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The population mean, often represented by the symbol \( \mu \), is the average of all the values in a population. Think of it as the point of center that represents the entire dataset. But, it's tough to measure because it requires data from every member of a population, which is usually unrealistic. Instead, we estimate this mean using a sample from the population, which is a smaller, manageable subset. Understanding this, a sample mean \( \bar{x} \), calculated from a sample, serves as an approximation of the population mean. For instance, in exercise part a, the sample mean is \(0.84\), and part b gives us \(21.9\). These values provide our best guesses of the true average tendencies of their respective populations.

A z-score tells us how far, in terms of standard deviations, a data point is from the mean of a data set. It provides a way to understand and communicate where a particular data point sits, relative to the entire distribution. When constructing confidence intervals, the z-score is used to determine how many standard deviations away we need to be to achieve a specific confidence level. For a \(90\%\) confidence level, we use a z-score of \(1.645\) because it encompasses \(90\%\) of the distribution under the standard normal curve. This number forms a critical part of the confidence interval calculation, as it helps determine the width of the interval around our sample mean, ensuring we cover the desired confidence level.

The sample standard deviation, symbolized as \(s\), measures how much the values in a sample differ from the sample mean. It's vital because it gives us an idea of the dispersion or spread of the sample data set.The formula to calculate it is \(s = \sqrt{s^2}\), where \(s^2\) is the sample variance. In step 1 of the provided solution, for example part a, \(s^2\) is \(0.086\), yielding a sample standard deviation of about \(0.2931\).Incorporating it into the confidence interval formula \(\bar{x} \pm z \frac{s}{\sqrt{n}}\), the standard deviation plays a critical role in determining the interval's precision. The smaller the standard deviation, the more precise the estimate of the population mean is likely to be.