The prices of a certain model of new homes are normally distributed with a mean of $150,000 and a standard deviation of $900. What percentage of homes exceed $152,700?
The Correct Answer and Explanation is:
To find the percentage of homes that exceed \$152,700, we use the normal distribution with the given:
- Mean (μ) = \$150,000
- Standard deviation (σ) = \$900
- X = \$152,700
Step 1: Calculate the Z-score
The Z-score tells us how many standard deviations a value is from the mean. The formula is:
$$
Z = \frac{X – \mu}{\sigma}
$$
$$
Z = \frac{152,700 – 150,000}{900} = \frac{2,700}{900} = 3
$$
Step 2: Find the area to the left of Z = 3
Using a standard normal distribution table or calculator:
$$
P(Z < 3) = 0.9987
$$
This means that 99.87% of the homes cost less than or equal to \$152,700.
Step 3: Find the area to the right (homes that exceed \$152,700)
$$
P(Z > 3) = 1 – P(Z < 3) = 1 – 0.9987 = 0.0013
$$
$$
\text{Percentage} = 0.0013 \times 100 = \boxed{0.13\%}
$$
✅ Correct Answer: 0.13%
Explanation (300+ words)
In statistics, when values are normally distributed, they form a bell-shaped curve where most data points cluster around the mean. In this case, the home prices are centered around \$150,000, with a standard deviation of \$900, which measures how spread out the prices are.
The question asks for the percentage of homes priced above \$152,700. To answer this, we use the concept of Z-scores, which convert any normal distribution into the standard normal distribution (mean = 0, standard deviation = 1), allowing us to use standard Z-tables.
By calculating the Z-score of 3, we found that \$152,700 is 3 standard deviations above the mean. Statistically, values that lie this far above the mean are extremely rare. According to the standard normal distribution table, 99.87% of values fall below this point. That leaves 0.13% of the values above \$152,700.
This small percentage confirms that it’s highly unusual for a home of this model to exceed that price. In real estate, this could help buyers or sellers understand whether a price is typical or unusually high.
Understanding such probabilities helps in decision-making, price evaluation, and market analysis. For example, a buyer may recognize that a home priced above \$152,700 is statistically rare and may need further justification (like better location or upgrades). For developers, this insight helps them set realistic pricing strategies.?