The average home price in Massachusetts is $410,000, with a standard deviation of $65,000. A city planner is studying the distribution across the state of home prices in the bottom 10%. 

Use a calculator to find the price that will indicate that a home is in the bottom 10% if the city planner examines 200 homes around the state.  

Round σ¯x and ¯x  to two decimal places. 

To solve this problem, we'll find the price that corresponds to the bottom 10% of home prices, using the normal distribution.

Given:
- Mean home price ($\mu$) = \$410,000
- Standard deviation ($\sigma$) = \$65,000
- Number of homes ($n$) = 200

We need to find the 10th percentile of the distribution of home prices. Here are the steps to find the solution:

1. **Find the Z-score for the 10th percentile:**

   The Z-score corresponding to the 10th percentile (bottom 10%) is approximately -1.28. This can be found using a Z-table or standard normal distribution calculator.

2. **Calculate the standard error of the mean ($\sigma_{\bar{x}}$):**

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

   \[
   \sigma_{\bar{x}} = \frac{65,000}{\sqrt{200}} = \frac{65,000}{14.1421} \approx 4596.19
   \]

3. **Calculate the mean of the sampling distribution ($\bar{x}$):**

   \[
   \bar{x} = \mu = 410,000
   \]

4. **Find the price corresponding to the 10th percentile:**

   Using the Z-score formula:
   
   \[
   X = \mu + Z \cdot \sigma_{\bar{x}}
   \]

   Substituting the values:
   
   \[
   X = 410,000 + (-1.28) \cdot 4596.19
   \]

   \[
   X = 410,000 - 5883.13 \approx 404,116.87
   \]

So, the price that indicates a home is in the bottom 10% is approximately \$404,116.87.

### Summary:
- $\sigma_{\bar{x}} \approx 4596.19$
- $\bar{x} = 410,000$
- The price that indicates a home is in the bottom 10% is approximately \$404,116.87.

Would you like more details or have any questions?

Here are five related questions you might want to ask:
1. How do you find Z-scores for other percentiles?
2. What does the standard error of the mean represent?
3. How does the sample size affect the standard error?
4. What assumptions are made when using the normal distribution in this context?
5. How can the central limit theorem be applied to this problem?

**Tip:** Understanding the Z-score and its role in the normal distribution can greatly enhance your ability to interpret statistical data and make predictions based on sample means.

Learn how to find the price that indicates a home is in the bottom 10% of home prices in Massachusetts using statistical methods. Given a mean home price of $410,000 and a standard deviation of $65,000, this problem involves calculating the 10th percentile using the normal distribution. Understand the Z-score and standard error of the mean concepts involved. Suitable for college-level statistics and mathematics enthusiasts.

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