools in its service area that use natural gas as fuel for heating. If 35 of the 50 sampled schools indicate the use of natural gas, the point estimate of the proportion of the 500 schools in the population that use natural gas is p 35/ Using equation (20.9), we can compute an estimate of the standard error of p (1.7) s p Therefore, using equation (20.10), we find that an approximate 95% confidence interval for the population proportion is.7 2(.0621) or.5758 to As this example has shown, the width of the confidence interval can be rather large when one is estimating a population proportion. In general, large sample sizes are needed to obtain precise estimates of population proportions. For large populations, samples of n 1200 or more are often used. We assume the Excel procedure described in Chapter 7 has been used to select the sample. Using Excel for Simple Random Sampling To show how Excel can be used to summarize the results for sample surveys that use simple random sampling, we will construct an approximate 95% confidence interval of the population mean, population total, and population proportion for the NEG study. We begin with the population total case. Refer to Figure 20.1 as we describe the tasks involved. The formula worksheet is in the background; the value worksheet appears in the foreground.

10 20-10 Chapter 20 Sample Survey FIGURE 20.1 EXCEL WORKSHEET FOR THE POPULATION TOTAL IN THE NEG EXAMPLE A B C D E F 1 School Square Feet Interval Estimate of a Population Total Population Size Sample Size =COUNT(B2:B51) Mean =AVERAGE(B2:B51) Standard Deviation =STDEV(B2:B51) Standard Error of Mean =(SQRT((E3-E4)/E3))*(E6/SQRT(E4)) Standard Error of Total =E3*E Bound on Sampling Error =2*E Point Estimate =E3*E Lower Limit =E12-E Upper Limit =E12+E Note: Rows are hidden. A B C D E F 1 School Square Feet Interval Estimate of a Population Total Population Size Sample Size Mean Standard Deviation Standard Error of Mean Standard Error of Total Bound on Sampling Error Point Estimate Lower Limit Upper Limit Enter Data: Column A identifies the school in the sample, and column B shows the number of square feet in the school. Enter Functions and Formulas: The population size is entered into cell E3 and the descriptive statistics needed are provided in cells E4:E6. Excel s COUNT, AVERAGE, and STDEV functions are used to compute the sample size, sample mean, and sample standard deviation, respectively. Cells E8:E10 are used to compute the standard error of the mean, standard error of the total, and the bound on the sampling error. The standard error of the mean was computed using equation (20.2) by entering the formula (SQRT((E3-E4)/E3))*(E6/SQRT(E4)) into cell E8. The standard error of the total was computed using equation (20.6) by entering the formula E3*E8 into cell E9, and the bound on the sampling error was computed in cell E10 by entering the formula 2*E9. The value worksheet shows that the standard error of the mean is , the standard error of the total is , and the bound on the sampling error is Cells E12:E14 provide the point estimate and the lower and upper limits for the confidence interval. Because the point estimate of the population total is just the population size times the sample mean, we entered the formula E3*E5 into cell E12. To compute the lower limit of the 95% confidence interval, the formula E12-E10 was entered into cell E13; to compute the upper limit, the formula E12 E10 was entered into cell E14. The value worksheet shows that the point estimate of the population total is 11,000,000 and that the 95% approximate interval is 10,463,342 to 11,536,658. The differences in the Excel values and the hand calculations we computed previously are due to the fact that in doing the hand calculations we rounded the intermediate calculations to simplify the presentation.

11 20.4 Simple Random Sampling It is easy to compute an approximate 95% confidence interval for the population mean in the NEG study by modifying the worksheet developed to compute the approximate 95% confidence interval of the population total. For this case, the point estimate shown in cell E12 is just the sample mean in cell E5; thus, we would enter the formula E5 into cell E12. The bound on the sampling error would be 2 times the standard error of the mean in cell E8; thus, we would enter the formula 2*E8 into cell E10. The resulting lower and upper limits in cells E13:E14 would be the upper and lower limits of the approximate 95% confidence interval of the population mean. To show how Excel can be used to compute an approximate 95% confidence interval for the population proportion, we will compute an approximate 95% confidence interval of the proportion of schools in the NEG example that use natural gas as fuel for heating. Refer to Figure 20.2 as we describe the tasks involved. The formula worksheet is in the background; the value worksheet appears in the foreground. Enter Data: Column A identifies the school, and column B indicates whether the school uses natural gas as fuel for heating. Enter Functions and Formulas: The descriptive statistics needed are provided in cells E4, E6, and E7. The population size and response of interest are typed into cells E3 and E5. Excel s COUNTA and COUNTIF functions are used to compute the sample size and the number of schools in the sample that use natural gas. The value worksheet shows that 35 schools use natural gas and that the sample proportion is.7. Cells E9:E10 are used to compute the standard error of the proportion and the bound on the sampling error. The standard error of the proportion was computed using equation (20.9) by entering the formula SQRT(((E3-E4)/E3)*(E7*(1-E7))/(E4-1)) into cell E9. The bound FIGURE 20.2 EXCEL WORKSHEET FOR THE NEG STUDY: POPULATION PROPORTION A B C D E F 1 School Natural Gas? Interval Estimate of a Population Proportion 2 1 No 3 2 Yes Population Size Yes Sample Size =COUNTA(B2:B51) 5 4 Yes Response of Interest Yes 6 5 No Count for Response =COUNTIF(B2:B51,E5) 7 6 Yes Sample Proportion =E6/E4 8 7 Yes 9 8 Yes Standard Error of Proportion =SQRT(((E3-E4)/E3)*(E7*(1-E7))/(E4-1)) 10 9 Yes Bound on Sampling Error =2*E Yes Yes Point Estimate =E Yes Lower Limit =E12-E No Upper Limit =E12+E No Yes Yes 52 Note: Rows are hidden. A B C D E F 1 School Natural Gas? Interval Estimate of a Population Proportion 2 1 No 3 2 Yes Population Size Yes Sample Size Yes Response of Interest Yes 6 5 No Count for Response Yes Sample Proportion Yes 9 8 Yes Standard Error of Proportion Yes Bound on Sampling Error Yes Yes Point Estimate Yes Lower Limit No Upper Limit No Yes Yes 52

12 20-12 Chapter 20 Sample Survey on the sampling error was computed in cell E10 by entering the formula 2*E9. The value worksheet shows that the value of the standard error of the proportion is.0621 and that the bound on the sampling error is Cells E12:E14 provide the point estimate and the lower and upper limits for the confidence interval. Because the point estimate of the population proportion is just the sample proportion, we entered the formula E7 into cell E12. To compute the lower limit of the 95% confidence interval, the formula E12-E10 was entered into cell E13; to compute the upper limit, the formula E12 E10 was entered into cell E14. The value worksheet shows that the point estimate of the population proportion is.7 and that the 95% approximate interval is.5758 to Determining the Sample Size An important consideration in sample design is the choice of sample size. The best choice usually involves a trade-off between cost and precision. Larger samples provide greater precision (tighter bounds on the sampling error), but are more costly. Often the budget for a study will dictate how large the sample can be. In other cases, the size of the sample must be large enough to provide a specified level of precision. A common approach to choosing the sample size is to first specify the precision desired and then determine the smallest sample size providing that precision. In this context, the term precision refers to the size of the approximate confidence interval; smaller confidence intervals provide more precision. Because the size of the approximate confidence interval depends on B, the bound on the sampling error, choosing a level of precision amounts to choosing a value for B. Let us see how this approach works in choosing the sample size necessary to estimate the population mean. Equation (20.2) showed that the estimate of the standard error of the mean is s x N n N n s Recall that the bound on the sampling error is 2 times the estimate of the standard error of the point estimator. Thus, B 2 N n N n s (20.11) Solving equation (20.11) for n will provide a bound on the sampling error equal to B. Doing so yields n (20.12) Once a desired level of precision has been selected (by choosing a value for B), equation (20.12) can be used to find the value of n that will provide the desired level of precision. Using equation (20.12) to choose n for a practical study presents problems, however. In addition to specifying the desired bound on the sampling error B, one must know the sample variance s 2. But, s 2 will not be known until the sample is actually taken. Cochran* suggests several ways to develop an estimate of s 2 in practice. Three of them are stated as follows: 1. Take the sample in two stages. Use the value of s 2 found in stage 1 in equation (20.12); the resulting value of n is what the size of the total sample must be. Then, select the number of additional units needed at stage 2 to provide the total sample size determined in stage 1. Ns 2 N B 2 4 s2 *William G. Cochran, Sampling Techniques, 3rd ed. (New York: John Wiley & Sons, 1977).

13 20.4 Simple Random Sampling Use the results of a pilot survey or pretest to estimate s Use information from a previous sample. Let us now consider an example involving the estimate of the population mean for starting salaries of graduates of a particular university. Suppose N 5000 graduates, and we want to develop an approximate 95% confidence interval with a width of at most $1000. To provide such a confidence interval, B 500. Before using equation (20.12) to determine the sample size, we need an estimate of s 2. Suppose a study of starting salaries conducted last year found that s $3000. We can use the data from this previous sample to estimate s 2. Using B 500, s 3000, and N 5000, we can now use equation (20.12) to determine the sample size. 5000(3000) 2 n (500) (3000) Rounding up, we see that a sample size of 140 will provide an approximate 95% confidence interval of width $1000. Keep in mind, however, that this calculation is based on the initial estimate of s $3000. If s turns out to be larger in this year s sample survey, the resulting approximate confidence interval will have a width greater than $1000. Consequently, if cost considerations permit, the survey designer might choose a sample size of, say, 150 to provide added assurance that the final approximate 95% confidence interval will have a width less than $1000. The formula for determining the sample size necessary for estimating a population total with a bound B on the sampling error is as follows. Ns 2 n B 2 s2 4N (20.13) In the previous example, we wanted to estimate the mean starting salary with a bound on the sampling error of B 500. Suppose we are also interested in estimating the total salary of all 5000 graduates with a bound of $2 million. We can use equation (20.13) with B 2,000,000 to find the sample size needed to provide such a bound on the population total. 5000(3000) 2 n (2,000,000) 2 (3000) 2 4(5000) Rounding up, we see that a sample size of 216 is necessary to provide an approximate 95% confidence interval with a bound of $2 million. We note here that if the same survey is expected to provide a bound of $500 on the population mean and a bound of $2 million on the population total, a sample size of at least 216 must be used. This size will provide a tighter bound than necessary on the population mean, while providing the minimum desired precision for the population total. To choose the sample size for estimating a population proportion, we use a formula similar to the one for the population mean. We simply substitute p (1 p ) for s 2 in equation (20.12) to obtain Np (1 p ) n N B2 4 p (1 p ) (20.14)

14 20-14 Chapter 20 Sample Survey To use equation (20.14), we must specify the desired bound B and provide an estimate of p. If a good estimate of p is not available, we can use p.5; it will ensure that the resulting approximate confidence interval will have a bound on the sampling error at least as small as desired. Exercises SELF test Methods 1. Simple random sampling has been used to obtain a sample of n 50 elements from a population of N 800. The sample mean was x 215, and the sample standard deviation was found to be s 20. a. Estimate the population mean. b. Estimate the standard error of the mean. c. Develop an approximate 95% confidence interval for the population mean. 2. Simple random sampling has been used to obtain a sample of n 80 elements from a population of N 400. The sample mean was x 75, and the sample standard deviation was found to be s 8. a. Estimate the population total. b. Estimate the standard error of the population total. c. Develop an approximate 95% confidence interval for the population total. 3. Simple random sampling has been used to obtain a sample of n 100 elements from a population of N The sample proportion was p.30. a. Estimate the population proportion. b. Estimate the standard error of the proportion. c. Develop an approximate 95% confidence interval for the population proportion. 4. A sample is to be taken to develop an approximate 95% confidence interval estimate of the population mean. The population consists of 450 elements, and a pilot study has resulted in s 70. How large must the sample be if we want to develop an approximate 95% confidence interval with a width of 30? SELF test Applications 5. In 1996, the Small Business Administration (SBA) granted 771 government-guaranteed loans to small businesses in North Carolina (The Wall Street Journal Almanac, 1998). Suppose a sample of 50 small businesses that received an SBA loan showed an average loan of $149,670 with a standard deviation of $73,420 and that 18 of the businesses in the sample were manufacturing companies. a. Develop an approximate 95% confidence interval for the population mean value of a loan. b. Develop an approximate 95% confidence interval for the total value of all 771 SBA loans in North Carolina. c. In the sample, 18 of the 50 businesses were involved in manufacturing. Develop an approximate 95% confidence interval for the proportion of loans involving manufacturing companies. 6. A county in California had 724 corporate tax returns filed. The mean annual income reported was $161,220 with a standard deviation of $31,300. How large a sample will be necessary next year to develop an approximate 95% confidence interval for mean annual corporate earnings? The precision required is an interval width of no more than $5000.

15 20.5 Stratified Simple Random Sampling Stratified Simple Random Sampling In stratified simple random sampling, the population is first divided into H groups, called strata. Then for stratum h a simple random sample of size n h is selected. The data from the H simple random samples are combined to develop an estimate of a population parameter such as the population mean, total, or proportion. If the variability within each stratum is smaller than the variability across the strata, a stratified simple random sample can lead to greater precision (narrower interval estimates of the population parameters). The basis for forming the various strata depends on the judgment of the designer of the sample. Depending on the application, a population might be stratified by department, location, age, product type, industry type, sales levels, and so on. As an example, suppose the College of Business at Lakeside College wants to conduct a survey of this year s graduating class to learn about their starting salaries. The five majors in the college are accounting, finance, information systems, marketing, and operations management. Of the N 1500 students who graduated this year, there were N accounting majors, N finance majors, N information systems majors, N marketing majors, and N operations management majors. Analysis of previous salary data suggests more variability in starting salaries across majors than within each major. As a result, a stratified simple random sample of n 180 students was selected; 45 of the 180 students majored in accounting (n 1 45), 40 majored in finance (n 2 40), 30 majored in information systems (n 3 30), 35 majored in marketing (n 4 35), and 30 majored in operations management (n 5 30). Population Mean In stratified sampling an unbiased estimate of the population mean is obtained by computing a weighted average of the sample means for each stratum. The weights used are the fraction of the population in each stratum. The resulting point estimator, denoted x st, is defined as follows. POINT ESTIMATOR OF THE POPULATION MEAN x st H N h N x h (20.15) H number of strata x h sample mean for stratum h N h number of elements in the population in stratum h N total number of elements in the population; N N 1 N 2... N H For stratified simple random sampling, the formula for computing an estimate of the standard error of the mean is a function of s h, the sample standard deviation for stratum h. The resulting point estimator, denoted, is defined as follows. s x st s x st 1 N 2 H s 2 h N h (N h n h ) n h (20.16)

16 20-16 Chapter 20 Sample Survey Using these results, we see that an approximate 95% confidence interval estimate of the population mean is given by the following expression. APPROXIMATE 95% CONFIDENCE INTERVAL ESTIMATE OF THE POPULATION MEAN x st 2 s x st (20.17) CD file Lakeland The survey of 180 graduates of the College of Business at Lakeland College provided the sample results shown in the data set named Lakeland; a summary of the starting salaries is shown in Table The sample means for each major, or stratum, are $35,000 for accounting, $33,500 for finance, $41,500 for information systems, $32,000 for marketing, and $36,000 for operations management. Using these results and equation (20.15), we can compute a point estimate of the population mean. In Table 20.2 we show a portion of the calculations needed to estimate the standard error of ; note that x st Thus, x st 1500 (35,000) (33,500) (41,500) s x st (32,000) 1500 (36,000) 35,017 5 s 2 h N h (N h n h ) 1 (1500) 2 (42,909,037,698) 19, Hence, using expression (20.17), we find that an approximate 95% confidence interval estimate of the population mean is 35,017 2(138) 35, , or $34,741 to $35,293. Using Excel: Population Mean n h 42,909,037,698 To show how Excel can be used for sample surveys that use stratified simple random sampling we will construct an approximate 95% confidence interval of the population mean for the Lakeland College sample survey. In a separate worksheet for each stratum (major), we have TABLE 20.1 LAKELAND COLLEGE SAMPLE SURVEY OF STARTING SALARIES OF GRADUATES Major (h) x h s h N h n h Accounting $35, Finance $33, Information systems $41, Marketing $32, Operations management $36,

17 20.5 Stratified Simple Random Sampling TABLE 20.2 PARTIAL CALCULATIONS FOR THE ESTIMATE OF THE STANDARD ERROR OF THE MEAN FOR THE LAKELAND COLLEGE SAMPLE SURVEY Major h Accounting 1 Finance 2 Information systems 3 Marketing 4 Operations management 5 N h (N h n h ) 500(500 45) (2000) (350 40) (1700) (200 30) (2300) (300 35) (1600) (150 30) (2250)2 30 s 2 h n h 20,222,222,222 7,839,125,000 5,995,333,333 5,814,857,143 3,037,500,000 42,909,037,698 5 N h (N h n h ) s2 h n h computed the sample mean and sample standard deviation for that stratum. Another worksheet is used to summarize the calculations needed to compute an approximate 95% confidence interval of the population mean. A copy of this summary worksheet is shown in Figure The formula worksheet is in the background; the value worksheet appears in the foreground. Cells F3:F8 computed x st using equation (20.15). Enter Data: The worksheet shown in Figure 20.3 summarizes the starting salary data that appears in the worksheets for the separate strata of the Lakeland data set. Note that the standard deviations shown in cells C3:C7 were rounded to the nearest dollar. Enter Functions and Formulas: The descriptive statistics needed are provided in cells B3:E7. The strata sample means and standard deviations were calculated in the separate worksheets for each major and entered as constants in the summary worksheet. Cells F3:G7 represent partial calculations required to compute the point estimate of the population mean and the estimate of the standard error of the mean. To compute the weighted sample mean for the accounting majors we entered the formula (D3/$D$8)*B3 into cell F3; we then copied this formula into cells F4:F7 to compute the weighted sample means for each of the other majors. The formula SUM(F3:F7) was entered into cell F8 to compute the point estimate of the population mean. The computations in cells G3:G8 correspond to the calculations shown in Table For instance the formula D3*(D3-E3)*(C3^2/E3) was entered into cell G3 to compute the value of for the accounting majors; copying this formula into cells G4:G7 provided similar values for each of the other majors. In cell G8 we entered the cell formula SUM(G3:G7) to compute 5 s 2 h N h (N h n h ) n h s 2 h N h (N h n h ) n h

18 20-18 Chapter 20 Sample Survey FIGURE 20.3 SUMMARY WORKSHEET FOR THE LAKELAND COLLEGE STRATIFIED SAMPLE: POPULATION MEAN A B C D E F G H 1 Standard Population Strata Partial Calculations for 2 Major Mean Deviation Size Size Mean Standard Error 3 Accounting =(D3/$D$8)*B3 =D3*(D3-E3)*(C3^2/E3) 4 Finance =(D4/$D$8)*B4 =D4*(D4-E4)*(C4^2/E4) 5 Inf. Systems =(D5/$D$8)*B5 =D5*(D5-E5)*(C5^2/E5) 6 Marketing =(D6/$D$8)*B6 =D6*(D6-E6)*(C6^2/E6) 7 Oper. Mgt =(D7/$D$8)*B7 =D7*(D7-E7)*(C7^2/E7) 8 =SUM(D3:D7) =SUM(E3:E7) =SUM(F3:F7) =SUM(G3:G7) 9 10 Standard Error of Mean =SQRT(1/D8^2*G8) 11 Bound on Sampling Error =2*B Point Estimate =F8 14 Lower Limit =B13-B11 15 Upper Limit =B13+B11 16 A B C D E F G H 1 Standard Population Strata Partial Calculations for 2 Major Mean Deviation Size Size Mean Standard Error 3 Accounting ,222,222,222 4 Finance ,839,125,000 5 Inf. Systems ,995,333,333 6 Marketing ,814,857,143 7 Oper. Mgt ,037,500, ,909,037, Standard Error of Mean Bound on Sampling Error Point Estimate Lower Limit Upper Limit x st H N h N x h 5 s 2 h N h (N h n h ) n h Note that the value worksheet shows that this partial calculation is 42,909,037,698. To compute the standard error of the mean we entered the formula SQRT(1/D8^2*G8) into cell B10; we then computed the bound on the sampling error by entering the formula 2*B10 into cell B11. The value worksheet shows for the standard error of the mean and for the bound on the sampling error. Cells B13:B15 provide the point estimate and the lower and upper limits for the confidence interval. Because the point estimate of the population mean was computed in cell F8, we entered the formula F8 into cell B13. To compute the lower limit of the 95% confidence interval, the formula B13-B11 was entered into cell B14; to compute the upper limit, the formula B13 B11 was entered into cell B15. The value worksheet shows that the point estimate of the population mean is 35, and that the 95% approximate confidence interval is 34, to 35, Population Total The point estimator of the population total (X) is obtained by multiplying N times x st. POINT ESTIMATOR OF THE POPULATION TOTAL Xˆ Nx st (20.18) The point estimator of the standard error of Xˆ is s Xˆ Ns x st (20.19)

19 20.5 Stratified Simple Random Sampling Thus, an approximate 95% confidence interval for the population total is given by the following expression. APPROXIMATE 95% CONFIDENCE INTERVAL ESTIMATE OF THE POPULATION TOTAL Nx st 2 s Xˆ (20.20) Now suppose the College of Business at Lakeland College would also like to estimate the total earnings of the 1500 business graduates in order to estimate their impact on the economy. Using equation (20.18), we obtain an unbiased estimate of the total earnings. Using equation (20.19), we obtain an estimate of the standard error of the population total. Thus, using equation (20.20), we find that an approximate 95% confidence interval estimate of the total earnings of the 1500 graduates is 52,525,500 2(207,000) 52,525, ,000 or $52,111,500 to $52,939,500. Using Excel: Population Total The worksheet that we used to compute an approximate 95% confidence interval of the population mean for the Lakeland College sample survey can be easily modified in order to compute an approximate 95% confidence interval estimate of the population total. Refer to the worksheet in Figure 20.3 as we describe the changes. The point estimator of the population total is just N, the total number of elements in the population, times the point estimate of the population mean, x st. Thus, we would enter the formula D8*F8 into cell B13 to compute the point estimate of the population total. The point estimator of the standard error of the population total is N times s x, the standard error of the mean. Thus, we would enter the st formula D8*SQRT(1/D8^2*G8) into cell B10 to compute the estimate of the standard error of the population total. The resulting upper and lower limits in cells B14:B15 will then provide the approximate 95% confidence interval of the population total. Population Proportion Xˆ (1500)35,017 52,525,500 s Xˆ 1500(138) 207,000 An unbiased estimate of the population proportion, p, for stratified simple random sampling is a weighted average of the sample proportions for each stratum. The weights used are the fraction of the population in each stratum. The resulting point estimator, denoted p st, is defined as follows. POINT ESTIMATOR OF THE POPULATION PROPORTION p st H N h N p h (20.21)