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Determining the sample size is crucial in statistics to ensure that the sample accurately represents the population. In the context of this exercise, we're tasked with finding out how many corporations must be sampled for the sample mean's standard deviation to match our specified value of \( \$ 15.50 \) million.

To achieve this, we rearrange the formula for the standard deviation of the sample mean. The formula is:

• \( \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} \)

By solving this equation for \( n \), we can find the sample size necessary to meet the criteria:

• \( n = \left( \frac{\sigma}{\sigma_{\bar{x}}} \right)^2 \)

With \( \sigma \) being the population standard deviation and \( \sigma_{\bar{x}} \) the desired standard deviation of the sample mean, this formula guides us to the appropriate sample size.

Population standard deviation is a measure of how much individual data points in a full population differ from the overall population mean. In this exercise, the population standard deviation is given as \( \$ 139.50 \) million.

This value provides insight into the general variability of corporate sales figures in 2011. A large standard deviation indicates significant variability in sales figures among corporations.

Understanding population standard deviation is important as it establishes the baseline for any statistical measures performed on samples taken from this population. Knowing this figure allows us to accurately calculate other statistics, like the sample standard deviation or mean.

The mean of a sample, often denoted as \( \bar{x} \), provides the average value within a subset of the population. When we talk about the mean of sample in the context of this exercise, we are dealing with the average gross sales from our sampled corporations in 2011.

The sample mean gives us a snapshot of the population mean but on a smaller scale, making it easier to analyze and understand trends without needing to calculate data for every single entity in the population. It acts as a reliable estimator for the population mean when the sample size is adequately large.

Understanding the formula for sample standard deviation is key to grasping sample variability. The sample standard deviation helps determine how much the sample's values deviate from the sample mean. The formula used in this context is:

• \( \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} \)

Here, \( \sigma \) is the population standard deviation, and \( n \) is the sample size. This formula is central to controlling and predicting the variability of the sample mean.

In practical applications, calculating the sample standard deviation enables statisticians and researchers to allocate resources effectively. It considers how likely the sample mean will represent the population mean, allowing for more accurate predictions and decisions.