**Introduction to Sample Surveys**

*The results of surveys are presented almost daily in newspapers, over the radio, and on television. From surveys, the proportion p of the population with a certain trait or opinion can be estimated. In fact, if the sample size is 1,500, we can be almost sure that our estimate is within 0.03 of the population proportion. Remarkably, being able to estimate the population proportion with this precision does not depend on the size of the population. A sample of 1,500 people is sufficient whether we are drawing inference to the people living in a particular state, to the people living within the United States, or to the people living on Earth, provided the sample is taken properly. Taking a proper sample is challenging. In this lesson, we will learn more about conducting sample surveys.*

**Margin of Error**

From June 24 through 26, 2005, the Gallup Organization contacted 1,009 adults nationally and asked them, "How patriotic are you? Would you say —extremely patriotic, very patriotic, somewhat patriotic, or not especially patriotic?" Of the respondents, 72% said "extremely or very patriotic." Thus, = 0.72 is the estimate of the proportion *p* of adults in the United States who would state they are extremely or very patriotic. The sample proportion is a point estimate of the population proportion. A *point estimate* of a population parameter is a single number that is based on sample data and represents a plausible value of the parameter.

The Gallup Organization also reported that there was a ±3 percentage point margin of error associated with the survey. The margin of error provided by this and other media descriptions of survey results has two important characteristics. First, the difference between the sample proportion and the population proportion *p* is less than the margin of error about 95% of the time; that is, for about 19 of every 20 random samples of the same size from the same population, the sample proportion will be within the margin of error of the population proportion. Second, the sample proportion will differ from the population proportion by more than the margin of error about 5% of the time; that is, for about one in every 20 samples of the same size from the same population, the difference in the sample proportion and the population proportion will be greater than the margin of error.

The margin of error can be used to obtain an interval of plausible values for the parameter of interest. For the survey on patriotism, the point estimate was 0.72, and the margin of error was 0.03. Thus, the interval of plausible values based on this sample is 0.69 to 0.75.

**Example**

A sample of high school students was randomly selected from a very large city. Each student was asked, "Are you employed either part time or full time during the school year?" Of those sampled, 38% reported that they had a part-time or a full-time job during the school year. The margin of error was reported to be 5%. Give a point estimate and an interval of reasonable values for the proportion of this city's high school students having employment that, with 95% certainty, includes the true proportion.

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