91Ó°ÊÓ

A confidence interval is a statistical tool used to estimate the range in which a population parameter lies based on sample data. For example, in our exercise, we're focused on estimating the population proportion \( p \).

This interval provides a range that we are fairly confident includes the true population parameter. The confidence level, here \(98\%\), indicates the degree of certainty that the interval contains the population parameter. A \(98\%\) confidence level means that if we were to take 100 different samples and compute a confidence interval for each, we'd expect about 98 of them to contain the true value of the parameter.

Confidence intervals are valuable because they provide more information than a simple point estimate, like a sample mean or proportion. They reflect the uncertainty and variability that come with taking a sample instead of examining the entire population.

The margin of error represents the range within which the true population parameter is expected to lie from the estimated value. In our exercise, the margin of error \( E \) is \(0.045\).

This means the sample proportion could be higher or lower by 4.5% from the true population proportion. The margin of error depends on the confidence level and the variability within the population, which is often estimated using preliminary sample data.

The smaller the margin of error, the closer our sample estimate is expected to be to the actual population parameter. However, to achieve a smaller margin of error, we may need a larger sample size, as seen in this exercise.

The z-score is a statistical measurement that describes a value's relation to the mean of a group of values, measured in terms of standard deviations from the mean. For confidence intervals, the z-score (\(Z_{\alpha/2}\)) is crucial because it adjusts the width of the confidence interval based on the desired confidence level.

For a 98% confidence interval, the z-score \(Z_{\alpha/2}\) is approximately 2.33. This value can be found using a standard normal distribution table. The higher the confidence level, the larger the z-score, and subsequently, the wider the confidence interval, which affects the margin of error.

In summary, a higher z-score indicates a broader interval, thus providing more certainty but requiring a larger sample size to maintain the same margin of error.

Statistical formulas are key to determining the optimal sample size. In our exercise, we use the formula for determining sample size:
\[ n = \frac{(Z_{\alpha/2})^2 \times p \times (1-p)}{E^2}\]

Where

• \(n\) is the required sample size
• \(Z_{\alpha/2}\) is the z-score associated with the confidence level
• \(p\) is the sample proportion
• \(E\) is the margin of error

For the most conservative sample size, \(p\) is set to 0.5, as this value maximizes the product \(p(1-p)\), ensuring sufficient sample size regardless of the actual proportion.

This formula helps balance the precision of the estimate against the effort needed to obtain a sufficiently large sample.