Final answer:
The confidence level and margin of error are inversely related when sample size is held constant. Increasing the confidence level results in a larger margin of error, and decreasing the confidence level allows for a smaller margin of error. Sample size adjustments are necessary to maintain the desired margin of error at a given confidence level.
Step-by-step explanation:
When considering the relationship between confidence level and margin of error, with a constant sample size, we are discussing a fundamental concept in statistics, particularly in the construction of confidence intervals. The confidence level represents how certain we are that the true parameter lies within the calculated interval, while the margin of error determines the range around the point estimate.
To maintain the same sample size, if we wish to increase the confidence level, the margin of error must increase. This is because a higher confidence level implies a greater certainty that the interval encloses the true parameter, thus demanding a wider range. Conversely, if we decrease the confidence level, we can afford a smaller margin of error, resulting in a narrower interval.
For instance, if we construct a 90% confidence interval with a given margin of error and sample size, and we want to increase the confidence to 99% while keeping the same sample size, our margin of error will increase. This is a reflection of balancing the trade-off between certainty (confidence level) and precision (margin of error). On the other hand, decreasing the confidence level to, say 80%, would allow us to have a smaller margin of error, because we are accepting a lower level of certainty.
It is also important to note that the determination of sample size is intricately related to these two concepts. If, at a given confidence level, we aim for a smaller margin of error, we would need to increase the sample size to achieve a narrower confidence interval. This ties back to the original summary, indicating that increasing the sample size reduces the error bound, whereas decreasing it increases the error bound.