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MajorIronMouse30
1. Samples Inductive arguments often rely on data gathered from…
1. Samples
Inductive arguments often rely on data gathered from samples as evidence. If a sample possesses a certain characteristic or property, you can argue that the group or population as a whole, from which the sample was drawn, has that property also. Such arguments are inductive because a sample’s possessing a property does not guarantee that the whole population has the same property, but it does give at least some reason for thinking so.
A conclusion about a larger group, based on properties of a sample from that group, follows more strongly from the evidence if the sample is a representative sample. A representative sample really does reflect the properties of the larger group from which it is drawn. A sample that is not representative of the group from which it is drawn is called a biased sample. Several factors contribute to the determination of whether a sample is biased. These factors are as follows:
Random sampling: A random sample is a sample in which every member of a group has an equal chance of being selected. A random sample helps to ensure (but does not guarantee) that the sample is representative of the whole population.
Sample size: Assuming a randomly selected sample, a larger sample is more likely to be an accurate representation of the population from which it was drawn. Sampling error is the difference between the relative frequency with which a characteristic occurs in the sample and the relative frequency with which the same characteristic occurs in the population from which the sample is drawn. A larger population usually requires a larger sample to maintain the same degree of sampling error.
Psychological factors: Psychological factors are relevant when the population being sampled consists of human beings. For example, if survey questions are phrased (whether accidentally or deliberately) to elicit a particular response, the results of the survey may be biased and may not accurately reflect the beliefs of those people being surveyed. To prevent psychological factors from affecting the results of a scientific experiment, “double blind” conditions are often used to prevent either the surveyor or the respondent from influencing or being influenced by psychological factors that could skew the results.
Consider the following scenario, which involves a biased sample. Determine which one of these factors is the best explanation for why the sample is not representative of the whole population from which it is drawn.
To determine whether the students at a college (with a total population of 10,000 students) tend to consume too much alcohol on the weekends, college administrators surveyed a randomly chosen group of 7,000 members of the student body. On the survey, students were asked, “Do you think you drink excessive alcohol on the weekends?” The results of this survey were not anonymous and the results were to be disclosed to the students’ professors. Of the 7,000 students surveyed, 99 percent replied that they do not consume too much alcohol on the weekends. From this survey, the college administration concluded that excessive drinking is not a problem at this particular college.
What is the best reason for thinking that the sample involved in this scenario is a biased sample?
The sample is possibly biased due to psychological factors.
The sample is possibly biased because it is not a random sample.
The sample is possibly biased because it is too small.
2. Sampling Error and Very Large Populations
Sampling error is the difference between the relative frequency with which some property or characteristic occurs in a sample and the relative frequency with which the same characteristic or property appears in the population from which the sample is taken. Sampling error is often expressed as a percentage. For example, suppose that a survey is given to a sample of a population and that 45 percent of the people surveyed indicate that they support campaign finance reformbut 50 percent of the people in the population from which the sample is taken actually support campaign finance reform. In this case, the sampling error is 5 percent.
In order to obtain a sample that has the greatest likelihood of being representative of the whole, you must have a sample that is large enough to minimize sampling error to an acceptable degree. In general, a larger population requires a larger sample size in order to obtain results with the same degree of sampling error, although the change in the ratio between sample size and sampling error is not linear. For very large populations, the sample size needed to ensure the same degree of precision levels off, as the following chart demonstrates:
Sample Size and Sampling Error
Number of Interviews Margin of Error (in percentage points)
4,000 ± 2%
1,500 ± 3%
1,000 ± 4%
750 ± 4%
600 ± 5%
400 ± 6%
200 ± 8%
100 ± 11%
Source: Charles W Roll Jr. and Albert H. Cantril, Polls: Their Use and Misuse in Politics (New York: Basic Books, 1972), p. 72.
The chart applies to very large populations (above 100,000). For populations of this size, a random sample of 1,000 people usually yields a sampling error of ±4%, regardless of how large the population is.
Use the chart to answer the following questions about the relationship between sample size and sampling error for very large populations.
According to the chart, what is the minimum number of interviews needed to obtain a sampling error of 3 percent for a very large population?
1,500
600
200
400
100
4,000
750
1,000
According to the chart, what is the minimum number of interviews needed to obtain a sampling error of 6 percent for a very large population?
100
200
400
4,000
750
600
1,000
1,500
