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To understand the concept of sample proportion, let's start by considering the example of First USA's credit card campaign. Of the 50,000 cardholders surveyed, 1,184 took the offer. To make informed decisions, we calculate the sample proportion \( \hat{p} \), which represents the estimated percentage of the entire population who will register. The formula used is simply the number of successes (registrations) divided by the number of cases (total cardholders). In this case, the sample proportion is calculated as \( \hat{p} = \frac{1184}{50000} = 0.02368 \). This means around 2.368% of the sample accepted the offer.

By using the sample proportion, companies like First USA can predict how successful their campaign might be, without needing to survey every customer. It acts as a miniature representation of the whole customer base, providing a glimpse of the potential acceptance rate.

The standard error is crucial in statistics, especially when dealing with sample data. It measures the variability or dispersion of the sample proportions from the sample mean. In simpler terms, it tells us how much we can expect the sample proportion to differ from the true proportion of the entire population. The standard error for the sample proportion is calculated using the formula:
\[ SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \]
For First USA, this equates to:
\[ SE = \sqrt{\frac{0.02368(1-0.02368)}{50000}} \approx 0.000685 \]

The small standard error of 0.000685 suggests that there's little variation, indicating the sample proportion is likely a good estimate of the population proportion.

In practical terms, a smaller standard error reflects more precise estimates, while a larger one points to greater uncertainty about the sample being accurate to the actual population.

The Z-score plays a key role in determining confidence intervals. It represents how many standard deviations away a data point is from the mean of the data set. In the context of building confidence intervals, the Z-score is used to establish the range of data within which the true parameter (such as a proportion) likely falls. For a 95% confidence interval, the widely accepted Z-score is 1.96.

This value implies that if we were to draw multiple samples and calculate their proportions, approximately 95% of those samples' proportions would fall within 1.96 standard deviations of the true proportion.

Applying this to First USA's campaign, the Z-score is critical to widening or narrowing the interval around the sample proportion. It helps in understanding the level of certainty we have about the sample proportion representing the population.

Acceptance rate is the proportion of individuals who take part in an offer or activity out of the total approached or qualified. In business or marketing terms, measuring the acceptance rate helps in evaluating the success of promotional campaigns.

For First USA, the acceptance rate in question was whether at least 2% of cardholders would register for the double airline miles offer. This threshold is important since it determines if the campaign's costs are justified by the proportion of participants.

Given the calculated confidence interval of (0.02234, 0.02502) in this exercise, this entire range sits above the 2% mark. Such insight strongly suggests that the campaign is indeed likely to exceed the minimum acceptance threshold, demonstrating likely success and cost-effectiveness for First USA.