The monthly returns for a financial advisory service can be modeled by a Normal distribution with a mean of $140 and standard deviation of $58, per $10,000 invested. Find the following boundaries: (use 4 decimals for all answers)

(a) the highest 25% of monthly returns: 

(b) the lowest 25% of monthly returns: 

(c) the middle 40% of monthly returns: 
 and 
 (Enter the lower value first.)

The problem requires calculating the boundaries of a normal distribution with a mean ($\mu$) of 140 and a standard deviation ($\sigma$) of 58, which represents the monthly returns for a financial advisory service.

We need to find the values corresponding to specific percentiles or percentages of the distribution.

### Step-by-Step Breakdown:

For a standard normal distribution, we use the **z-score** formula:

\[
Z = \frac{X - \mu}{\sigma}
\]

Where:
- $Z$ is the standard score (z-score),
- $X$ is the value in the normal distribution,
- $\mu$ is the mean,
- $\sigma$ is the standard deviation.

We will use the cumulative distribution function (CDF) of the standard normal distribution to determine the z-scores that correspond to certain percentiles, and then convert those z-scores back to actual values.

### (a) The highest 25% of monthly returns

The top 25% corresponds to the 75th percentile (since the highest 25% means everything greater than the 75th percentile).

From the z-score table, the z-score corresponding to the 75th percentile is approximately **0.6745**.

Now, we convert the z-score back to an actual value using the z-score formula:

\[
X = Z \cdot \sigma + \mu = 0.6745 \cdot 58 + 140
\]

Let's calculate that.

### (b) The lowest 25% of monthly returns

The lowest 25% corresponds to the 25th percentile.

From the z-score table, the z-score corresponding to the 25th percentile is approximately **-0.6745**.

Now, we convert this z-score back to an actual value using the z-score formula:

\[
X = Z \cdot \sigma + \mu = -0.6745 \cdot 58 + 140
\]

Let's calculate that.

### (c) The middle 40% of monthly returns

The middle 40% means the range between the 30th percentile and the 70th percentile (since 100% - 40% = 60%, and 60%/2 = 30% on either side).

From the z-score table:
- The z-score for the 30th percentile is approximately **-0.5244**.
- The z-score for the 70th percentile is approximately **0.5244**.

Now, we convert these z-scores to actual values:

For the 30th percentile:
\[
X_{\text{lower}} = -0.5244 \cdot 58 + 140
\]

For the 70th percentile:
\[
X_{\text{upper}} = 0.5244 \cdot 58 + 140
\]

Let's calculate all of these values.Here are the boundaries for the monthly returns based on the normal distribution with a mean of $140 and standard deviation of $58:

(a) The highest 25% of monthly returns is **$179.1210** or greater.

(b) The lowest 25% of monthly returns is **$100.8790** or less.

(c) The middle 40% of monthly returns falls between **$109.5848** and **$170.4152**.

Do you have any further questions or need additional details?

Here are 5 related questions to expand on this topic:
1. How would the boundaries change if the standard deviation were increased?
2. What percentage of returns are expected to fall between $120 and $160?
3. How would you calculate the probability that the monthly return is less than $90?
4. What is the z-score for a monthly return of $200?
5. How would the boundaries change if the mean were $160 instead of $140?

**Tip**: When calculating percentiles for a normal distribution, you can always reference z-tables or use statistical software for greater accuracy!

The monthly returns for a financial advisory service can be modeled by a Normal distribution with a mean of $140 and standard deviation of $58, per $10,000 invested. Find the following boundaries: (a) the highest 25% of monthly returns, (b) the lowest 25% of monthly returns, (c) the middle 40% of monthly returns.

Explore how to calculate the highest 25%, lowest 25%, and middle 40% of monthly returns modeled by a normal distribution with a mean of $140 and standard deviation of $58. This problem involves key statistical concepts such as Z-scores and percentiles. Suitable for college-level statistics students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation