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The z-score is an essential concept when working with normal distributions, like in our exercise about mortgage payments. Z-score helps us understand how far away a particular data point is from the mean, in terms of standard deviations. For instance, if the mean monthly mortgage is \(2850, and you're interested in a payment of \)1200, you can use the z-score formula:

• \( Z = \frac{X - \mu}{\sigma} \)

Here, \(X\) is the data point in question, \(\mu\) is the mean, and \(\sigma \) is the standard deviation. Calculating a z-score is helpful because it translates values into a common scale: the number of standard deviations below or above the mean.
The z-score allows us to compare data points from different normal distributions or to identify how uncommon or common a particular data point is in the context of its specific distribution.

Standard deviation is a critical measure when analyzing normal distribution data. It helps us understand the amount of variation or spread in a set of values. In our exercise, the standard deviation of the monthly mortgage payments is \( \\(420 \). This indicates that most homeowners' payments won't deviate too far from the average of \( \\)2850 \).
The closer the data points are to the mean, the smaller the standard deviation, indicating that most people pay around the average amount. Conversely, a larger standard deviation would imply more significant differences among individuals' payments.A practical understanding of standard deviation helps with interpreting a normal distribution curve on a graph:

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

This helps you predict the likelihood of seeing a particular data point in the distribution.

In the context of normal distribution and our exercise, the mean value is the sum of all data points divided by the number of data points, often referred to as the average. For these mortgage payments, the mean value is \( \$2850 \).
The mean serves as the "central" point of our normal distribution curve, with half of the values lying above it and the other half below it. Understanding the mean is essential as it represents the typical value of the data set, providing a baseline for comparison.
In a nicely symmetrical normal distribution, the mean also coincides with the peak of the curve. For data analysis, the mean is critical in calculating z-scores and understanding what is "ordinary" versus "extraordinary" within the given data set.

Calculating probabilities within a normal distribution is a valuable skill, especially when determining the likelihood of certain events occurring. Once you have the z-scores for particular data points, as we saw with the mortgage payment ranges provided, they can be used to look up probabilities on the Standard Normal Distribution table.
This table gives the probability that a standard normal variable \( Z \) is less than or equal to a specific z-score:

• To find the probability that a value is less than a certain number, locate the z-score in the table.
• To determine probability between two values, find their z-scores and subtract their respective probabilities.

Being skilled in probability calculation allows for making informed predictions and decisions based on statistical data to handle queries like those in the exercise effectively.