According to the National Center for Health Statistics, 22.4% of adults are smokers. A random sample of 300 adults is obtained.
a. Describe the sampling distribution of p̂, the sample proportion of adults who smoke.

Verified step by step guidance

1. Identify the population proportion \(p\) given in the problem, which is 22.4%, or \(p = 0.224\).
2. Recognize that \(\hat{p}\), the sample proportion, is a random variable representing the proportion of smokers in the sample of size \(n = 300\).
3. State that the sampling distribution of \(\hat{p}\) can be approximated by a normal distribution if the sample size is large enough, which is generally true if both \(np \geq 10\) and \(n(1-p) \geq 10\).
4. Calculate the mean (expected value) of the sampling distribution of \(\hat{p}\), which is \(E(\hat{p}) = p\).
5. Calculate the standard deviation (standard error) of the sampling distribution of \(\hat{p}\) using the formula \(\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}\).

Key Concepts

Sampling Distribution of a Sample Proportion

The sampling distribution of a sample proportion (p̂) describes the probability distribution of p̂ over many random samples of the same size from a population. It shows how p̂ varies from sample to sample and is approximately normal if the sample size is large enough.

Mean and Standard Deviation of the Sampling Distribution

The mean of the sampling distribution of p̂ equals the true population proportion (p). The standard deviation, called the standard error, is calculated as sqrt[p(1-p)/n], where n is the sample size. This measures the typical variability of p̂ around p.

Conditions for Normal Approximation

For the sampling distribution of p̂ to be approximately normal, both np and n(1-p) must be at least 10. This ensures the sample size is large enough for the Central Limit Theorem to apply, allowing use of normal probability methods.