2.1 Introduction

In Chapter 1, we briefly introduced the concept of statistical inference-the process of inferring information about a population from a sample. Because information about populations can usually be described by parameters, the statistical technique used generally deals with drawing inferences about population parameters from sample statistics. (Recall that a parameter is a measurement about a population, and a statistic is a measurement about a sample.)

Working within the covers of a statistics textbook, we can assume that population parameters are known. In real life, however, calculating parameters become prohibitive because populations tend to be quite large. As a result, most population parameters are unknown. For example, in order to determine the mean annual income of blue-collar workers, we would have to ask each blue-collar worker what his or her income is and then calculate the mean of all the responses. Because this population consists of several million people, the task is both expensive and impractical. If we are willing to accept less than 100% accuracy, we can use statistical inference to obtain an estimate.

Rather than investigating the entire population, we select a sample of workers, determine the annual income of the workers in this group, and calculate the sample mean. While there is very little chance that the sample mean and the population mean are identical, we would expect them to be quite close. However, for the purposes of statistical inference, we need to be able to measure how close the sample mean is likely to be to the population mean. In this chapter, however, we will discuss the basic concepts and techniques of sampling itself. But first we will take a look at various sources for collecting data.

2.2 Sources of Data

The validity of the results of a statistical analysis clearly depends on the reliability and accuracy of the data used. Whether you are actually involved in collecting the data; performing a statistical analysis on the data, or simply reviewing the results of such an analysis, it is important to realize that the reliability and accuracy of the data depend on the method of collection. Three of the most popular sources of statistical data are published data, data collected from observational studies, and data collected from experimental studies.

2.2.1 Published Data

The use of published data is often preferred due to its convenience, relatively low cost, and reliability (assuming that it has been collected by a reputable organization). There is an enormous amount of published data produced by government agencies and private organizations, available in printed form, on data tapes and disks, and increasingly on the Internet. Data published by the same organization that collected them are called primary data. An Example of primary data would be the data published by Egypt Bureau of the Census, which collects data on numerous industries as well as conducts the census of the population every ten years. Statistics agency is the central statistical agency, collecting data on almost every aspect of social and economic life in the country. These primary sources of information are invaluable to decision makers in both the government and private sectors.

Secondary data refers to data that are published by an organization different from the one that originally collected and published the data. A popular source of secondary data is The Statistical Book of Egypt, which compiles data from several primary government sources and is updated annually. Another Example of a secondary data source is Central Bank which has a variety of financial data tapes that contain data compiled from such primary sources as the Stock Exchange. Care should be taken when using secondary data, as errors may have been introduced as a result of the transcription or due to misinterpretation of the original terminology and definitions employed.

An interesting Example of the importance of knowing how data collection agencies define their terms appeared in an article in The Globe and Mail (February 12, 1996). The United States and Canada had similar unemployment rates up until the 1980s, at which time Canada's rate started to edge higher than the U.S. rate. By February 1996, the gap had grown to almost four percentage points (9.6% in Canada compared with 5.8% in the United States). Economists from the United States and Canada met for two days to compare research results and discuss possible reasons for this puzzling gap in jobless rates. The conference organizer explained that solving this mystery matters because "we have to understand the nature of unemployment to design policies to combat it." An Ohio State University economist was the first to notice a difference in how officials from the two countries define unemployment. "If jobless people say they are searching for work, but do nothing more than read job advertisements in the newspaper, Canada counts them as unemployed. U.S. officials dismiss such 'passive' job hunters and count them as being out of the labor force altogether, so they are not counted among the jobless." Statistics Canada reported that this difference in definitions accounted for almost one-fifth of the difference between the Canadian and U.S. unemployment rates.

2.2.2 Observational and Experimental Studies

If relevant data are not available from published sources, it may be necessary to generate the data by conducting a study. This will especially be the case when data are needed concerning a specific company or situation. The difference between two important types of studies – observational and experimental – is best illustrated by means of an example.

Example (1)

Six months ago, the director of human resources for a large mutual fund company announced that the company had arranged for its salespeople to use a nearby fitness center free of charge. The director believes that fitter salespeople have more energy and an improved appearance, resulting in higher productivity. Interest and participation in the fitness initiative were high initially, but after a few months had passed, several employees stopped participating. Those who continued to exercise were committed to maintaining a good level of fitness, using the fitness center about three times per week on average.

The director recently conducted an observational study and determined that the average sales level achieved by those who regularly used the fitness center exceeded that of those who did not use the center. The director was tempted to use the difference in productivity levels to justify the cost to the company of making the fitness center available to employees. But the vice-president of finance pointed out that the fitness initiative was not necessarily the cause of the difference in productivity levels. Because the salespeople who exercised were self-selected - they determined themselves whether or not to make use of the fitness center-it is quite likely that the salespeople who used the center were those who were more ambitious and disciplined. These people would probably have had higher levels of fitness and productivity even without the fitness initiative. We therefore cannot necessarily conclude that fitness center usage led to higher productivity. It may be that other factors, such as ambition and discipline, were responsible both for higher fitness center usage and higher productivity.

The director and vice-president then discussed the possibility of conducting an experimental study, designed to control which salespeople made regular use of the fitness center. The director would randomly select 60 salespeople to participate in the study. Thirty of these would be randomly selected and persuaded to use the fitness center on a regular basis for six months. The other 30 salespeople selected would not be approached, but simply would have their sales performances monitored along with those using the fitness center regularly. Because these two groups were selected at random, we would expect them to be fairly similar in terms of original average fitness level, ambition, discipline, age, and other factors that might affect performance. From this experimental study, we would be more confident that any significantly higher level of productivity by the group using the fitness center regularly would be due to the fitness initiative rather than other factors.

The point of the preceding Example is to illustrate the difference between observational study and an experimental (or controlled) study. In the observational study, a survey simply was conducted to observe and record the average level for each group, without attempting to control any of the factors that might influence the sales levels. In the experimental study, the director controlled one factor (regular use of the fitness center) by randomly selecting who would be persuaded to use the center regularly, thereby reducing the influence of other factors on the difference between the sales levels of the two groups.

Although experimental studies make it easier to establish a cause–and–effect relationship between two variables, observational studies are used predominately in business and economics. More often than not, surveys are conducted to collect business and economic data (such as consumer preferences or unemployment statistics), with no attempt to control any factors that might affect the variable of interest.

Surveys

One of the most familiar methods of collecting primary data is the survey, which solicits information from people concerning such things as their income, family size, and opinions on various issues. We're all familiar, for example, with opinion that accompany each political election. The Gallup poll and the Harris survey are two well-known surveys of public opinion whose results are often reported in the media. But the majority of surveys are conducted for private use. Private surveys are used extensively by market researchers to determine the preferences and attitudes of consumers and voters. The results can be used for a variety of purposes, from helping to determine the target market for an advertising campaign to modifying a candidate's platform in an election campaign. As an illustration, consider a television network that has hired a market research firm to provide the network with a profile of owners of luxury automobiles, including what they watch on television and at what times. The network could then use this information to develop a package of recommended time slots for Cadillac commercials including costs; that it would present to General Motors. It is quite likely that many, students reading this notes will one day be marketing executives who will "live and die" by such market research data.

Many researchers feel that the best way to survey people is by means of a personal interview, which involves an interviewer soliciting information from respondent by asking prepared questions. A personal interview has the advantage of having a higher expected response rate than other methods of data collection. In addition, there will probably be fewer incorrect responses resulting from respondents misunderstanding some questions, because the interviewer can clarify misunderstandings when asked to. But the interviewer must also be careful not to say too much, for fear of biasing the response. To avoid introducing such biases, as well as to reap the potential benefits of a personal interview, interviewer must be well trained in proper interviewing techniques and well informed on the purpose of the study. The main disadvantage of personal interviews is that they are expensive, especially when travel is involved. A telephone interview is usually less expensive, but it is also less personal and has a lower expected response rate.

A third popular method of data collection is the self - administered questionnaire, which is usually mailed to a sample of people selected to be surveyed. This is a relatively inexpensive method of conducting a survey and is therefore attractive when the number of people to be surveyed is large. But self-administered questionnaires usually have a low response and may have a relatively high number of incorrect responses due to respondents misunderstanding some questions.

Whether a questionnaire is self – administered or completed by an interviewer, it must be well designed. Proper questionnaire design takes knowledge, experience, time, and money. Some basic points to consider regarding questionnaire design follow.

1-       First and foremost, the questionnaire should be kept as short as possible to encourage respondents to complete it. Most people are unwilling to spend much time filling out a questionnaire.

2-       The questions themselves should also be short, as well as simply and clearly worded, to enable respondents to answer quickly, correctly, and without ambiguity. Even familiar terms, such as "unemployed" and "family," must be defined carefully because several interpretations are possible.

3-       Questionnaires often begin with simple demographic questions to help respondents get started and become comfortable quickly.

4-       Dichotomous questions (questions with only two possible responses, such as "yes" and "no') and multiple – choice questions are useful and popular because of their simplicity, but they, too, have possible shortcomings. For example, a respondent's choice of yes or no to a question may depend on certain assumptions not stated in the question. In the case of a multiple- choice question, a respondent may feel that none of the choices offered is suitable.

5-       Open-ended questions provide an opportunity for respondents to express opinions more fully, but they are time-consuming and more difficult to tabulate and analyze.

6-       Avoid using leading questions, such as "Wouldn't you agree that the statistics exam was too difficult?" These types of questions tend to lead the respondent to a particular answer.

7-       Time permitting, it is useful to pretest a questionnaire on a small number of people in order to uncover potential problems, such as ambiguous wording.

8-       Finally, when preparing the questions, think about how you intend to tabulate and analyze the responses. First determine whether you are soliciting values (i.e., responses) for a qualitative variable or a quantitative variable. Then consider which type of statistical techniques – descriptive or inferential – you intend to apply to the data to be collected, and note the requirements of the specific techniques to be used. Thinking about these questions will help to assure that the questionnaire is designed to collect the data you need.

Whatever method is used to collect primary data, we need to know something about sampling, the subject of the next section.

2.3 Sampling

The chief motive for examining a sample rather than a population is cost. Statistical inference permits us to draw conclusions about a population parameter based on a sample that is quite small in comparison to the size of the population. For example, television executives want to know the proportion of television viewers who watch a network's programs. Because 100 million people may be watching television in the world on a given evening, determining the actual proportion of the population that is watching certain programs is impractical and prohibitively expensive. The ratings provide approximations of the desired information by observing what is watched by a sample of 1,000 television viewers. The proportion of households watching a particular program can be calculated for the households in the sample. This sample proportion is then used as an estimate of the proportion of all households (the population proportion) that watched the program.

Another illustration of sampling can be taken from the field of quality control. In order to ensure that a production process is operating properly, the operations manager needs to know what proportion of items being produced is defective. If the quality - control technician must destroy the item in order to determine whether it is defective, then there is no alternative to sampling: a complete inspection of the product population would destroy the entire output of the production process.

We know that the sample proportion of television viewers or of defective items is probably not exactly equal to the population proportion we want to estimate. Nonetheless, the sample statistic can come quite close to the parameter it is designed to estimate if the target population (the population about which we want to draw inferences) and the sampled population (the actual population from which the sample has been taken) are the same. In practice, these may not be the same.

2.4 Sampling Plans

Our objective in this section is to introduce three different sampling plans: simple random sampling, stratified random sampling, and cluster sampling. We begin our presentation with the most basic design.

2.4.1 Simple Random Sampling

Simple Random Sample

A simple random sample is a sample selected in such a way that every possible sample with the same number of observations is equally likely to be chosen.

One way to conduct a simple random sample is to assign a number to each element in the population, write these numbers on individual slips of paper, toss them into a hat, and draw the required number of slips (the sample size, n) from the hat. This is the kind of procedure that occurs in raffles, when all the ticket stubs go into a large, rotating drum from which the winners are selected.

Sometimes the elements of the population are already numbered. For example, virtually all adults have Social Security numbers or Social Insurance numbers; all employees of large corporations have employee numbers; many people have driver's license numbers, medical plan numbers, student numbers, and so on. In such cases, choosing which sampling procedure to use is simply a matter of deciding how to select from among these numbers.

In other cases, the existing form of numbering has built-in flaws that make it inappropriate as a source of samples. Not everyone has a phone number, for example, so the telephone book does not list all the people in a given area. Many households have two (or more) adults, but only one phone listing. Couples often list the phone number under the man's name, so telephone listings are likely to be disproportionately male. Some people do not have phones, some have unlisted phone numbers, and some have more than one phone; these differences mean that each element of the population does not have an equal probability of being selected.

After each element of the chosen population has been assigned a unique number, sample numbers can be selected at random. A random - number table can be used to select these sample numbers. Alternatively, we can employ a software package to generate random numbers. Both Minitab and Excel have this capability.

Example (2)

A government income-tax auditor has been given responsibility for 1,000 returns. A computer is used to check the arithmetic of each return. However, to determine if the returns have been completed honestly, the auditor must check each entry and confirm its veracity. Because it takes, on average, one hour to completely audit a return and she has only one week to complete the task, the auditor has decided to randomly select 40 returns. The returns are numbered from 1 to 1,000. Use a computer random -number generator to select the sample for the auditor.

Solution:

There are several software packages that can produce the random numbers we need. Minitab and Excel are two of these.

Minitab Output for Example (2)

We generated 50 numbers between 1 and 1,000 and stored them in column 1. Although we needed only 40 random numbers, we generated 50 numbers because it is likely that some of them will be duplicated. We will use the first 40 unique random number to select our sample.

Excel Output for Example (2)

We generated 50 numbers between 1 and 1,000 and stored them in column 1. Although we needed only 40 random numbers, we generated 50 numbers because it is likely that some of them will be duplicated. We will use the first 40 unique random number to select our sample.

Stratified Random Sampling

In making inferences about a population, we attempt to extract as much information as possible from a sample. The basic sampling plan, simple random sampling, often accomplishes this goal at low cost. Other methods, however, can be used to increase the amount of information about the population. One such procedure is stratified random sampling.

Stratified Random Sample

A stratified random sample is obtained by separating the population into mutually exclusive sets, or strata, and then drawing simple random sample from each stratum.

Examples of criteria for separating a population into strata (and of the strata themselves) follow.

1 - Sex

Male

Female

2 - Age

Under 20

20-30

31-40

41-50

51-60

Over 60

3 - Occupation

Professional

Clerical

Blue-collar other

4 - Household income

Under $15,000 $15,000-$29,999

$30,000-$50,000

Over $50,000

To illustrate, suppose a public opinion survey is to be conducted in order to determine how many people favor a tax increase. A stratified random sample could be obtained by selecting a random sample of people from each of the four income groups described above. We usually stratify in a way that enables us to obtain particular kinds of information. In this example, we would like to know if people in the different income categories differ in their opinions about the proposed tax increase, since the tax increase will affect the strata differently. We avoid stratifying when there is no connection between the survey and the strata. For example, little purpose is served in trying to determine if people within religious strata have divergent opinions about the tax increase.

One advantage of stratification is that, besides acquiring information about the entire population, we can also make inferences within each stratum or compare strata. For instance, we can estimate what proportion of the lowest income group favors the tax increase, or we can compare the highest and lowest income groups to determine if they differ in their support of the tax increase.

Any stratification must be done in such a way that the strata are mutually exclusive: each member of the population must be assigned to exactly one stratum.

After the population has been stratified in this way, we can employ simple random sampling to generate the complete sample. There are several ways to do this. For example, we can draw random samples from each of the four income groups according to their proportions in the population. Thus, if in the population the relative frequencies of the four groups are as listed below, our sample will be stratified in the same proportions. If a total sample of 1,000 is to be drawn, we will randomly select 250 from stratum 1,400 from stratum 2,300 from stratum 3, and 50 from stratum 4.

The problem with this approach, however, is that if we want to make inferences about the last stratum, a sample of 50 may be too small to produce useful information. In such cases, we usually increase the sample size of the smallest stratum (or strata) to ensure that the sample data provide enough information for our purposes. An adjustment must then be made before we attempt to draw inferences about the entire population. This procedure is beyond the level of these notes. We recommend that anyone planning such a survey consult an expert statistician or a reference book on the subject. Better still, become an expert statistician yourself by taking additional statistics courses. 

2.4.2 Cluster Sampling

Cluster Sample

A cluster sample is a simple random sample of groups or clusters of elements. Cluster sampling is particularly useful when it is difficult or costly to develop a complete list of the population members (making it difficult and costly to generate a simple random sample). It is also useful whenever the population elements are widely dispersed geographically. For example, suppose we wanted to estimate the average annual household income in a large city. To use simple random sampling, we would need a complete list of households in the city from which to sample. To use stratified random sampling, we would need the list of households, and we would also need to have each household categorized by some other variable (such as age of household head) in order to develop the strata. A less expensive alternative would be to let each block within the city represent a cluster. A sample of clusters could then be randomly selected, and every household within these clusters could be questioned to determine income. By reducing the distances the surveyor must cover to gather data, cluster sampling reduces the cost.

But cluster sampling also increases sampling error, because households belonging to the same cluster are likely to be similar in many respects, including household income. This can be partially offset by using some of the cost savings to choose a larger sample than would be used for a simple random sample.

Sample Size

Whichever type of sampling plan you select, you still have to decide what size of sample to use. In determining the appropriate sample size, we can rely on our intuition, which tells us that the larger the sample size is, the more accurate we can expect the sample estimates to be.

2.5 Errors Involved in Sampling

Two major types of errors can arise when a sample of observations is taken from a population: sampling error and nonsampling error. Managers reviewing the results of sample surveys and studies, as well as researchers, who conduct the surveys and studies, should understand the sources of these errors.

Sampling Error

Sampling error refers to differences between the sample and the population that exist only because of the observations that happened to be selected for the sample. Sampling error is an error that we expect to occur when we make a statement about a population that is based only on the observations contained in a sample taken from the population. To illustrate, consider again the Example described in which we wish to determine the mean annual income of blue-collar workers. As was stated there, we can use statistical inference to estimate the mean income  of the population if we are willing to accept less than 100% accuracy. If we record the incomes of a sample of the workers and find the mean  of this sample of incomes, this sample mean is an estimate of the desired population mean. But the value of  will deviate from the population mean  simply by chance, because the value of the sample mean depends on which incomes just happened to be selected for the sample. The difference between the true (unknown) value of the population mean  and its sample estimate  is the sampling error. The size of this deviation may be large simply due to bad luck that a particularly unrepresentative sample happened to be selected. The only way we can reduce the expected size of this error is to take a larger sample.

Given a fixed sample size, the best we can do is to state the probability that the sampling error is less than a certain amount. It is common today for such a statement to accompany the results of an opinion poll. If an opinion poll states that, based on sample results, Candidate Kreem has the support of 54% of eligible voters in an upcoming election, that statement may be accompanied by the following explanatory note: This percentage is correct to within percentage points, 29 times out of 30. This statement means that we have a certain level of confidence (95%) that the actual level of support for Candidate Kreem is between 51 % and 57%.

Nonsampling Error

Nonsampling error is more serious than sampling error, because taking a larger sample won't diminish the size, or the possibility of occurrence, of this error. Even a census can (and probably will) contain nonsampling errors. Nonsampling errors are due to mistakes made in the acquisition of data or due to the sample observations being selected improperly.

Three Types of Nonsampling Errors

1- Errors in Data Acquisition. These types of errors arise from the recording of incorrect responses. This may be the result of incorrect measurements being taken because of faulty equipment, mistakes made during transcription from primary sources, inaccurate recording of data due to misinterpretation of terms, or inaccurate responses to questions concerning sensitive issues such as sexual activity or possible tax evasion.

2- Nonresponse Error. Nonresponse error refers to error (or bias) introduced when responses are not obtained from some members of the sample. When this happens, the sample observations that are collected may not be representative of the target population, resulting in biased results. Nonresponse can occur for a number of reasons. An interviewer may be unable to contact a person listed in the sample, or the sampled person may refuse to respond for some reason. In either case, responses are not obtained from a sampled person, and bias is introduced. The problem of nonresponse is even greater when self-administered questionnaires are used rather than an interviewer, who can attempt to reduce the nonresponse rate by means of callbacks. As noted earlier, a high nonresponse rate, resulting in a biased, self-selected sample.

3- Selection Bias. Selection bias occurs when the sampling plan is such that some members of the target population cannot possibly be selected for inclusion in the sample. Together with nonreponse error, selection biases played a role in pool being so wrong, as voters without telephones or without a subscription were excluded from possible inclusion in the sample taken.

2.6 Summary

Because most populations are very large, it is extremely costly and impractical to investigate each member of the population to determine the values of the parameters. As a practical alternative, we take a sample from the population and use the sample statistics to draw inferences about the parameters. Care must be taken to ensure that the sampled population is the same as the target population.

We can choose from among several different sampling plans, including simple random sampling, stratified random sampling, and cluster sampling. Whatever sampling plans used, it is important to realize that both sampling error and nonsampling error will occur, and to understand what the sources of these errors are.