Final answer:
Increasing the standard deviation leads to a larger margin of error in a survey. To maintain a specific level of confidence with a larger standard deviation, the sample size must be increased. This adjustment ensures the margin of error remains at the target size and the confidence interval remains accurate.
Step-by-step explanation:
In the context of surveys and confidence intervals, the standard deviation is a key factor that impacts the margin of error. When considering the effects of increasing the standard deviation on the margin of error, it is crucial to understand that the margin of error is positively related to the standard deviation. This means that as the standard deviation increases, the margin of error also increases.
Considering the scenario where a firm needs to be at least 96 percent confident of the population mean to within a certain accuracy, an increased standard deviation requires the firm to either increase the number of people surveyed or to accept a larger margin of error. Similarly, if the Census wants to increase its level of confidence and keep the error bound the same, the sample size must increase.
If the firm only surveyed 49 people, but wanted to maintain the same error bound, the level of confidence would decrease unless the margin of error was allowed to increase, illustrating the inverse relationship between sample size and confidence level when the margin of error is constant.
Summary: Increasing the sample size while maintaining or increasing the confidence level without inflating the margin of error requires a smaller error bound, leading to a narrower confidence interval. If a firm or surveying body needs to compensate for an increase in the standard deviation without adjusting the error bound, then a larger sample size is necessary to maintain the same level of confidence.
This concept further elaborates the relationship between sample size, confidence level, and margin of error as represented in various formulae like the error bound formula.