Question

assume the average selling price for houses in a certain county is $294,000 with a standard deviation of $48,000.
a) determine the coefficient of variation.
b) calculate the z - score for a house that sells for $317,000.
c) using the empirical rule, determine the range of prices that includes 95% of the homes around the mean.
d) using chebychev’s theorem, determine the range of prices that includes at least 83% of the homes around the mean.

Explanation:

Part a)

Step1: Recall Coefficient of Variation Formula

The coefficient of variation (CV) is calculated as \( CV = \frac{\text{Standard Deviation}}{\text{Mean}} \times 100\% \).

Step2: Substitute Values

Given mean (\(\mu\)) = \$294,000 and standard deviation (LXI1) = \$48,000.
\( CV = \frac{48000}{294000} \times 100\% \)
Simplify \( \frac{48000}{294000} \approx 0.1633 \), then \( 0.1633 \times 100\% = 16.33\% \).

Step1: Recall Z - Score Formula

The z - score is calculated as \( z=\frac{x - \mu}{\sigma} \), where \( x \) is the value, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.

Step2: Substitute Values

Given \( x = 317000 \), \( \mu = 294000 \), and \( \sigma = 48000 \).
\( z=\frac{317000 - 294000}{48000}=\frac{23000}{48000}\approx0.479 \)

Step1: Recall Empirical Rule for 95%

The Empirical Rule states that for a normal distribution, approximately 95% of the data lies within \( \mu\pm2\sigma \).

Step2: Calculate Lower and Upper Bounds

Lower bound: \( \mu - 2\sigma=294000 - 2\times48000 = 294000 - 96000 = 198000 \)
Upper bound: \( \mu + 2\sigma=294000 + 2\times48000 = 294000 + 96000 = 390000 \)

Answer:

Approximately \( 16.33\% \)

Part b)