Credit Cards According to creditcard.com, 29% of adults do not own a credit card.
a. Suppose a random sample of 500 adults is asked, “Do you own a credit card?” Describe the sampling distribution of p̂, the proportion of adults who do not own a credit card.

Verified step by step guidance

1. Identify the population proportion \( p \) of adults who do not own a credit card. According to the problem, \( p = 0.29 \).
2. Recognize that \( \hat{p} \) is the sample proportion of adults who do not own a credit card in a sample of size \( n = 500 \).
3. Since \( \hat{p} \) is a sample proportion, its sampling distribution can be approximated by a normal distribution if the sample size is large enough. Check the conditions: \( np \geq 10 \) and \( n(1-p) \geq 10 \).
4. Calculate the mean of the sampling distribution of \( \hat{p} \), which is equal to the population proportion: \[ \mu_{\hat{p}} = p \].
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 due to random sampling and is approximately normal if the sample size is large enough and certain conditions are met.

Conditions for Normal Approximation

For the sampling distribution of p̂ to be approximately normal, the sample size must be large enough so that both np and n(1-p) are at least 10. This ensures the binomial distribution of successes and failures is well-approximated by a normal distribution, allowing use of normal probability methods.

Mean and Standard Deviation of the Sampling Distribution

The mean of the sampling distribution of p̂ equals the population proportion p (here, 0.29). The standard deviation (standard error) is calculated as sqrt[p(1-p)/n], measuring the typical distance p̂ is from p in repeated samples, which quantifies sampling variability.