Problem 26 - Basic Statistics for Business and Economics

Question

Refer to Exercise \(20,\) where the amount requested for home loans followed the normal distribution with a mean of \(\$ 70,000\) and a standard deviation of \(\$ 20,000\). a. How much is requested on the largest 3 percent of the loans? b. How much is requested on the smallest 10 percent of the loans?

Step-by-Step Solution

Answer

a. \$107,600 or more; b. Less than \$44,400.

Step 1: Identify Parameters

From the problem statement, we know that the loan amounts are normally distributed with a mean \( \mu = 70,000 \) and a standard deviation \( \sigma = 20,000 \).

Step 2: Largest 3 Percent: Find Z-score

To find the amount for the largest 3% of loans, we need the corresponding Z-score. We seek a Z-score such that the cumulative distribution function (CDF) equals 0.97 (since we want the top 3%). Using a Z-table or calculator, the Z-score is approximately 1.88.

Step 3: Compute Loan Amount for Largest 3 Percent

Use the Z-score of the normal distribution formula: \[X = \mu + Z \cdot \sigma = 70,000 + 1.88 \cdot 20,000 = 107,600.\]Thus, the largest 3% of loans are approximately \$107,600 or more.

Step 4: Smallest 10 Percent: Find Z-score

To determine the amount for the smallest 10% of loans, find the Z-score where the CDF equals 0.10. The corresponding Z-score is approximately -1.28.

Step 5: Compute Loan Amount for Smallest 10 Percent

Again, use the Z-score formula: \[X = \mu + Z \cdot \sigma = 70,000 + (-1.28) \cdot 20,000 = 44,400.\]Therefore, the smallest 10% of loans are less than \$44,400.

Key Concepts

Z-score

A Z-score, also known as a standard score, is a concept from statistics used to describe how far away a value is from the mean in terms of standard deviations. This means it tells us how many standard deviations an element is from the average. The formula for calculating a Z-score is:\[ Z = \frac{X - \mu}{\sigma} \]where:

• \(Z\) is the Z-score.
• \(X\) represents the value of the element.
• \(\mu\) is the mean of the data set.
• \(\sigma\) is the standard deviation.

Calculating Z-scores helps us understand where a particular value stands within a distribution.
For example, in the exercise, to find the loan amounts in the top 3%, a Z-score of approximately 1.88 was used, indicating that these amounts are 1.88 standard deviations above the mean.
Similarly, for the smallest 10% of loans, a negative Z-score of -1.28 shows those amounts are 1.28 standard deviations below the mean.

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. It tells us how much the values in a data set deviate from the average (mean) value.
The formula for standard deviation is:\[ \sigma = \sqrt{\frac{1}{N}\sum_{i=1}^{N}(X_i - \mu)^2} \]where:

• \(\sigma\) is the standard deviation.
• \(N\) is the number of data points.
• \(X_i\) represents each data point.
• \(\mu\) is the mean of the data set.

In simpler terms, a smaller standard deviation means the data points tend to be closer to the mean,
whereas a larger standard deviation indicates that the data spread out over a wider range of values.
In the presented exercise, the standard deviation \(\sigma\) was \(20,000\) dollars, showing how much the loan amounts deviate from the mean of \(70,000\) dollars.

Cumulative Distribution Function

The cumulative distribution function (CDF) is a function used in statistics to describe the probability that a real-valued random variable \(X\), with a given probability distribution, will take a value less than or equal to \(x\). For the normal distribution, the CDF is imperative for finding probabilities and Z-scores.
The CDF of the normal distribution increases from zero to one across the data set.
Thus, if we want to find a percentile, like the top 3% or bottom 10%, we use the CDF in reverse.

• Largest 3%: We find 0.97 in the CDF, as this encompasses all but the largest 3% of the distribution.
• Smallest 10%: We use a CDF value of 0.10 to find where the smallest 10% of values lie.

In this context, the CDF tells us the probability that the loan amount is less than or equal to a particular number.
This allows us to determine thresholds for specific percentages of the distribution, helping in data analysis and prediction.