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# Samples for AP Statistics

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By McGraw-Hill Professional
Updated on Feb 5, 2011

Practice problems for these concepts can be found at:

In lesson deals with statistical inference—making statements about a population based on samples drawn from the population. In both data analysis and inference, we would like to believe that our analyses, or inferences, are meaningful. If we make a claim about a population based on a sample, we want that claim to be true. Our ability to do meaningful analyses and make reliable inferences is a function of the data we collect. To the extent that the sample data we deal with are representative of the population of interest, we are on solid ground. No interpretation of data that are poorly collected or systematically biased will be meaningful. We need to understand how to gather quality data before proceeding on to inference. In this chapter, we study techniques for gathering data so that we have reasonable confidence that they are representative of our population of interest.

### Census

The alternative to sampling to gather data about a population is to conduct a census, a procedure by which every member of a population is selected for study. Doing a census, especially when the population of interest is quite large, is often impractical, too time consuming, or too expensive. Interestingly enough, relatively small samples can give quite good estimates of population values if the samples are selected properly. For example, it can be shown that approximately 1500 randomly selected voters can give reliable information about the entire voting population of the United States.

The goal of sampling is to produce a representative sample, one that has the essential characteristics of the population being studied and is free of any type of systematic bias. We can never be certain that our sample has the characteristics of the population from which it was drawn. Our best chance of making a sample representative is to use some sort of random process in selecting it.

### Probability Sample

A list of all members of the population from which we can draw a sample is called a sampling frame. A sampling frame may or may not be the same set of individuals we are studying. A probability sample is one in which each member of the population has a known probability of being in the sample. Each member of the population may or may not have an equal chance of being selected. Probability samples are used to avoid the bias that can arise in a nonprobability sample (such as when a researcher selects the subjects she will use). Probability samples use some sort of random mechanism to choose the members of the sample. The following list includes some types of probability samples.

• random sample: Each member of the population is equally likely to be included.
• simple random sample (SRS): A sample of a given size is chosen in such a way that every possible sample of that size is equally likely to be chosen. Note that a sample can be a random sample and not be a simple random sample (SRS). For example, suppose you want a sample of 64 NFL football players. One way to produce a random sample would be to randomly select two players from each of the 32 teams. This is a random sample but not a simple random sample because not all possible samples of size 64 are possible.
• systematic sample: The first member of the sample is chosen according to some random procedure, and then the rest are chosen according to some well-defined pattern. For example, if you wanted 100 people in your sample to be chosen from a list of 10,000 people, you could randomly select one of the first 100 people and then select every 100th name on the list after that.
• stratified random sample: This is a sample in which subgroups of the sample, strata, appear in approximately the same proportion in the sample as they do in the population. For example, assuming males and females were in equal proportion in the population, you might structure your sample to be sure that males and females were in equal proportion in the sample. For a sample of 100 individuals, you would select an SRS of 50 females from all the females and an SRS of 50 males from all the males.
• cluster sample: The population is first divided into sections or "clusters." Then we randomly select an entire cluster, or clusters, and include all of the members of the cluster(s) in the sample.
example: You are going to conduct a survey of your senior class concerning plans for graduation. You want a 10% sample of the class. Describe a procedure by which you could use a systematic sample to obtain your sample and explain why this sample isn't a simple random sample. Is this a random sample?
solution: One way would be to obtain an alphabetical list of all the seniors. Use a random number generator (such as a table of random digits or a scientific calculator with a random digits function) to select one of the first 10 names on the list. Then proceed to select every 10th name on the list after the first.
Note that this is not an SRS because not every possible sample of 10% of the senior class is equally likely. For example, people next to each other in the list can't both be in the sample. Theoretically, the first 10% of the list could be the sample if it were an SRS. This clearly isn't possible.
Before the first name has been randomly selected, every member of the population has an equal chance to be selected for the sample. Hence, this is a random sample, although it is not a simple random sample.
example: A large urban school district wants to determine the opinions of its elementary schools teachers concerning a proposed curriculum change. The district administration randomly selects one school from all the elementary schools in the district and surveys each teacher in that school. What kind of sample is this?
solution: This is a cluster sample. The individual schools represent previously defined groups (clusters) from which we have randomly selected one (it could have been more) for inclusion in our sample.
example: You are sampling from a population with mixed ethnicity. The population is 45% Caucasian, 25% Asian American, 15% Latino, and 15% African American. How would a stratified random sample of 200 people be constructed?
solution: You want your sample to mirror the population in terms of its ethnic distribution. Accordingly, from the Caucasians, you would draw an SRS of 90 (that's 45%), an SRS of 50 (25%) from the Asian Americans, an SRS of 30(15%) from the Latinos, and an SRS of 30 (15%) from the African Americans.
• Of course, not all samples are probability samples. At times, people try to obtain samples by processes that are nonrandom but still hope, through design or faith, that the resulting sample is representative. The danger in all nonprobability samples is that some (unknown) bias may affect the degree to which the sample is representative. That isn't to say that random samples can't be biased, just that we have a better chance of avoiding systematic bias. Some types of nonrandom samples are:

• self-selected sample or voluntary response sample: People choose whether or not to participate in the survey. A radio call-in show is a typical voluntary response sample.
• convenience sampling: The pollster obtains the sample any way he can, usually with the ease of obtaining the sample in mind. For example, handing out questionnaires to every member of a given class at school would be a convenience sample. The key issue here is that the surveyor makes the decision whom to include in the sample.
• quota sampling: The pollster attempts to generate a representative sample by choosing sample members based on matching individual characteristics to known characteristics of the population. This is similar to a stratified random sample, only the process for selecting the sample is nonrandom.

Practice problems for these concepts can be found at:

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