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The Central Limit Theorem (CLT) is a fundamental concept in statistics. It explains how, for sufficiently large sample sizes, the distribution of the sample mean will approximate a normal distribution. This happens even if the original population distribution is not normal. In our problem with farm worker incomes that are skewed to the right, the CLT assures us that the sample mean distribution will still be approximately normal as long as the sample size is large enough, like the sample of 100 workers here.
\[ \bar{x} \sim N(\mu, \frac{\sigma^2}{n}) \]

Because of this theorem, we can confidently predict the behavior of the sample mean, which means we can use tools and properties of the normal distribution, such as the Empirical Rule, and convert distances to z-scores for calculations, despite the original skew in the data.
The CLT is essential in making inferences about a population mean based on a sample mean, as it allows us to estimate probabilities and make predictions about where the sample mean is likely to fall relative to the population mean.

The Empirical Rule is a key principle related to normally distributed datasets. It provides a quick way to understand the variability of data in itself and how much of the data falls within certain intervals. According to this rule:

• Approximately 68% of the data falls within one standard deviation of the mean.
• About 95% falls within two standard deviations.
• Almost 99.7% falls within three standard deviations.

In the context of our farm workers' income study, we use the Empirical Rule to predict that nearly all sample means will fall within three standard deviations of the true population mean. Here, with a calculated standard deviation of 16 for the sample mean (from our large sample size calculation), this means the sample mean will fall within \(500 \pm 48\) New Zealand dollars, thanks to this rule.
By using the Empirical Rule, we can support the idea that observing a sample mean within \(\pm48\) of the population mean is almost certain. This simplifies understanding how variable or consistent our sample mean might be, giving us confidence in its reliability.

Standard deviation is a critical measure in statistics, indicating how spread out the data points are from the mean. In the case of a normal distribution, it tells us how much individual data points differ from the average.
For the farm workers' income distribution, the given standard deviation is \(\sigma = 160\) dollars. When dealing with a sample mean, like in this example, we adjust the original standard deviation based on the sample size using the formula:
\[ \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} \]

For a sample of 100 workers, this leads to a standard deviation of \(\sigma_{\bar{x}} = 16\). This lower value for the sample means reflects less variability in the average of the sample compared to individual values, indicating how more data results in a more stable average.
Understanding standard deviation in this context helps clarify why our predictions about the sample mean using the Empirical Rule and CLT remain reliable. It emphasizes the sample's statistical stability, rendering our estimates both practical and interpretable.