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The concept of a sample proportion is crucial when working with data from a survey or a sample group. It represents the fraction of the sample that exhibits a certain characteristic or condition. In our exercise, the sample proportion (\( \hat{p} \)) is calculated by dividing the number of customers who have more than 12 items by the total sample size. For instance:

• Number of customers with more than 12 items = 74
• Total number of customers sampled = 200

The sample proportion is thus \( \hat{p} = \frac{74}{200} = 0.37 \).
This value helps in estimating the true proportion in the entire population. It forms the basis for constructing confidence intervals, as it informs us how many in our sample adhere to a specific condition, allowing us to project this to the broader group.

The standard error is an important statistical concept that measures how much our sample proportion is expected to vary from the true population proportion. The formula for standard error when dealing with proportions is \( SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{N}} \), where:

• \( \hat{p} \) is the sample proportion
• \( N \) is the sample size

In our situation, using \( \hat{p} = 0.37 \) and the total sample size of 200, the standard error helps determine the precision of our sample proportion.
It gives us an idea about the range in which the true proportion might fall within the population, and is central to constructing confidence intervals.

The Z-score plays a pivotal role in determining how far our sample proportion estimate is from the true population parameter in terms of standard deviations. This score is particularly useful in finding confidence intervals to quantify the certainty of our estimate. For a given confidence level, the Z-score signifies the number of standard errors to include on either side of the sample proportion. In most statistical problems, higher confidence levels correspond to larger Z-scores. For the problem at hand, a 98% confidence level corresponds to a Z-score of approximately 2.33. To construct the confidence interval:

• Multiply the standard error by the Z-score
• Add and subtract this from the sample proportion

This results in a range where the true proportion is likely to be found.

The confidence level is a measure of how certain we are that the true parameter lies within the confidence interval. A higher confidence level provides greater assurance, but it also leads to a wider interval. In our problem, a 98% confidence level was used, indicating strong confidence in the bounds produced.

• This means if you were to repeat the sample 100 times, about 98 of those intervals would contain the true population proportion.
• It is directly related to the Z-score used in calculating the confidence interval.

To reduce the width of the confidence interval, we could consider lowering the confidence level.
However, this might compromise the accuracy of our estimate, seeming less advisable than increasing the sample size, which allows for a more precise estimate without losing confidence.