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The population mean, often represented by the Greek letter \( \mu \), is a key statistical concept that refers to the average value of a set of measurements in the entire population. When we talk about a population, we're referring to the complete collection of data points or individuals that we're interested in studying. For example, if we are looking at the heights of all adult women in a city, the population mean would be the average height of all these women.

To calculate the population mean, we simply sum up all the values and divide by the total number of observations in the population. It's important to note that when we take a random sample from this population and calculate the average, we're seeking an estimate of this population mean. This estimated average is known as the sample mean.

The standard deviation of the sample mean, sometimes called the standard error, describes how much variation or 'spread' exists from the sample mean to the true population mean. A smaller standard error indicates that the sample mean is likely to be close to the population mean. This concept is critical when we make inferences about the population from our sample.

In mathematical terms, the standard deviation of the sample mean \( \sigma_{\bar{x}} \) is calculated by dividing the population standard deviation \( \sigma \) by the square root of the sample size \( n \) (\( \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} \) ). The smaller the standard error, the more confident we can be that our sample mean is a good representation of the population mean.

The Central Limit Theorem is a fundamental principle in statistics that has far-reaching implications. This theorem states that, given a sufficiently large sample size, the sampling distribution of the mean will be approximately normally distributed, irrespective of the shape of the original population distribution.

This theorem proves especially useful when the population distribution is unknown or not normally distributed. As a rule of thumb, a sample size of 30 or more is often considered sufficient for the Central Limit Theorem to hold, which means that it can be reliably used to approximate the sampling distribution with a normal distribution, as seen in our example with a sample size of 64.

A normal distribution, also known as a Gaussian or bell curve, is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. In a perfectly normal distribution, the mean, median, and mode of the distribution are all equal.

The normal distribution is important in statistics due to its recurrence in numerous natural phenomena. For instance, heights, blood pressure, and test scores are often found to be approximately normally distributed. When a distribution is normal, we can fully describe it with just two parameters: the mean and the standard deviation. This makes it simpler to calculate probabilities for the occurrence of certain events within the distribution, as showcased in the calculation of the probabilities for the sample mean being within a specific range from the population mean.