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A confidence interval provides a range of values which is likely to contain the population parameter with a certain level of confidence. When Katrina and Matthew construct a confidence interval for the proportion of adult Americans who read at least 10 books last year, they are essentially creating a range that is expected to include the true proportion for the entire population.

Confidence intervals have two main components: the sample statistic (e.g., sample proportion) and the margin of error. The sample statistic is the point estimate derived from the sample, while the margin of error accounts for how much the sample statistic is expected to vary from the true population parameter.

To construct a confidence interval, you need:

• The sample proportion \( \hat{p} \)
• The z-score corresponding to the desired confidence level
• The sample size \( n \)

Using these, the confidence interval is typically formulated as:

\[ \hat{p} \pm E \] where \( E \) is the margin of error.

Sample size () is a critical factor in statistical analysis as it directly influences the margin of error and the reliability of the estimated population parameter. The larger the sample size, the more accurately it represents the population. In our exercise, Katrina's sample size is 100, while Matthew's is 400.

When it comes to margin of error, a larger sample size results in a smaller margin of error. This is because the variability or uncertainty in the estimate decreases with larger sample sizes. The relationship can be seen in the formula for the margin of error:

\[ E = z \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \]

Here, increasing \( n \) leads to a smaller value of \( E \), making the confidence interval narrower. Thus, Matthew’s estimate has a smaller margin of error compared to Katrina's due to his larger sample size.

The z-score represents the number of standard deviations a data point is from the mean. In the context of confidence intervals, the z-score is used to determine the critical value that corresponds to the desired confidence level.

For common confidence levels, typical z-scores are:

• 90% confidence level: z = 1.645
• 95% confidence level: z = 1.96
• 99% confidence level: z = 2.576

In our exercise, Katrina uses a 95% confidence level, giving her a z-score of 1.96. Matthew, on the other hand, uses a 99% confidence level, giving him a z-score of 2.576.

A higher z-score corresponds to a wider confidence interval because it reflects more certainty in the estimate. However, this also increases the margin of error. Although Matthew's z-score is higher, his larger sample size significantly reduces the margin of error, making his confidence interval more precise than Katrina's.