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In statistics, the concept of a normal distribution is fundamental. It often mirrors real-world situations. Imagine it as a bell-shaped curve where most occurrences take place near the mean. This mean is an average value around which data is distributed.
The essential properties of a normal distribution include:

• Symmetry: distribution is symmetric about the mean.
• Mean, median, and mode are equal.
• Curve is bell-shaped and approaches but never touches the horizontal axis.

A normal distribution is characterized by its mean and standard deviation. These two components give a complete description of the distribution. For example, in the PDI score case, the distribution has a mean of 100 and a standard deviation of 15. This setup allows us to make inferences about where a particular score falls relative to the average.

The Z-score is a statistical measurement that tells us how far a particular value is from the mean. It's expressed in terms of standard deviations.
To compute a Z-score for a value, use the formula:\[ z = \frac{x - \mu}{\sigma} \]where:

• \(x\) is the particular score.
• \(\mu\) is the mean of the distribution.
• \(\sigma\) is the standard deviation.

For a PDI value of 90, the Z-score is \(-0.67\), indicating that this score is below the mean but within the typical range one might expect in a normal distribution. In simple terms, a Z-score like \(-0.67\) indicates the value isn't unusual since it's less than one standard deviation away from the mean.

A sampling distribution refers to the probability distribution of a statistic obtained from a large number of samples drawn from a specific population.
In practice, we create a sampling distribution of the sample mean to understand how the mean of data samples from the population varies. When you take a large enough sample, the sampling distribution of the sample mean will tend to have a normal distribution, even if the original data is not perfectly normal. This is known as the Central Limit Theorem.
The sampling distribution helps us understand the precision of our sample mean in representing the population mean. For instance, with a sample size of 225 from the PDI distribution, we can create a distribution of sample means. This distribution has its own mean (same as the population mean) and a smaller standard deviation, known as the standard error.

The standard error provides a way to measure the variability of the sample mean compared to the population mean. It essentially tells us how precisely the sample mean estimates the population mean.
The formula to calculate standard error is:\[ SE = \frac{\sigma}{\sqrt{n}} \]where:

• \(\sigma\) is the standard deviation of the population.
• \(n\) is the sample size.

In the given exercise, with a sample size of 225 and a standard deviation of 15, the standard error becomes 1. This small standard error means the sample mean is a very precise estimate of the population mean. Hence, when a sample mean has a Z-score of \(-10\), it becomes a surprising result, indicating that this sample mean is remarkably far from what we would expect for the average sample mean, making it an unusual occurrence.