91Ó°ÊÓ

The Z-score is a statistical measurement that describes a value's position relative to the mean of a group of values. It is expressed in terms of standard deviations from the mean. For example, a Z-score of 0.674, as seen in this problem, indicates that the value is 0.674 standard deviations above the mean. Understanding Z-scores is crucial because they allow us to determine the probability of a value occurring in a normal distribution.

In this case, the Z-score helps identify the point at which 25% of data lies above it—this is what we call the 75th percentile. Calculating the Z-score involves referencing a Z-score table or using a calculator to find the corresponding value for a given percentile, providing a way to standardize different distributions for comparison.

Standard deviation (\( \sigma \)) measures the amount of variation or spread in a set of values. In a normal distribution, about 68% of values lie within one standard deviation of the mean. The formula to calculate standard deviation from a given Z-score is:

• \( Z = \frac{X - \mu}{\sigma} \)

Here, \(X\) is the value of interest, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.

If we know a specific value \(X\) and its Z-score, we can rearrange the formula to solve for \( \sigma \), as demonstrated in the solution:

• \( \sigma = \frac{X - \mu}{Z} \)

In the example given, the calculated standard deviation is 222.69, highlighting how much car insurance premiums can differ around the average in California.

A percentile indicates the relative standing of a value in a dataset by showing what percent of the data points fall below a particular value. For example, being in the 75th percentile means that 75% of observations are below that value. Percentiles are useful in understanding individual data points in context of the entire dataset and are often used to make comparisons.

In this problem, we are interested in determining the cutoff for insurance premiums that 75% of premiums are less than \(\\(1800\). Finding the Z-score corresponding to the 75th percentile allows us to see the position of \(\\)1800\) in the distribution. Using the concept of percentiles is essential in making informed decisions, especially when assessing how extreme a value is.

The mean, often referred to as the average, is the sum of all values divided by the number of values in a dataset. It is a measure of central tendency and provides a central reference point for where most values in a data set lie. In a normal distribution, the mean is right at the center of the curve.

In this exercise, the mean insurance premium is given as \(\$1650\). This mean is crucial because it is the point around which the standard deviation is calculated. It provides the necessary foundation for identifying percentiles such as where the cutoff for the 75th percentile lies. Understanding the mean is fundamental for interpreting various measures of distribution such as standard deviation and Z-scores.