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In statistical terms, the population mean is the average of a set of characteristics or numbers for the entire group being studied. When trying to discover how long 12th-grade students take to get to school, the population mean gives us the central tendency or typical travel time across all students. However, instead of surveying every single student, we often rely on a sample.
For the given exercise, our sample mean is 16.2 minutes. This sample mean serves as an estimate of the actual population mean, which would be the true average travel time if we were able to collect data from every student in the population.
Using a sample mean provides a reasonable estimate, but with a level of uncertainty. This is why constructing a confidence interval is helpful, as it gives us a range where the true population mean is likely to lie.

The sample standard deviation is a measure that indicates how much variability or dispersion exists in a set of sample data. In this context, it helps us understand how travel times differ among the 50 sampled 12th-grade students.
In our exercise, the sample standard deviation is 12.3 minutes. This tells us that there is quite a bit of variation in the time it takes different students to reach school. Some might take much less or much more time than the average.
Understanding the standard deviation is crucial because it plays a directly influential role in calculating the standard error. A higher standard deviation would lead to greater uncertainty about the location of the population mean, leading to a wider confidence interval.

A z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. When calculating confidence intervals, the z-score represents how many standard deviations away from the sample mean our true population mean is likely to fall.
For different confidence levels, different z-scores are used. For a 95% confidence interval, the z-score is typically 1.96, which reflects that our range will cover 95% of the potential data distribution.
Similarly, at a 90% confidence level, the z-score drops to 1.645. This change impacts the width of the confidence interval, making it narrower at 90% than at 95%. This occurs because we are accepting more risk of the mean lying outside our interval boundaries at a lower confidence level.

The standard error estimates the variability of the sample mean by considering the sample size and the sample standard deviation. It essentially measures how much the sample mean is expected to fluctuate from the true population mean.
In the given exercise, the standard error is calculated using the formula:

• Standard Error = \( \frac{\text{Sample Standard Deviation}}{\sqrt{\text{Sample Size}}} \)

Substituting our values, it becomes \( \frac{12.3}{\sqrt{50}} \approx 1.74 \) minutes.
A smaller standard error implies that the sample mean is a more accurate estimate of the population mean. Hence, it's pivotal in accurately determining the span of the confidence interval for estimating the population mean.