Final answer:
The margin of error is inversely related to the sample size; increasing the sample size reduces the margin of error, resulting in a narrower confidence interval.
Step-by-step explanation:
To find the margin of error for a given confidence level, we use the formula for the error bound in a proportion, which involves the critical value (z-score) for the given confidence level, the sample size (n), and the sample proportion (x/n) for successes. When we change the sample size while keeping the same confidence level, the margin of error is affected.
For example, if we increase the sample size to n = 100, the margin of error will decrease, leading to a narrower confidence interval. This occurs because increasing the sample size provides more data, reducing the variability and uncertainty in our estimate. Conversely, if we decrease the sample size to n = 25, the margin of error increases, resulting in a wider confidence interval due to increased variability and uncertainty.
In summary, as the sample size increases, the margin of error decreases, and as the sample size decreases, the margin of error increases, directly affecting the width of the confidence interval.