Questions: 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 σₓ̄ and x̄ to two decimal places. Provide your answer below: σₓ̄= x̄=

Solution Steps

To find the standard error of the mean (\( \sigma_{\bar{x}} \)), we use the formula:

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

where:

• \( \sigma = 65000 \) (standard deviation of home prices)
• \( n = 200 \) (sample size)

Calculating this gives:

\[ \sigma_{\bar{x}} = \frac{65000}{\sqrt{200}} \approx 4596.19 \]

Thus, rounding to two decimal places, we have:

\[ \sigma_{\bar{x}} = \$ 4596.19 \]

The z-score corresponding to the bottom 10% of a standard normal distribution is approximately:

\[ z = -1.2816 \]

To find the price (\( \bar{x} \)) that indicates a home is in the bottom 10%, we use the formula:

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

where:

• \( \mu = 410000 \) (mean home price)

Substituting the values, we get:

\[ \bar{x} = 410000 + (-1.2816) \cdot 4596.19 \approx 404109.52 \]

Thus, rounding to two decimal places, we have:

\[ \bar{x} = \$ 404109.52 \]

Final Answer