ISCAM: Investigating Statistical Concepts, Applications, and Methods

Combining what we have learned in the last few investigations, we can expand the Central Limit Theorem for a sample proportion: When drawing random samples from a large but finite population with a large enough sample size, then the sampling distribution of the sample proportion \(\hat{p}\) will be well modeled by a normal distribution with mean equal to \(\pi\text{,}\) the population proportion of successes, and standard deviation equal to \(SD(\hat{p}) = \sqrt{\frac{\pi(1-\pi)}{n}}\text{.}\)

We consider the sample size large enough if \(n \times \pi > 10\) and \(n \times (1 - \pi) > 10\text{.}\)

We consider the population size (\(N\)) large if it is more than 20 times the size of the sample.

We have no reason to doubt that the NHANES sample will be representative of the larger population of U.S. teens. Although we are not told how they selected the sample, they claim it was nationally representative, and presumably was collected using a probability sampling method. The population size (~21 million teens) is quite large compared to the sample size of nearly 1,800 teens, so that condition for using the normal model for the sampling distribution of the sample proportions is met. The sample size is also very large relative to the hypothesized population proportion of 0.20 (0.20 × 1771 = 354.2 and 0.80 × 1771 = 1416.8; both easily exceed 10). Therefore, calculations from the binomial probability model and the normal probability model will be quite similar. Under the null hypothesis that 20% of U.S. teens have some hearing loss, the sampling distribution of the sample proportion will be approximately normal, with a mean of 0.20 and a standard deviation of 0.0095. Therefore, our sample proportion, \(\hat{p} = 333/1771 = 0.188\) lies 1.26 standard deviations below the hypothesized population proportion, yielding a two-sided p-value of 0.2068 (Theory-Based Inference (TBI) applet). This p-value is not small and does not provide convincing evidence against the claim that \(\pi = 0.20\text{.}\) We would get sample proportions as or more extreme as 0.188 (meaning at least as far from 0.20) in about 21% of random samples from a population where 20% of teens have some hearing loss. Therefore, the headline seems appropriate. The confidence interval (TBI applet) tells us that we are 95% confident that between 17.0% and 20.6% of all U.S. teens have some form of hearing loss. (Do keep in mind that 0.20 is just one plausible value of \(\pi\text{.}\)) It is difficult to evaluate these values without knowing whether this percentage is increasing over time which we will investigate later.