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The mean of sample means, often referred to as the expected value of the sampling distribution, is a cornerstone concept in sampling distribution theory. To determine this, we use the mean of the population from which samples are drawn. In our exercise, the population mean (\( \mu \) is 516. This value doesn't change when we calculate the average of multiple samples from this population. Here’s why it remains 516:

• The expected value of the sample means (\( \mu_{\bar{x}} \) should reflect the true mean of the whole population.
• Repeatedly drawing multiple samples of the same size will fluctuate around this population mean, but their average will eventually converge to it.

By understanding this, even if each of those samples gives slightly different averages due to random variability, overall, they are centered around the population mean. Thus, finding the mean of sample means is straightforward: it is equal to the population mean, here, 516.

The standard deviation of the sample means, commonly known as the standard error, provides insight into the variability or spread of sample means around the population mean. It tells us how much the average of a sample of observations is expected to vary from sample to sample, given the sample size.The formula to calculate this standard deviation is:\[ \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} \] where \( \sigma \) is the population standard deviation, and \( n \) is the sample size. In our context:

• \( \sigma \), the population standard deviation, is 116.
• \( n \), the number of observations within each sample, is 100.

Plugging in these numbers, the detailed calculation follows:\[ \sigma_{\bar{x}} = \frac{116}{\sqrt{100}} = \frac{116}{10} = 11.6 \] This number, 11.6, highlights that the sample means are expected to typically vary by about 11.6 units from the true population mean.

The Central Limit Theorem (CLT) is a foundational principle in statistics that explains why the means of samples drawn from a population tend to follow a normal distribution, even if the original population distribution is not normal, given a sufficiently large sample size.Key aspects of the Central Limit Theorem include:

• Sample Size: The theorem holds as long as the sample size is large enough, typically \( n \geq 30 \), making the shape of the sampling distribution approximately normal.
• Universality: Regardless of how skewed or non-normal the original population distribution is, the distribution of the sample mean approaches normality as the sample size increases.
• Application: This characteristic allows statisticians to make inferences about the population mean using the properties of the normal distribution (mean and standard deviation) derived from sample data.

In our exercise, with a sample size of 100, which is quite large, the CLT assures us that the distribution of these sample means will be normal. This is crucial for making reliable statistical predictions and analyses based on the sample data.