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The population mean, often represented by the Greek letter \( \mu \), is a measure of the central tendency of an entire population. It is the average of all the values in the population. In the given exercise, the population mean \( \mu \) is 64. This means that when you sum up all the individual values in the population and divide by the number of values, you get 64.

Understanding the population mean is essential because it helps us know the central value around which the data points are distributed. It serves as a benchmark when comparing sample data to the overall population.

In a real-world scenario, the population could be a large set of data like the heights of all students in a school, the annual income of all citizens in a city, or any other comprehensive dataset.

The sample mean, denoted as \( \mu_x \), is the average of a subset (sample) taken from the population. One important aspect of the sample mean is that it is an unbiased estimator of the population mean. This means that over many samples, the average of the sample means will equal the population mean.

In our exercise, calculating the sample mean is straightforward because it is given that \( \mu = 64 \). Therefore, \( \mu_x \) is also 64. This equality holds because the sample mean is just an average of randomly picked values from a population, and in expectation, it should equal the population mean.

In practical terms, the sample mean allows us to make inferences about the population mean when it's not feasible to collect data from every member of the population. For instance, if you wanted to estimate the average height of students at a school, you might measure a sample of students rather than the entire student body.

Standard deviation, represented by the Greek letter \( \sigma \), measures the dispersion or spread of values in a dataset relative to its mean. The population standard deviation provides insight into how much individual data points differ from the mean. In this exercise, the population standard deviation \( \sigma \) is 18.

When dealing with samples, we often calculate the standard deviation of the sampling distribution of the sample mean, denoted as \( \sigma_x \). This is obtained by dividing the population standard deviation by the square root of the sample size: \( \sigma_x = \frac{\sigma} {\sqrt{n}} \)

In our example, given the population standard deviation of 18 and a sample size \( n = 36 \), we find the sample standard deviation \( \sigma_x \) as follows:
\[ \sigma_x = \frac{18}{\sqrt{36}} = \frac{18}{6} = 3. \]

This calculation is crucial because the smaller standard deviation in the sample mean's distribution indicates that sample means tend to be closer to the population mean, enhancing the reliability of statistical inferences.