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The sample proportion, often denoted as \( \hat{p} \), is a vital concept in statistics when you're working with proportions. It's a straightforward way to estimate a population proportion based on a sample from that population. In the context of our exercise, we're looking at credit card customers who incurred an interest charge. From a sample of 200 customers, 136 incurred interest, giving us a sample proportion \( \hat{p} = \frac{136}{200} = 0.68 \). This means 68% of the sampled credit card users had unpaid balances leading to interest charges. By pointing to a portion of the whole, sample proportions help formulate expectations about an entire population based on smaller groups.

The standard error (SE) is a measure in statistics that provides an estimate of the variability or dispersion of a sample statistic from its expected value. When we deal with proportions, the standard error is calculated using the formula \[ SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}, \]where \(\hat{p}\) is the sample proportion, and \(n\) is the sample size. In our scenario, with \(\hat{p} = 0.68\) and a sample size of 200, the standard error becomes approximately 0.0337. The smaller the standard error, the more precise our estimate of the population proportion is. It essentially helps in assessing how much the sample proportion might vary from the true population proportion.

The Z-score is a statistical measure that tells us how many standard deviations an element is from the mean of a set of data. When constructing confidence intervals, the Z-score determines how far out we extend from the sample statistic to capture the true population parameter with a given level of confidence. For a 90% confidence interval, the Z-score is approximately 1.645. This specific Z-score is pivotal as it represents the point on the standard normal distribution where the area under the curve matches the desired confidence level. Essentially, the Z-score helps us decide the range around our sample statistic, such that we are confident the true population parameter falls within this range 90% of the time.

Calculating the necessary sample size is an important step when aiming for a specific confidence interval width. The formula used is \[ n = \left(\frac{Z^*}{d}\right)^2 \hat{p}(1-\hat{p}) \],where \(d\) is half the desired width of the confidence interval, \(Z^*\) is the Z-score, and \(\hat{p}\) is the sample proportion. In our problem, we desire a confidence interval width of 0.05, translating to \(d = 0.025\), and use a Z-score of 1.645. Substituting these values into the formula provides us with a required sample size of approximately 1512. This calculation is critical for designing surveys and experiments to ensure that they are cost-effective and meet the precision requirements.