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 find the price that indicates a home is in the bottom 10%, we need to use the properties of the normal distribution.

Given:
- The average home price ($\mu$) = \$410,000
- The standard deviation ($\sigma$) = \$65,000
- Number of homes examined ($n$) = 200

We need to find the 10th percentile of the distribution of home prices. 

### Step-by-Step 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 value 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.

Do you want 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 calculate the home price that indicates the bottom 10% of the distribution using normal distribution properties. Given an average home price of $410,000 and a standard deviation of $65,000, find the price for 200 homes in the state. Detailed solution with formulas and step-by-step explanation.

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