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The margin of error is a vital component in statistics when we deal with confidence intervals. It represents the amount by which the sample estimate might differ from the true population parameter. Think of it as the wiggle room around our estimate.
When we say we have a margin of error, we imply that our confidence interval is plus or minus a certain range around the sample mean.
To calculate it, we multiply the Z-score by the standard error:

• Formula: Margin of Error (ME) = \( Z_{\alpha/2} \times \text{Standard Error} \)
• This range helps us capture the true population mean with a certain level of confidence.

Hence, knowing the margin of error is crucial as it determines the range of our confidence interval.

Standard Error (SE) indicates how much the sample mean is expected to vary from the true population mean. It's a way of measuring the accuracy of our sample estimate.
This is particularly important for a confidence interval, which is computed using the standard error.
To compute it, you use the following formula:

• Standard Error (SE) = \( \frac{\sigma}{\sqrt{n}} \)
• Where \( \sigma \) represents the population standard deviation and \( n \) is the sample size.

Using this formula, a larger sample size results in a smaller standard error, meaning more precision in our estimates.
This concept is essential as it measures the likely deviation of the sample mean from the population mean.

The Z-score is crucial when calculating confidence intervals, as it helps us determine how many standard deviations a data point is from the mean.
For a 95% confidence interval, the critical Z-score is 1.96, representing the cut-off for the middle 95% of the data in a standard normal distribution.

Why is the Z-score important?

• It helps define the width of our confidence interval.
• It's a fixed value drawn from a standard normal distribution curve.
• Different confidence levels will have different Z-scores.

Understanding the Z-score helps us determine how confident we are that the population parameter lies within a particular interval.

Population Standard Deviation (\(\sigma\)) quantifies the amount of variation or dispersion in a dataset. It measures the average distance of each data point from the mean of the population.
In your problem, it's provided as a known value: \(\$15\).

Why is it important?

• Having a known population standard deviation enables us to use the standard normal distribution (Z-distribution) for calculating the standard error and confidence intervals, making our results more accurate.
• It helps in understanding the spread of the entire population's data.

The population standard deviation is integral in forming the standard error formula, further affecting the margin of error and the construction of the confidence interval.