The average amount of financial losses in a divorce is $21,457, according to one survey. Assume that the variable is approximately normally distributed and the standard deviation is $3,000. If a randomly selected divorced person is chosen, find the probability that a person has spent less than $26,000?

We can solve this problem by finding the probability that a randomly selected divorced person has spent less than $26,000. Since the distribution of financial losses is approximately normal, we will use the standard normal distribution (Z-distribution) to solve this.

### Steps:

1. **Identify the given information:**
   - Mean ($\mu$) = $21,457
   - Standard deviation ($\sigma$) = $3,000
   - We are looking for $ P(X < 26,000) $.

2. **Convert the value to a Z-score:**

   The Z-score is calculated by the formula:
   \[
   Z = \frac{X - \mu}{\sigma}
   \]
   where:
   - $ X $ is the value we are interested in ($26,000),
   - $ \mu $ is the mean ($21,457),
   - $ \sigma $ is the standard deviation ($3,000).

   Substituting the values:
   \[
   Z = \frac{26,000 - 21,457}{3,000} = \frac{4,543}{3,000} \approx 1.514
   \]

3. **Find the probability associated with the Z-score:**

   Now we use the Z-score table or a calculator to find the probability corresponding to $ Z = 1.514 $. From the Z-table:
   \[
   P(Z < 1.514) \approx 0.9351
   \]

4. **Interpret the result:**

   The probability that a randomly selected divorced person has spent less than $26,000 is approximately 0.9351, or 93.51%.

### Conclusion:
There is about a 93.51% chance that a randomly selected divorced person has financial losses of less than $26,000.

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Do you need any further details or have any questions?

Here are 5 related questions to explore:

1. What is the probability that a randomly selected divorced person has spent more than $26,000?
2. What is the probability that a person has spent between $20,000 and $26,000?
3. How would the probability change if the standard deviation was $4,000 instead of $3,000?
4. What financial amount corresponds to the top 10% of losses in a divorce?
5. What is the Z-score for someone who lost exactly $30,000?

**Tip:** To quickly find probabilities for a normal distribution, you can always use Z-scores to standardize the data!

Discover how to calculate the probability that a randomly selected divorced person has financial losses less than $26,000. This problem covers key statistical concepts, including normal distribution, Z-scores, and the standard normal distribution theorem. The mean loss is $21,457 with a standard deviation of $3,000, and the solution is ideal for students in Grades 11-12.

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