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Population proportion refers to the fraction of the entire group (population) that holds a particular characteristic. In the given problem, the population proportion, denoted as \(p\), is 29%, or 0.29. This means that among all adults, 29% do not own a credit card. Understanding population proportion helps us estimate how a sample from this population might behave. It is a starting point for predicting outcomes and calculating probabilities in statistics.
When you take a random sample from a population, the sample will have its own proportion which usually approximates the population proportion. For example, if we randomly survey 500 adults out of all the adults, the fraction of those who do not own a credit card (sample proportion) will likely be close to 0.29.

Sample size, represented as \(n\), is the number of observations or data points in the sample. In this problem, the sample size is 500 adults. Choosing a large sample size is crucial because it tends to give more reliable and representative results of the population.
A larger sample size reduces the variability of the sample proportion. This means the standard deviation (spread) of the sampling distribution will be smaller, leading to more precise estimates of the population proportion. For instance, with a sample size of 500, we have more confidence in our estimates compared to a smaller sample size. The variability in smaller samples may lead to larger differences between the sample proportion and the population proportion, making predictions less reliable.
The formula used to calculate the standard deviation of the sampling distribution houses the sample size in its denominator, indicating the direct influence of sample size on the result.

A Z-score measures how many standard deviations an element is from the mean. In terms of sample proportions, it helps standardize the value to understand its position relative to the expected proportion. The Z-score is calculated using the formula:
\[ Z = \frac{ \hat{p} - \mu_{\hat{p}} }{ \sigma_{\hat{p}} } \]
Here, \( \hat{p} \) is the sample proportion, \( \mu_{\hat{p}} \) is the mean of the sampling distribution (which is the same as the population proportion \( p \)), and \( \sigma_{\hat{p}} \) is the standard deviation of the sampling distribution.
In the example provided, if we want to determine how unusual it is to find a sample proportion of 30% adults without a credit card, we can calculate the Z-score. This translates the observed proportion into a standard scale, making it easier to find the associated probability from standard Z-tables. For instance, a Z-score of 0.48 means the observed sample proportion is 0.48 standard deviations above the mean. Using Z-tables, we can find that the probability of observing such a value (or more extreme) in a sample is around 31.56%.