Sample Size

Precision

To understand how sample size is involved in the accuracy of an effect, consider a basic political poll. Suppose that 20 individuals sampled at random were interviewed about their choice between two different candidates. This type of study is modeled with the binomial distribution. If 12 of the 20 individuals reported a preference for Candidate A, is this sample proportion of .6 enough evidence to conclude that Candidate A has a majority lead in the population from which individuals were randomly sampled? To answer this question, the concept of standard error is needed. Right now, there is only a single sample of voters in the poll. Another sample of 20 different individuals could be collected, and the researcher can imagine that the new sample will probably not result in the exact same proportion of .6 preferring Candidate A. If an infinite number of polls were to be conducted, each based on a random sample of 20 individuals, each sample proportion could be plotted on a graph to create a sampling distribution. A large number of sample proportions from these polls would be clustered around the unknown true population proportion representing preference for Candidate A. But in reality, there is usually only one sample available. Where does this first sample proportion of .6 fall on this sampling distribution? This is where standard error comes in.

Standard error is the standard deviation of the sampling distribution or the square root of its variance. If the standard error is very large, and thus the sampling distribution is very wide, the single sample proportion could be a really inaccurate estimate of the preference for Candidate A. This is because this particular sample proportion could fall in the tails (extremes) of the sampling distribution, and in a wide sampling distribution, these tails are quite far away from the center. On the other hand, if the standard error is small, and thus the sampling distribution is narrow, the researcher can be more confident that this single sample proportion is a good guess at the true proportion of individuals in the population who prefer Candidate A, even if it does fall near the tails.

Thankfully, researchers do not actually need to conduct an infinite number of polls or plot a sampling distribution to determine the standard error. Only the single sample of 20 individuals is needed to calculate the estimated standard error for a proportion. The formula for the estimated standard error of a sample proportion is as [Page 1442]follows: $SE=\sqrt{\frac{\widehat{p}(1-\widehat{p})}{n}}$ where $\widehat{p}$ is the sample proportion. So, how can researchers make their standard error small so the single sample can do a good job of accurately estimating the true population proportion? Notice the “n” in the denominator of the formula. The letter n is often used to denote sample size. Thus, by increasing the sample size, the standard error will decrease, and in return the estimate of the true population proportion will be more accurate.

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