https://file.aiquickdraw.com/mathbot/file/1729472410007em1kbn4l.jpg

The problem asks us to calculate two values related to the rental prices of apartments based on a sample of 50 apartments, focusing on how much a client should expect to pay for a one-bedroom apartment that falls within the top 15% of rentals. The following steps outline the solution:

### 1. **Calculate the Standard Error of the Mean (σₓ̅):**
   
   The standard error (SE) of the sample mean is calculated by dividing the population standard deviation (σ) by the square root of the sample size (n).

   \[
   \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}
   \]

   Given:
   - Population standard deviation (σ) = \$400
   - Sample size (n) = 50

   \[
   \sigma_{\bar{x}} = \frac{400}{\sqrt{50}} \approx \frac{400}{7.071} \approx 56.57
   \]

   So, the standard error of the mean is approximately **\$56.57**.

### 2. **Calculate the z-score for the Top 15%:**

   The z-score corresponding to the top 15% of the normal distribution can be found using statistical tables or a calculator. The z-score for the 85th percentile (since we are interested in the top 15%, which is the complement of 85%) is approximately **1.036**.

### 3. **Calculate the Expected Rent (x̅) for the Top 15%:**

   To find the expected rent for apartments in the top 15%, we use the formula for the mean:

   \[
   \bar{x} = \mu + z \cdot \sigma_{\bar{x}}
   \]

   Where:
   - Population mean (μ) = \$2,000
   - z = 1.036 (for the top 15%)
   - σₓ̅ ≈ 56.57 (calculated above)

   \[
   \bar{x} = 2000 + (1.036 \times 56.57) \approx 2000 + 58.62 = 2058.62
   \]

   So, the expected rent for a one-bedroom apartment in the top 15% is approximately **\$2058.62**.

### Final Answers:
- **σₓ̅ ≈ \$56.57**
- **Expected Rent ≈ \$2058.62**

Would you like more detailed steps or clarifications? Here are some related questions to further your understanding:

1. What is the significance of the z-score in this calculation?
2. How is the standard error different from the standard deviation?
3. Why do we use the z-score for the 85th percentile in this problem?
4. How would the result change if we surveyed 100 apartments instead of 50?
5. What does the standard error tell us about the distribution of rental prices?

**Tip:** When calculating for top percentages, always ensure that you're looking at the complement (e.g., top 15% uses the 85th percentile).

The average one-bedroom apartment rents for $2,000 a month in Washington, D.C., with a standard deviation of $400. A real estate agent is looking to find a one-bedroom apartment for a client that is in the top 15% of all rentals in the city. Use a calculator to find how much a client should expect to pay for a one-bedroom apartment in the top 15% of rentals if the agent only surveys 50 apartments in the city. Round σₓ̅ and x̅ to two decimal places.

Calculate the expected rent for a one-bedroom apartment in Washington, D.C. that falls in the top 15% of rentals. Learn how to apply the standard error formula and z-scores to estimate rental prices, using a sample size of 50 apartments. This problem involves normal distribution and standard error calculations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation