91Ó°ÊÓ

The Z-score is a statistical measurement that describes a value's relationship to the mean of a data set. If you're tackling problems involving normal distributions, understanding Z-scores is essential. The mean is the middle point of your data, and the standard deviation tells us how spread out the data is. Together, they help us find out how far away a specific point (like 60,000 miles driven in a lease) is from the average.

To calculate a Z-score, you use the formula:

• \( Z = \frac{X - \mu}{\sigma} \)
• Where \( X \) is the value of interest (like those 60,000 miles), \( \mu \) is the mean (52,000 miles in our example), and \( \sigma \) is the standard deviation (5,000 miles).

Using these to calculate the Z-score for 60,000 miles, you find that:

• \[ Z = \frac{60,000 - 52,000}{5,000} = 1.6 \]

This Z-score tells us how many standard deviations away the 60,000 miles is from the mean. Essentially, a Z-score of 1.6 indicates that 60,000 miles is 1.6 standard deviations above the average mileage.

Once you have calculated a Z-score, you can use it to find probabilities related to the normal distribution. That's where the mileage penalty probability comes into play. By integrating the Z-score into a normal distribution table or using a calculator, you can identify the proportion of data above a certain threshold.

For the case where you exceed 60,000 miles, it's essential to calculate the chances of this happening. With a Z-score of 1.6:

• The cumulative probability of \( Z \leq 1.6 \) is approximately 0.9452, meaning that 94.52% of all samples fall below 60,000 miles.
• Conversely, the probability of being above 60,000 miles is \( 1 - 0.9452 = 0.0548 \) or 5.48%.

Thus, only about 5.48% of leases result in a penalty due to excess mileage, making these higher-mileage leases rather rare. Calculating this probability helps lessees understand their financial risk.

Percentiles tell you the relative position of a particular value within a data set. They are especially useful in the context of setting targets or limits in lease terms. Let's say an automobile company wants to adjust lease limits so that only a certain percentage of cars exceed a mileage and thus incur penalties.

In our problem, if the company desires that 25% of leases surpass the mileage cap, then 25% should exceed this new limit. This means we want to find the 75th percentile of the normal distribution.

• To find the 75th percentile, you look up the corresponding Z-score, which is approximately 0.674.
• Reversing the Z-score formula gives us the mileage limit:\[ X = \mu + Z \times \sigma = 52,000 + 0.674 \times 5,000 \]
• This results in a new limit of approximately 55,370 miles.

Setting the limit at 55,370 miles will ensure that 25% of the cars will exceed it, aligning with the policy intent and making it a calculable target for the lease agreement.