762 CHAPTER 12 Statistics called a sample , to make predictions concerning the population. It is important to understand the difference between a population and a sample. A population includes all items of interest. A sample includes some of the items in the population. When a statistician draws a conclusion from a sample, there is always the possibility that the conclusion is incorrect. For example, suppose that a jar contains 90 blue marbles and 10 red marbles, as shown in Fig. 12.1. If the statistician selects a random sample of five marbles from the jar and they are all blue, the statistician may wrongly conclude that the jar contains all blue marbles. If the statistician takes a larger sample, say, 15 marbles, the statistician is likely to select some red marbles. At that point, the statistician may make a prediction about the contents of the jar based on the sample selected. Of course, the most accurate result would occur if every object in the jar, the entire population, were observed. However, in most statistical experiments, observing the entire population is not practical. Statisticians use samples instead of the entire population for two reasons: (a) It is often impossible to obtain data on an entire population, and (b) sampling is less expensive because collecting the data takes less time and effort. For example, suppose that you wanted to determine the number of each species of all the fish in a lake. To do so would be almost impossible without using a sample. If you did try to obtain this information from the entire population, the cost would be considerable. Or suppose that you wanted to test soup cans for spoilage. If every can produced by the company was opened and tested, the company wouldn’t have any product left to sell. Instead of testing the entire population of soup cans, a sample is selected. The results obtained from the sample of soup cans selected are used to make conclusions about the entire population of soup cans. Consider the task of determining the political strength of a certain candidate running in a national election. It is not possible for pollsters to ask each of the approximately 245.5 million eligible voters their preference of a candidate. Thus, pollsters must select and use a sample of the population to obtain their information. How large a sample do you think they use to make predictions about an upcoming national election? You might be surprised to learn that pollsters use only about 1600 registered voters in their national sample. How can a pollster using such a small percentage of the population make an accurate prediction? The answer is that when pollsters select a sample, they use sophisticated statistical techniques to obtain an unbiased sample. An unbiased sample is one that is a small replica of the entire population with regard to income, education, gender, race, religion, political affiliation, age, and so on. The procedures statisticians use to obtain unbiased samples are quite complex. The following sampling techniques will give you a brief idea of how statisticians obtain unbiased samples. Random Sampling If a sample is drawn in such a way that each time an item is selected each item in the population has an equal chance of being drawn, the sample is said to be a random sample. When using a random sample, one combination of a specified number of items has the same probability of being selected as any other combination. When all the items in the population are similar with regard to the specific characteristic we are interested Figure 12.1 MATHEMATICS TODAY Tune In Tomorrow The A. C. Nielsen Company, which has been measuring the viewing population of TV shows since 1950, uses a sample of about 41,000 metered households in the United States to draw conclusions about more than 118 million American households that have a television. An electronic measurement system, called the People Meter, is placed on each TV in the sample household. Each household member is assigned a personal viewing button on the People Meter to keep track of which channels they watch and for how long. Nielsen then computes the rating of the show, using the data obtained from the sample. Nielson tracks viewership for broadcast and cable networks along with streaming services. You can learn more about the sampling techniques used by A. C. Nielsen at Nielson.com. A. C. Nielsen is involved in many other areas of statistical testing and measurement. Why This Is Important Television programs that have a large audience can charge more for commercial advertising. A. C. Nielsen is only one of a large number of companies that use a sample to make a prediction about an entire population. Andrey_Popov/Shutterstock
RkJQdWJsaXNoZXIy NjM5ODQ=