What is the key factor that affects the width of a confidence interval?

The margin of error is indeed the key factor that affects the width of a confidence interval. The confidence interval is designed to estimate a population parameter (like a mean or proportion) and implies a range of values within which the parameter is likely to fall. The width of this interval is directly influenced by the margin of error, which combines the critical value (based on the desired confidence level) with the standard deviation and the sample size.

As the margin of error increases, the confidence interval widens, reflecting greater uncertainty about the population parameter. Conversely, a smaller margin of error results in a narrower confidence interval, indicating more precision in the estimate of the parameter.

While the sample mean, population size, and sample proportion contribute to the calculation of the confidence interval, they do not directly impact its width in the way that the margin of error does. The key takeaway is that the margin of error encapsulates the variability and the degree of confidence in the estimate, making it the principal determinant of the confidence interval's width.