To address this problem, we are utilizing the empirical rule, also known as the 68-95-99.7 rule. This rule tells us about the distribution of data in a normal distribution.
We are provided with:
- Mean price (μ) = \[tex]$159,000
- Standard deviation (σ) = \$[/tex]8,000
According to the empirical rule:
- Approximately 68% of the data falls within one standard deviation of the mean (μ ± 1σ).
- Approximately 95% of the data falls within two standard deviations of the mean (μ ± 2σ).
- Approximately 99.7% of the data falls within three standard deviations of the mean (μ ± 3σ).
For our question, we are interested in the range that covers approximately 99.7% of the housing prices, which is within three standard deviations of the mean.
To find the low and high prices:
1. Low Price: Subtract three times the standard deviation from the mean:
[tex]\[
\text{Low Price} = \mu - 3\sigma = 159,000 - 3 \times 8,000 = 159,000 - 24,000 = 135,000
\][/tex]
2. High Price: Add three times the standard deviation to the mean:
[tex]\[
\text{High Price} = \mu + 3\sigma = 159,000 + 3 \times 8,000 = 159,000 + 24,000 = 183,000
\][/tex]
Therefore, approximately 99.7% of housing prices in this small town are between \[tex]$135,000 and \$[/tex]183,000.
