91Ó°ÊÓ

For borrowers with good credit scores, the mean debt for revolving and installment accounts is \(\$ 15,015\) (BusinessWeek, March 20,2006 ). Assume the standard deviation is \(\$ 3540\) and that debt amounts are normally distributed. a. What is the probability that the debt for a randomly selected borrower with good credit is more than \(\$ 18,000 ?\) b. What is the probability that the debt for a randomly selected borrower with good credit is less than \(\$ 10,000 ?\) c. What is the probability that the debt for a randomly selected borrower with good credit is between \(\$ 12,000\) and \(\$ 18,000 ?\) d. What is the probability that the debt for a randomly selected borrower with good credit is no more than \(\$ 14,000 ?\)

Step by step solution

01

Define the Given Parameters

The problem states: mean debt \( \mu = 15,015 \), standard deviation \( \sigma = 3540 \). The debt is normally distributed. We will use this information to find the probability for each scenario by calculating the z-scores.

02

Step 2a: Calculate Z-score for More than $18,000

To find the probability that the debt is more than \$18,000, calculate the z-score using the formula: \[ z = \frac{X - \mu}{\sigma} \]Where \( X = 18,000 \). Substituting the given values:\[ z = \frac{18,000 - 15,015}{3540} \approx 0.84 \]

03

Step 3a: Find Probability for Z-score in Step 2a

Using the standard normal distribution table, a z-score of 0.84 corresponds to a cumulative probability of approximately 0.8008. Since we are looking for the probability of more than \$18,000, we find:\[ P(X > 18,000) = 1 - P(Z < 0.84) = 1 - 0.8008 = 0.1992 \]

04

Step 2b: Calculate Z-score for Less than $10,000

To find the probability that the debt is less than \$10,000, calculate the z-score:\[ z = \frac{10,000 - 15,015}{3540} \approx -1.42 \]

05

Step 3b: Find Probability for Z-score in Step 2b

Using the standard normal distribution table, a z-score of -1.42 corresponds to a cumulative probability of approximately 0.0778. Thus, \[ P(X < 10,000) = 0.0778 \]

06

Step 2c: Calculate Z-scores for $12,000 and $18,000

For \\(12,000:\[ z = \frac{12,000 - 15,015}{3540} \approx -0.85 \]For \\)18,000 (already calculated in Step 2a):\[ z = 0.84 \]

07

Step 3c: Find Probability Between $12,000 and $18,000

Using the standard normal distribution table, find the cumulative probabilities:- For \( z = -0.85 \), probability is approximately 0.1977.- For \( z = 0.84 \), probability is 0.8008.Thus, \[ P(12,000 < X < 18,000) = 0.8008 - 0.1977 = 0.6031 \]

08

Step 2d: Calculate Z-score for No More than $14,000

To find the probability that the debt is no more than \$14,000, calculate the z-score:\[ z = \frac{14,000 - 15,015}{3540} \approx -0.29 \]

09

Step 3d: Find Probability for Z-score in Step 2d

Using the standard normal distribution table, a z-score of -0.29 corresponds to a cumulative probability of approximately 0.3859. Thus, \[ P(X \leq 14,000) = 0.3859 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Normal Distribution

Normal distribution is a fundamental concept in statistics, describing how a set of data is spread out. Imagine it as a bell-shaped curve, where most of the data points cluster around the middle, and fewer points appear as you move away from the center. For example, in our exercise about credit scores, the debt amounts for borrowers are normally distributed. This means that most people have debts around the average, which is the mean debt of $15,015.
Normal distribution has a few key properties:

• It is symmetric around the mean.
• The mean, median, and mode are all equal.
• The area under the curve represents the total probability, which equals 1.
• Approximately 68% of data falls within one standard deviation of the mean.

The solutions to our probability problems are derived from this distribution, allowing us to understand how likely certain debt amounts are.

Z-score Calculation

A Z-score helps you determine how far a specific data point is from the mean, in terms of standard deviations. This is a very common technique in statistics whenever normal distribution is involved. For example, in our exercise, Z-scores were used to compare specific debts, like \(18,000 and \)10,000, to the mean debt.
Here is how Z-score calculation works:

• Use the Formula: \( z = \frac{X - \mu}{\sigma} \)
• \( X \) is the value for which you want to find the Z-score.
• \( \mu \) is the mean of the distribution.
• \( \sigma \) is the standard deviation.

A positive Z-score means the data point is above the mean, while a negative Z-score indicates it's below the mean. For our calculations, this was crucial in determining the probability of having a debt higher or lower than certain amounts.

Probability Concepts

Probability concepts are at the heart of questions involving normal distribution and Z-scores. In our exercise, we used these ideas to calculate the likelihood of various debt scenarios.
Key Probability Concepts include:

• Cumulative Probability: This is the probability that a random variable takes on a value less than or equal to a specific value. It's found in the standard normal distribution table.
• Complementary Probability: Sometimes we need to calculate the probability of something not happening. This is done by subtracting the cumulative probability from 1.

These probabilities help us answer questions like: What is the chance a debt is more than $18,000? By finding cumulative and complementary probabilities, we obtain precise answers to such queries.

Credit Score Analysis

Credit score analysis involves understanding the financial behaviors which contribute to a person's creditworthiness. In the context of our exercise, it's about analyzing debts held by people with good credit scores. The mean and standard deviation give us insights into how much debt is typically held. This understanding helps lenders assess risk and determine the terms of credit for individuals.
By employing statistical techniques, we can also predict potential future outcomes. For instance:

• Identify the likelihood of someone having a debt lower or higher than average.
• Understand the variance in financial behavior among different individuals.
• Aid in decision making for both consumers and lenders based on risk assessments.

Through credit score analysis, lenders can make informed decisions and consumers understand their financial standing better.