For a fixed sample size, if we aim to increase our degree of confidence, what must occur to the margin of error and the interval width?

When you aim to increase the degree of confidence in a statistical analysis, you are essentially looking to be more certain that the true parameter (like a population mean) falls within your calculated confidence interval. This increase in confidence typically requires using a wider range of values around the sample estimate to ensure that the true parameter is more likely to be included.

As the degree of confidence increases, such as moving from a 95% confidence level to a 99% confidence level, you need to accommodate a larger critical value from the standard normal distribution (or t-distribution, depending on the scenario). This adjustment directly leads to an increase in both the margin of error and the overall width of the confidence interval.

Consequently, if you're raising the confidence level while keeping the sample size fixed, the interval width expands to maintain that higher confidence, resulting in a larger margin of error. This understanding emphasizes that in statistical terms, higher confidence comes at the cost of having to accept a wider interval, which reflects a greater uncertainty in the exact parameter value.