Get Ready for the Postgraduate Entrance English Exam. Working with Texts. Часть 1. Учебное пособие

in the sample results. Why do you feel that increases in sample size increase the amount of confidence you should have in the sample results? Because, as the sample size increases, the probability that the sample result departs greatly from the corresponding result for the entire population becomes smaller and smaller. Thus, as the Gallup Poll takes a bigger and bigger sample of voters, it seems logical that we can place more and more confidence in the sample estimate of the proportion of votes that each candidate will receive.

But what exactly do we mean by a probability? And how can we measure a particular probability? These questions are important to the statistician and to the user of statistics alike. To go beyond vague, intuitive notions about the degree of accuracy of a particular sample result, we must draw on probability theory, a branch of mathematics distinct from, but closely related to, statistics. (To delve deeply into probability theory, one needs a considerable mathematical background. However, no mathematics beyond elementary high-school algebra is needed to understand the elements of probability theory required for an introductory course in statistics.) Until we present a more adequate definition of a probability, we shall treat probability intuitively. That is, if a particular event has a 50-50 chance of occurring, we shall say that the probability of its occurring is 0.5. Similarly, if a particular event has one chance in four of occurring we shall say that the probability of its occurring is 0.25. Or if a particular event has one chance in ten of occurring, we shall say that the probability of its occurring is 0.1.

Bias and Error

Because the field of statistics attempts to make inferences from a sample concerning a population, statistics must necessarily be concerned with error. Any result based on a sample is likely to depart in some measure from the corresponding result for the population.

The error in a particular sample result consists of two parts: experimental or sampling error (sometimes called random error) and bias. Experimental or sampling errors occur because of a large number of uncontrolled factors, which we can subsume under the term chance.

The essential characteristic of experimental or sampling errors is that they can reasonably be expected to cancel each other out over a period of time or over a large number of experiments or samples. Many uncontrolled factors operate to cause the results of a sample to be in error,

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sometimes on the high side and sometimes on the low side. However, if the sample or the experiment is repeated a large number of times, the errors tend to offset each other when the results are averaged. In contrast, bias consists of a systematic and persistent type of error which will not tend to cancel out.

To summarize, an important reason for distinguishing between experimental or sampling error and bias is that increases in a sample’s size tend to reduce experimental or sampling errors, while such increases do not reduce bias. Thus, although (as pointed out previously) the accuracy of the results of a sample tends to increase with the sample’s size, not all errors can be eliminated by increasing sample size. If there are serious biases, the results may be considerably in error even if the sample is huge.

Two important causes of bias

Bias can be particularly dangerous when, like termites or dry rot, its presence is undetected. Two types of methodological errors are commonly causes of serious bias. The first arises when a sample is taken from a population that differs in an important way from the population that the statistician or decision maker is really interested in. For example, suppose that you want to estimate the proportion of people in Chicago who are college graduates. To do so, you go to three Chicago suburbs and pick a sample of their residents. Such a sample is likely to result in substantial bias because you are sampling from the wrong population. Rather than sampling from the population of all Chicago residents, you are sampling from the population of Chicago suburban residents. And since the educational level is likely to be higher in the suburbs than in the inner city, the result is likely to be biased.

A second methodological mistake which can be the source of serious bias arises when the effect of the variable one wants to measure is mixed up inextricably with the effect of some other factor. In this case, if one finds that the variable in question seems to have a certain effect, this estimated effect may be biased because it also reflects the effect of another factor. Table 1 presents part of the results of a nationwide test of the effectiveness of the Salk antipolio vaccine. These data seem perfectly adequate to measure the effects of the vaccine on the incidence of polio in children, but in fact they contain a major bias because only those secondgraders who received their parents’ permission were given the vaccine, whereas all firstand third-graders were used to indicate what the inci-

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dence of polio would be without the vaccine. The bias arises because the incidence of polio was substantially lower among nonvaccinated children who did not receive permission than among those who did. Why? Because children of higher-income parents were more likely than children of low- er-income parents to receive permission – and they were also more likely to contract polio. It is worth taking some trouble to avoid the methodological errors that result in these biases – unless, of course, you don’t mind making serious mistakes.

II. Vocabulary Items

accuracy n – точность, правильность

average v – выводить среднее значение, усреднять bias n – смещение; ошибка, погрешность

cancel out v – уравновесить, свести на нет conclusion n – заключение, вывод

depart v – отклоняться

draw a sample – отбирать пробу entire a – полный, целый, весь error n – ошибка

estimate n – оценка mix up v – путать

notion n – понятие, представление occur v – случаться, происходить probability n – вероятность

subject (to) a – подверженный (чему-л.), склонный (кчему-л.) subsume v – включать в какую-л. категорию, группу vague a – неопределенный, нечеткий

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III. Exercises

1.Read the text.

2.Learn the vocabulary items by heart.

3.Translate the text in written form.

4.Retell the text using the following expressions and terms: inferential statistics, to draw the sample, to be subject to a certain amount of uncertainty, the bigger the sample, the less confidence you are likely to have, the entire population, to draw on probability theory, to treat probability intuitively, to be concerned with error, to depart from, on the high side, on the low side, to be in error, the source of serious bias.

IV. Test V (2)

1. Read the text again and decide which statements are true.

1.The smaller the sample, the more confidence you are likely to have in the sample results; the bigger the sample, the less confidence you are likely to have in the sample results.

2.The essential characteristic of experimental or sampling errors is that they can reasonably be expected to cancel each other out over a period of time or over a large number of experiments or samples.

3.Increases in a sample’s size tend to reduce experimental or sampling errors and bias.

4.A methodological mistake which can be the source of serious bias arises when the effect of the variable one wants to measure is mixed up inextricably with the effect of some other factor.

2. Match the words given in the left column with the words in the right column.

Unit 3

I. Information for study

Population and Sample Surveys

The population consists of the total collection of observations or measurements that are of interest to the statistician or decision maker. A sample is a subset of measurements taken from the population in which the statistician is interested. Social scientists, hospitals, firms, government agencies, and other organizations continually draw samples to obtain needed information because it would be too expensive and time-consuming to try to obtain complete data concerning all relevant units. Thus, the federal government’s unemployment statistics are based on a sample of persons rather than on a complete count of all workers without jobs and looking for work. Television ratings, which have so strong an influence over which programs survive and which are canceled, are based on the TV-viewing decisions of a sample of families. It is easy to find examples of the use of sampling in all areas of social and government activity.

Consider the case of the Gallup Poll, which has used sampling for many years to help predict election results. To predict whether Walter Mondale or Ronald Reagan would win the presidency in 1984, the Gallup Poll chose a sample of about 3,000 eligible voters. To choose this sample of voters, Gallup began by selecting a sample of geographical areas in the United States. Then a sample of precincts within each of the selected areas was chosen. Finally, a sample of households in each selected precinct was selected, and some members of each of these households were interviewed.

The interviewers asked each person a number of questions, some of which were designed to indicate whether the person was likely to vote. In this way, Gallup hoped to screen out those people who were unlikely to vote. To determine each person’s preferences, the interviewer asked each one to fill out a secret ballot which contained the names of each of the candidates. The interviewer instructed the person to indicate on the ballot the candidate he or she would vote for if the election were held at the time of the interview.

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How accurate has this sampling procedure been? The predicted percentage of the vote going to the winning candidate was generally in error by less than 2 percentage points. This does not mean that all sampling procedures are as accurate as the Gallup Poll or that it is always better to sample than to carry out a complete count. What this example does illustrate is that, if correct statistical principles are known and observed, it is generally possible to design samples that will be sufficiently accurate for the purposes at hand. In some cases, pinpoint accuracy is required; in other cases, rough estimates will do. The nature of any given sampling procedure should depend on the accuracy required, as well as on the type of population from which the sample is drawn.

Kinds of populations and samples

Let’s return now to the concept of the population. What was the population in the case of the Gallup poll? It was the set of observations indicating which candidate each eligible voter in the United States supported. Since there are millions of eligible voters in the United States, this population contained millions of observations. Many populations do not contain nearly so many observations. Using a quite different example, suppose that a psychologist is interested in the characteristics of famous scientists. She administers a Wechsler IQ test to all of the (living) Nobel Prize winners in a particular field. (According to Lubin, Larsen, and Matarazzo, the Wechsler Adult Intelligence Scale is the most frequently used psychological test.)

In this case, the population consists of the set of scores achieved by the Nobel Prize winners. To simplify, suppose that there were only 20 such Nobel Prize winners and that the score of each one is as shown in Table 2; then the population consists of the 20 numbers in Table 2.

Table 2

When carrying out sample surveys, a listing of all the elements or units in the population is often called a frame. In the case of the Gallup Poll, the frame is a list of all eligible voters. A census is a survey that at-

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tempts to include all the elements or units in the frame. For example, the Census Bureau’s decennial Census of Population attempts to include each person in the United States. If the frame is very large, it is generally impossible to include all the elements or units; nevertheless, surveys that do so with reasonable success are called censuses.

Populations are of various types. Some consist of quantitative information, whereas others consist of qualitative information. In the Nobel Prize winners example, the population consists of quantitative measurements; that is, ones that can be expressed numerically. Each observation or measurement in this population is the score that a particular Nobel Prize winner achieved on the Wechsler IQ test. Each such score is a number. Populations also can definite or infinite. If a population contains a finite number of members, it is called a finite population. Some finite populations, such as the hypothetical ones in Table 2, contain relatively few members; others have a very large number of members. If a population contains an unlimited number of members, it is called an infinite population. Typically, infinite populations are conceptual constructs.

As indicated above, a sample is a subset of measurements taken from the population. For example, suppose that we choose as a sample the Nobel Prize winners in the first column in Table 2; then the sample consists of 151, 143, 139, and 132. A sample can be drawn in a variety of ways: simple random sampling, stratified random sampling, systematic sampling, arid cluster sampling. In order to use sample data intelligently, it is essential that you have some familiarity with each of these methods.

II. Vocabulary Items

ballot n – голосование cluster a – кластерный designed a – предназначенный eligible a – подходящий estimate n – оценка

finite population – конечная генеральная совокупность household n – домохозяйство

indicate v – указывать, означать, свидетельствовать infinite population – бесконечная генеральная совокупность percentage n – процентное отношение

qualitative a – качественный

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quantitative a – количественный rating n – оценка

relevant a – относящийся к делу, релевантный rough a – приблизительный

sampling n – выборка

III. Exercises

1.Read the text.

2.Learn the vocabulary items by heart.

3.Translate the text in written form.

4.Retell the text using the following expressions and terms: the total collection of observation, a subset of measurements, to be of interest to the statistician, a subset of measurements, to draw samples to obtain needed information, to predict election results, to be time-consuming, to screen out, to determine preferences, to be in error by less than 2 percentage points, to carry out a complete count, sample surveys, eligible voters, quantitative information, qualitative information, finite population, infinite population, random sampling, cluster samples.

IV. Test V (3)

1. Read the text again and decide which statements are true.

1.A sample is a subset of measurements taken from the population in which the statistician is interested.

2.The federal government’s unemployment statistics are based on a complete count of all workers without jobs and looking for work.

3.If correct statistical principles are known and observed, it is generally possible to design samples that will be sufficiently accurate for the purposes at hand.

4.If a population contains a restricted number of members, it is called an infinite population.

2. Match the words given in the left column with the words in the right column.

Unit 4

I. Information for study

Factor analysis

Factor analysis is a statistical technique that originated in psychometrics. It is used in the social sciences and in marketing, product management, operations research, and other applied sciences that deal with large quantities of data. The objective is to explain the most of the variability among a number of observable random variables in terms of a smaller number of unobservable random variables called factors. The observable random variables are modeled as linear combinations of the factors, plus “error” terms.

The nature of factor analysis seen via an example

This oversimplified example should not be taken to be realistic. Suppose a psychologist proposes a theory that there are two kinds of intelligence, which let us call “verbal intelligence” and “mathematical intelligence”. Evidence for the theory is sought in the examination scores of 1000 students in each of 10 different academic fields. If a student is chosen randomly from a large population, then the student’s 10 scores are random variables. The psychologist’s theory may say that the average score in each of the 10 subjects for students with a particular level of “verbal intelligence” and a particular level of “mathematical intelligence” is a certain number times the level of “verbal intelligence” plus a certain number times the level of “mathematical intelligence”, i.e., it is a linear combination of those two “factors”. The numbers by which the two “intelligences” are multiplied are posited by the theory to be the same for all students, and are called “factor loadings”. For example, the theory may hold that the average student’s aptitude in the science of omphalology is

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{ 10 × the student’s verbal intelligence } + { 6 × the student’s mathematical intelligence }.

The numbers 10 and 6 would be the factor loadings associated with the field of omphalology. Other academic subjects would have factor loadings other than 10 and 6. Two students having identical degrees of verbal intelligence and identical degrees of mathematical intelligence would have different aptitudes in omphalology or any other subject because individual aptitudes differ from average aptitudes. That difference is the “error” – an unfortunate misnomer in statistics that means the amount by which an individual differs from what is average.

The observable data that go into factor analysis would be 10 scores of each of the 1000 students, a total of 10,000 numbers. The factor loadings and levels of the two kinds of intelligence of each student must be inferred from the data. Indeed, even the number of factors (two, in this example) must be inferred from the data.

Factor analysis in marketing

The basic steps are:

•Identify the salient attributes consumers use to evaluate products in this category.

•Use quantitative marketing research techniques (such as surveys) to collect data from a sample of potential customers concerning their ratings of all the product attributes.

•Input the data into a statistical program and run the factor analysis procedure. The computer will yield a set of underlying attributes (or factors).

•Use these factors to construct perceptual maps and other product positioning devices.

Information collection

The data collection stage is usually done by marketing research professionals. Survey questions ask the respondent to rate a product sample or descriptions of product concepts on a range of attributes. Anywhere from five to twenty attributes are chosen. They could include things like: ease of use, weight, accuracy, durability, colorfulness, price, or size. The attributes chosen will vary depending on the product being studied. The same question is asked about all the products in the study. The data for multiple products is coded and input into a statistical program such as SPSS or SAS.

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