91Ó°ÊÓ

In statistics, the Z-score is a crucial concept that helps us understand where a data point lies in relation to the mean of a distribution. The Z-score formula is given by: \[ Z = \frac{X - \mu}{\sigma} \]where:

• \(X\) is the value of the data point.
• \(\mu\) is the mean of the distribution.
• \(\sigma\) is the standard deviation of the distribution.

The Z-score expresses how many standard deviations a data point is from the mean. A positive Z-score indicates the value is above the mean, while a negative Z-score shows it is below. For example, in part (a) of our problem, we calculated the Z-score for 60,000 miles as 1.6, meaning it is 1.6 standard deviations above the mean. Knowing this helps us find probabilities associated with the normal distribution, which is particularly useful for determining penalties.

A probability distribution describes how the probabilities of various outcomes are distributed for a random variable. The normal distribution, or Gaussian distribution, is one of the most common probability distributions in statistics. It is characterized by its bell-shaped curve and is defined by two parameters: the mean (\(\mu\)) and the standard deviation (\(\sigma\)).
In our exercise, the distribution of miles driven falls under the normal distribution with a mean of 52,000 miles and a standard deviation of 5,000 miles. This information allows us to calculate specific probabilities, such as those needed for excess mileage penalties or defining low-mileage returns.
When we refer to a value like 60,000 miles, we can use the Z-score to determine what percentage of the total data lies beyond this point. Using a standard normal distribution table, we found that approximately 5.48% of leases will have penalties, corresponding to the excess miles driven.

The standard deviation is a measure of how spread out the numbers in a data set are. In a normal distribution, most data points will lie within one standard deviation of the mean. Standard deviation helps us to quantify the amount of variation or dispersion in a set of values.
In the context of our exercise, the standard deviation of 5,000 miles tells us how much the mileage on the leases typically varies from the mean of 52,000 miles. A smaller standard deviation would indicate that the mileages are closely packed around the mean, whereas a larger one would show more spread out values.

• It’s crucial for Z-score calculation, as the Z-score tells us how many standard deviations away a particular value is from the mean.
• The standard deviation directly impacts the shape of the normal distribution; a larger standard deviation results in a "flatter" bell curve.

Understanding standard deviation helps us make sense of the variability in the leases and their related probabilities, such as determining what constitutes a low-mileage return.