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Determining the right sample size is crucial when conducting any statistical study. In our case, we want to increase the confidence level of our survey results without changing the margin of error. This requires us to adjust the sample size appropriately.

The relationship between confidence level and sample size is directly proportional. If we want to be more confident that our survey results are reflective of the entire population, we must sample a larger group. This is because higher confidence levels are associated with a larger Z-score, which is the number that directly impacts how many people we should survey.

To find out the necessary sample size for keeping the margin of error constant while increasing confidence, we can use the formula:

• \( n_{new} = \left( \frac{Z_{new}}{Z_{old}} \right)^2 \times n \)

This equation helps us understand that as our Z-score climbs with a desire for increased confidence, the sample size \( n \) should also multiply accordingly. By following this rule, we ensure that we can increase our confidence without altering the error bounds.

Standard deviation is a measure that helps us understand the amount of variation or dispersion in a set of values. In the context of the Census Bureau study, the known standard deviation was 2.2 minutes. This number tells us how much the time people take to fill the form varies.

In simple terms, if the standard deviation is small, it means that most of the values are close to the mean, indicating less variability. Conversely, a larger standard deviation means there is a wider spread of data points from the mean.

Standard deviation is used in various calculations including computing the error bound of a confidence interval. It's pivotal in understanding our data's behavior and is crucial when determining sample size, influencing how confident we can be in our results.

The margin of error is one of the key components of constructing confidence intervals. It essentially determines the range within which we expect our true population parameter to fall. In our context, it’s the range around the sample mean where we estimate the true mean time to fill the form lies.

An error bound, or margin of error, can be computed using the formula:

• \( E = Z \times \frac{\sigma}{\sqrt{n}} \)

Where \( E \) is the margin of error, \( Z \) is the z-score correlating with our desired confidence level, \( \sigma \) is the standard deviation, and \( n \) is the sample size.

The margin of error is vital as it communicates the potential error in our survey estimate. A smaller margin of error indicates more precision while a larger margin indicates less precision. Adjusting the sample size is a primary method for altering the margin of error, but as in this exercise, if the confidence level changes, sample size becomes the tool to preserve it without changing our confidence in the results.