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A researcher wants to estimate the proportion of people who report the side effect of nausea when taking a drug to reduce anxiety. Of 25 people who take the drug, 8 report nausea. In this sample, therefore, 0.32 of the patients reported nausea. Most likely the researcher would construct a confidence interval on the population proportion. The procedure for constructing a confidence interval assumes that the sampling distribution of p is normal. Since the sample proportion, p, can be thought of as the mean of N scores, each score being either zero or one, the central limit theorem is applicable. This theorem states that as N increases, the sampling distribution of the mean (p in this case) approaches a normal distribution. But how large an N is big enough? The population proportion, Pi, is another factor that affects the shape of the distribution. The closer Pi is to 0.5, the more normal the sampling distribution.
This applet allows you to explore the validity of confidence intervals on a proportion with various values of N and Pi. After you specify N, Pi, the level of confidence, and the number of simulations you wish to perform, the applet samples data according to your specification and computes a confidence interval for each simulation. The proportion of simulations for which the confidence interval contains Pi is recorded. If the method for constructing confidence intervals is valid, then about 95% of the 95% confidence intervals should contain Pi.