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The mean is a fundamental statistical measure, often known as the average. It's calculated by adding up all the values in a data set and dividing by the number of values. In this way, the mean provides a central point for the data. In our case of monthly mortgages, the mean is given as \(\\(2365\), which means that when you consider all the homeowners in the town, their average mortgage payment per month is \(\\)2365\). This figure serves as a benchmark for analyzing how individual mortgage payments vary across homeowners.

The mean situates our data and acts as a helpful reference when applying other concepts like standard deviation and Chebyshev's theorem. By understanding the average mortgage, we can better grasp the spread of the data and identify how many homeowners pay significantly more or less than this standard.

Standard deviation measures the amount of variation or dispersion in a set of data points. In simpler terms, it tells us how much the individual data points deviate from the mean. For the town's mortgage payments, the standard deviation is \(\\(340\).

A smaller standard deviation means that the values tend to be closer to the mean, while a larger one indicates more spread out data. Here, the mortgage payments vary by about \(\\)340\) from the mean on average. This helps us understand the diversity in mortgage payments among homeowners.

• Small standard deviation: Data points are closer to the mean
• Large standard deviation: Data points are more spread out

Understanding standard deviation is crucial when applying Chebyshev's theorem. It allows us to calculate the range around the mean where most of the data points lie.

Probability distributions describe how values are distributed across possible outcomes. In the context of our exercise, we're particularly looking at how often mortgage payments fall within certain ranges.

Chebyshev's theorem applies to any data distribution, providing minimum certainty of data falling within a specific range of standard deviations from the mean. It states that at least \(1 - \frac{1}{k^2}\) of data points fall within \(k\) standard deviations of the mean. In the problem, we see this used to estimate the proportion of homeowners that fall within certain payment brackets:

• For payments between \(\\(1685\) and \(\\)3045\): At least \(75\%\) of homeowners.
• For payments between \(\\(1345\) and \(\\)3385\): At least \(88.9\%\) of homeowners.

While the probabilities given by Chebyshev’s theorem are quite broad, they are extremely useful for datasets without assuming any specific distribution like normal distribution.

Interval estimation involves finding a range of values that likely contain a parameter (like the mean) with a certain level of confidence. In many cases, this could be a confidence interval, but here we're using Chebyshev's theorem to determine the interval for mortgage payments.

In part b of the exercise, we sought the range which contains at least \(84\%\) of all mortgage payments using the theorem. By rearranging the theorem's formula, we determined that \(k ≈ 1.22\). From this, we calculated that the interval is approximately between \(\\(1909.02\) and \(\\)2820.98\).

This range estimation does not particularly depend on a normal distribution, unlike traditional confidence intervals. Instead, it assures us that a certain percentage of values falls within this calculated range regardless of the actual distribution shape. This flexibility makes Chebyshev's theorem valuable for real-world applications, where exact distribution forms aren't always known.