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A normal distribution is a common pattern found in statistics, which appears graphically as a symmetric bell-shaped curve. Most of the data points in a normal distribution lie around the mean, with fewer data points at the extremes. This means that if you randomly select a data point from a normally distributed dataset, it's more likely to be close to the mean, rather than far off in the tails. A normal distribution is fully characterized by its mean and standard deviation. It's used extensively in statistical analysis because many variables are naturally approximately normally distributed, such as heights, test scores, and yes, even labor costs of fixing heat pumps! In this context, understanding that labor costs follow a normal distribution allows us to use statistical tools like the Z-score to make sense of how unusual or typical specific costs are within the dataset.

Standard deviation is a key concept in statistics that measures the amount of variability or dispersion in a dataset. A small standard deviation means the data points tend to be close to the mean, while a large standard deviation means they are spread out over a wider range. For example, in the exercise, the standard deviation of the heat pump labor costs is \(22\). This tells us that most labor costs are within \(22\) dollars of the mean cost of \(90\) dollars.

When interpreting standard deviation, remember:

• A standard deviation of zero means no variation (all data points are the same).
• A low standard deviation suggests data points are generally similar.
• A high standard deviation indicates that the data points vary widely from the mean.

In the example given, the standard deviation helps to quantify how typical or atypical a particular repair cost is.

The mean, often referred to as the average, is a measure of the center of a data set. It's calculated by adding up all the values and dividing by the number of values. In statistical terms, it's a type of "expected value."

Here, the mean labor cost for heat pump repairs is given as \(90\) dollars. This tells us that if you were to gather all the costs of such repairs and average them, you would get \(90\) dollars. It serves as a point of comparison for individual data points to see how typical or atypical they are.
The mean is foundational in calculating the Z-score, which helps determine how far a specific value (like the cost of individual repairs) deviates from the typical or average scenario.

Statistical analysis involves collecting, reviewing, and drawing conclusions from data. In the exercise, statistical analysis helps us understand the variation between different costs of repairing heat pumps. One of the tools used in this analysis is the Z-score, which indicates how far and in what direction a data point deviates from the mean.

To calculate a Z-score:

Subtract the mean from the data point.

Divide by the standard deviation.

This converts the value to a "standardized" form, useful for comparison. In the exercise, calculating the Z-scores for labor costs of \(75\) and \(100\) helps us understand their relationship to the average cost of \(90\).
The Z-scores indicate that these costs are within one standard deviation of the mean, demonstrating that they are in the typical range observed by the service provider.