Problem 49 - Basic Statistics for Business and Economics

Question

Most four-year automobile leases allow up to 60,000 miles. If the lessee goes beyond this amount, a penalty of 20 cents per mile is added to the lease cost. Suppose the distribution of miles driven on four-year leases follows the normal distribution. The mean is 52,000 miles and the standard deviation is 5,000 miles. a. What percent of the leases will yield a penalty because of excess mileage? b. If the automobile company wanted to change the terms of the lease so that 25 percent of the leases went over the limit, where should the new upper limit be set? c. One definition of a low-mileage car is one that is 4 years old and has been driven less than 45,000 miles. What percent of the cars returned are considered low-mileage?

Step-by-Step Solution

Answer

a. 5.48% incur a penalty. b. New upper limit: 55,373 miles. c. 8.08% are low-mileage.

Step 1: Calculate Z-score for Penalty

To find the percentage of leases that result in a penalty, we calculate the Z-score for 60,000 miles using the formula: \( Z = \frac{X - \mu}{\sigma} \), where \( X = 60,000 \), \( \mu = 52,000 \), and \( \sigma = 5,000 \). Thus, \( Z = \frac{60,000 - 52,000}{5,000} = \frac{8,000}{5,000} = 1.6 \).

Step 2: Find Probability Above Z-score

Using the standard normal distribution table, find the probability corresponding to a Z-score of 1.6. The probability of a Z-score less than 1.6 is approximately 0.9452. Thus, the proportion exceeding 60,000 miles is \(1 - 0.9452 = 0.0548 \), or 5.48%.

Step 3: Calculate Z-score for New Upper Limit

Find the Z-score corresponding to 25% of leases exceeding the limit. This means 75% are within the limit, or 0.75 probability. From a standard normal table, the Z-score for 0.75 is approximately 0.6745.

Step 4: Calculate the New Upper Mileage Limit

Using the Z-score formula rearranged to solve for \( X \), \( X = Z \cdot \sigma + \mu \). Substitute the values: \( X = 0.6745 \cdot 5,000 + 52,000 = 3,372.5 + 52,000 = 55,372.5 \) miles. Round to 55,373 miles.

Step 5: Calculate Z-score for Low-Mileage Definition

Calculate the Z-score for 45,000 miles: \( Z = \frac{45,000 - 52,000}{5,000} = \frac{-7,000}{5,000} = -1.4 \).

Step 6: Find Probability Below Z-score for Low-Mileage

Using the standard normal distribution table, the probability for a Z-score of -1.4 is approximately 0.0808. Hence, 8.08% of cars are considered low-mileage.

Key Concepts

Z-score

The Z-score is a measurement that describes a value's relation to the mean of a group of values. Typically, it is used in normal distributions to determine how far away a number is from the mean, measured in units of standard deviation.
The Z-score is calculated using the formula: \( Z = \frac{X - \mu}{\sigma} \) where \( X \) is the value in question, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.

• A positive Z-score indicates the value is above the mean.
• A negative Z-score indicates it is below the mean.
• A Z-score of zero means the value is exactly at the mean.

Understanding the Z-score helps in knowing how rare a particular measurement is compared to the general data. It is especially useful when comparing values from different distributions.

Probability

Probability is the measure of the likelihood that a given event will occur. In the context of standard normal distribution and Z-scores, it is used to determine how likely it is for a value to be greater or less than a specific Z-score.
For example, once you know the Z-score, you can use a standard normal distribution table to find the probability associated with that score.

• The probability indicates the area under the normal curve up to a certain point.
• A probability close to 1 indicates a high likelihood, whereas close to 0 indicates a low likelihood.
• In the car leasing scenario, finding out what percent of leases result in a penalty is about finding the probability associated with exceeding a certain threshold.

This concept is foundational in statistics, helping us make critical decisions based on probabilistic outcomes.

Standard Deviation

Standard deviation is a key concept in statistics, representing a measure of the amount of variation or dispersion in a set of values. A lower standard deviation indicates that the values tend to be close to the mean, while a higher standard deviation indicates the values are spread out over a wider range.
Mathematically, the standard deviation is the square root of the variance, which is the average of the squared differences from the Mean.

• In our exercise, a standard deviation of 5,000 miles indicates how the actual miles deviate from the average.
• Larger deviations might indicate wider mileage variety among leased cars.
• It plays a pivotal role in defining the shape of the normal distribution, concentrating data around the mean.

Understanding the standard deviation helps in assessing the risk and variability, which is useful in making informed decisions.

Mean

The mean, often referred to as the average, is a central value of a set of numbers. It is found by adding up all the numbers and then dividing by the total count of the numbers.
Mathematically, the mean is expressed as \( \mu = \frac{\sum X}{N} \), where \( \sum X \) is the sum of all values and \( N \) is the number of values.

• In the context of the car leasing problem, the mean mileage \( \mu \) is 52,000 miles.
• It acts as the baseline from which other calculations, like the Z-score, are derived.
• The mean indicates what is typical or expected in a dataset.

Knowing the mean helps us understand the general trends present within a group of observations and serves as a point of comparison for individual data points.