Final answer:
Increasing sample size decreases the margin of error when the confidence level is held constant; decreasing sample size increases it. A higher margin of error necessitates a larger sample size to maintain confidence levels, and vice versa. Adjusting the confidence level impacts the error bound: lower confidence levels decrease the margin of error with the same sample size and statistics.
Step-by-step explanation:
How sample size and margin of error correlate, if the confidence level is held constant, is a significant concept in statistics. The margin of error represents the range within which we expect the true population parameter to lie, and this range is affected by both the sample size and the confidence level. If we increase the sample size while maintaining the same confidence level, the data's standard error decreases, leading to a narrower margin of error. This means the confidence interval becomes more precise, as the estimate is now closer to the true population parameter. Conversely, a smaller sample size increases the standard error and consequently increases the margin of error, implying a less precise confidence interval.
When we decrease the allowable margin of error but wish to keep the same level of confidence, we must increase the minimum sample size to offset the increased precision required. This explains why studies aiming for a high degree of accuracy must have large sample sizes.
Changes in the confidence level can also impact the margin of error and the sample size needed for an analysis. Should a firm, for instance, desire to increase its confidence level while maintaining the same error bound, it would need to increase its sample size. If it were to reduce the sample size but wanted to retain the same margin of error, the confidence level would consequently decrease due to the increased variability inherent in a smaller sample.
Additionally, decreasing the confidence level to 90 percent, keeping the mean, standard deviation, and sample size constant, would make the error bound smaller. It happens because a lower confidence level means we are accepting a higher probability that the true population parameter lies outside our confidence interval.
Ultimately, researchers must decide on the appropriate balance between sample size, confidence level, and margin of error to achieve reliable results within feasible resource constraints.