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The confidence level is a measure of how certain we can be that our sample accurately reflects the population. When we say a result is at a 97% confidence level, it means if we were to take 100 different samples, about 97 of them would contain the true population mean.

This level of confidence is crucial in research, especially when estimating population parameters like average water usage. Higher confidence levels result in wider intervals, which means more certainty, but at the cost of precision.

When deciding on a confidence level, the choice depends on how precise we want our estimate to be and how much uncertainty we are willing to accept. For the city planner in this task, a rather high confidence level of 97% is chosen, indicating the need for a very reliable estimate.

The margin of error (MOE) is the range within which we expect the true population parameter to lie. It gives us an idea about the extent of error we may encounter from our sampling. In the context of the water usage estimation, the MOE is set at 100 gallons.

A larger margin of error means a less precise estimate. Conversely, a smaller margin of error means we expect our sample to be more precise.

Here's how it impacts the calculation:

• Smaller MOE requires a larger sample size to achieve the same confidence level.
• Higher MOE allows for a smaller sample because you're accepting more potential error.

This knowledge helps in planning the size of the study sample effectively.

Population standard deviation (\(\sigma\)) measures how much the data in a population varies from the population mean. It is a critical element in determining sample size because it tells us how spread out the data points are.

In this exercise, the standard deviation is already provided as 389.60 gallons.

This value indicates significant fluctuation in monthly water usage among residents, which directly influences how many samples are needed to achieve a reliable estimate. A higher standard deviation means more variability and typically requires a larger sample to ensure that our sample mean reflects the population accurately. The formula for calculating sample size incorporates standard deviation as it ensures our results are representative of the population's variability.

The Z-value, also known as the Z-score, is a statistical metric that expresses the distance of a data point from the mean in terms of standard deviations. It is used in the context of sample size calculation to translate a desired confidence level into a numeric value.

The Z-value is derived from the standard normal distribution table and is specific to the chosen confidence level.

• For a 97% confidence level, like in this exercise, the Z-value is approximately 2.17.
• This means we are looking for a data point that is 2.17 standard deviations away from the mean, reflecting our chosen level of confidence.

The role of the Z-value is crucial as it affects the width of the confidence interval. Using our Z-value with the standard deviation and margin of error, we determine how many samples are necessary to attain our desired level of accuracy and reliability.