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A confidence interval, like the one in our penny example, gives a range of values that is likely to contain the true population parameter. In this case, the true population parameter is the actual proportion of Americans who support abolishing the penny. The confidence level (in our exercise, 98%) tells us how sure we can be about that range. The higher this percentage, the more confident we are that our range includes the true proportion.

A 98% confidence level means that if we were to take many samples and build a confidence interval from each one, approximately 98 out of 100 of those intervals would contain the true proportion. This confidence interval is critical for decision-makers, as it provides a quantified measure of uncertainty about the estimate. Using it helps ensure that conclusions drawn from the sample are as reliable as possible.

In the exercise, we need a specific level of accuracy (within 2 percentage points). This requirement defines the margin of error but also affects the sample size. Larger sample sizes usually result in smaller margins of error, making our confidence interval narrower and our estimate more precise.

A proportion estimate in statistics refers to the fraction or percentage of the sample that exhibits a particular characteristic. For instance, in the exercise, we wish to estimate the proportion of Americans who support abolishing the penny.

To make this estimate, we use the formula:

\[ \text{Sample Size} = \frac{{\left( Z \times \sqrt{p(1-p)} \right)^2 }}{{E^2}} \]
where:

• Z is the Z-score corresponding to our desired confidence level
• p is the estimated proportion from a prior study or guess
• E is the margin of error we can tolerate

In case (a) of the exercise, we use the prior estimate of 23% (\ p = 0.23\ ). In case (b), with no prior estimate, we use \ p = 0.5\ since it maximizes the sample size needed and thus ensures that our sample is adequately large.

Using these values, we can compute the number of people the researcher should include in the sample to get a reliable estimate within the desired accuracy.

The margin of error defines how much we expect our sample's proportion to differ from the true population proportion. It's like a buffer around our estimate. If the researcher wants to be within 2 percentage points, then the margin of error (E) is 0.02.

The formula to determine the sample size considers the margin of error directly. A smaller margin of error requires a larger sample size because we want our estimate to be close to the true value. Conversely, a larger margin of error allows for a smaller sample size but gives us less precision.

In the exercise, since we want a margin of error of 2 percentage points (0.02), that precision necessitates computing the sample size to keep the error margin in check. As the confidence interval formula shows, the margin of error affects the number of observations required, ensuring that with higher confidence levels (like our 98%), more observations are needed to keep the error margin small.