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When we estimate a population parameter such as the mean gasoline mileage of a car model, a confidence interval gives us a range within which we expect the true parameter to lie. For example, if we have a 95% confidence interval, it means there is a 95% chance that the true mean lies within this range. This helps us in making decisions based on the sample data
and gives us a quantified idea of the uncertainty in our estimation. To calculate a confidence interval, we use the formula: \[ \text{Confidence Interval} = \bar{x} \pm (Z \times \frac{\sigma}{\sqrt{n}}) \] where:

• \(\bar{x}\) is the sample mean
• \(Z\) is the Z-score corresponding to the desired confidence level
• \(\sigma\) is the standard deviation
• \(n\) is the sample size

This formula shows that the interval is built by adding and subtracting the margin of error from the sample mean, giving us an upper and lower bound of where the true mean is likely to be.

The margin of error indicates the range of uncertainty about a sample statistic, such as the sample mean. In this exercise, the margin of error is set at 0.3 mpg, meaning our estimation of the mean gasoline mileage could vary by this amount. A smaller margin of error results in a more precise estimate. However, reducing the margin of error requires increasing the sample size.To determine the margin of error, we use: \[ E = Z \times \frac{\sigma}{\sqrt{n}} \] Key factors affecting the margin of error include:

• The Z-score, which aligns with the confidence level
• The sample size (\(n\))
• The standard deviation (\(\sigma\)), representing data dispersion

By understanding how these variables impact the margin of error, we can quantitatively control the precision of our statistical estimates.

A Z-score is a statistical measurement that describes a value's relation to the mean of a group of values. In the context of confidence intervals, the Z-score is a multiplier that reflects how many standard deviations a certain value lies from the mean at a specific confidence level. For this exercise, a 95% confidence level corresponds to a Z-score of roughly 1.96. This number can be found using a normal distribution table, indicating that 95% of data under a normal curve falls within 1.96 standard deviations from the mean.
Key considerations with Z-scores include:

• A higher confidence level will result in a higher Z-score, which in turn will increase the range of the confidence interval.
• Each confidence level has a corresponding Z-score that is used in statistical calculations.

Understanding Z-scores helps us interpret how likely it is for a sample mean to reflect the actual population mean within our chosen margin of error.

Standard deviation is a measure of how spread out numbers are in a data set. In sample size determination, standard deviation affects both the margin of error and the overall precision of our estimates. In this problem, the standard deviation is given as 1.5 mpg, indicating the average deviation of each sample run's mileage from the mean. The greater the standard deviation, the more spread out the data points are, resulting in a wider confidence interval. A few core points include:

• Standard deviation helps in understanding variability in the data. Larger standard deviations indicate more variability.
• When calculating sample size, a larger standard deviation requires a larger sample to maintain the same margin of error.

By acknowledging the role of standard deviation, we can better prepare and understand the variability in our estimated confidence intervals.