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The margin of error is an important concept when estimating a population parameter, such as the mean amount spent per customer in a restaurant. It provides a range within which we can be fairly confident that the true population mean falls.
To calculate the margin of error, we use the formula:

• Margin of Error (ME) = \( Z \times \frac{\sigma}{\sqrt{n}} \)

Here, \( Z \) is the Z-score associated with the desired confidence level, \( \sigma \) is the population standard deviation, and \( n \) is the sample size.
In our scenario, with a 95% confidence level, the Z-score is 1.96. The margin of error tells us how much we expect the sample mean to vary from the true population mean.
As calculated in the step-by-step solution, the margin of error is approximately 1.40. This means we can be 95% confident that the true mean falls within about 1.40 units above or below the sample mean we observed.

Understanding the population mean is crucial in statistics, especially when it's not feasible to measure an entire population. We often want to estimate what the average, or mean, of a certain measurement is across a whole group.
The population mean is denoted by \( \mu \) and is often unknown because it involves measuring the entire population, which isn't usually possible.
Instead, we use a sample mean, \( \bar{x} \), calculated from a subset of the population. In our original exercise, this sample mean was given as \( 24.80 \).
To estimate the range where the actual population mean might lie, we create a confidence interval.In our case, the calculated confidence interval at a 95% confidence level was \((23.40, 26.20)\), as shown in the original solution. This indicates that with 95% confidence, the average amount spent per customer falls between these values.

Sample size (denoted as \( n \)) is the number of observations or data points collected in a study. It plays a crucial role in determining the reliability of our estimates. The larger the sample size, the more accurate our estimation of the population mean is likely to be, since more data gives a better representation of the entire population.
The formula for the margin of error includes \( n \), in the term \( \sqrt{n} \). This indicates that as the sample size increases, the margin of error typically decreases, leading to a more precise confidence interval. In our exercise, the sample size was 49 customers. The relatively large sample size helps reduce the uncertainty in our estimation of the population mean, making the interval more reliable.
When planning a study, selecting an appropriate sample size is crucial to ensure accurate results without unnecessary data collection efforts.