Assume the average selling price for houses in a certain county is ​$275 comma 000 with a standard deviation of ​$59 comma 000. 
​a) Determine the coefficient of variation.
​b) Caculate the​ z-score for a house that sells for ​$312 comma 000.
​c) Using the Empirical​ Rule, determine the range of prices that includes 68​% of the homes around the mean.
​d) Using​ Chebychev's Theorem, determine the range of prices that includes at least 87​% of the homes around the mean.
Question content area bottom
Part 1
​a) Determine the coefficient of variation.
CVequals
  
21.5​%  ​(Round to one decimal place as​ needed.)
Part 2
​b) Calculate the​ z-score for a house that sells for ​$312 comma 000.
zequals
  
. 63  ​(Round to two decimal places as​ needed.)
Part 3
​c) Using the Empirical​ Rule, determine the range of prices that includes 68​% of the homes around the mean.
upper bound x equals
​$
  
334000
lower bound x equals
​$
  
216000
​(Round to the nearest dollar as​ needed.)
Part 4
​d) Using​ Chebychev's Theorem, determine the range of prices that includes at least 87​% of the homes around the mean.
upper bound x equals
​$
  
438430
lower bound x equals
​$
  
111570
​(Round to the nearest dollar as​ needed.)

Let's solve each part of the problem step by step.

### Part 1: Coefficient of Variation (CV)
The coefficient of variation (CV) is calculated using the formula:

\[
\text{CV} = \left(\frac{\sigma}{\mu}\right) \times 100\%
\]

Where:
- $\sigma$ is the standard deviation.
- $\mu$ is the mean.

Given:
$\mu = 275,000$
$\sigma = 59,000$

\[
\text{CV} = \left(\frac{59,000}{275,000}\right) \times 100\% \approx 21.5\%
\]

### Part 2: Z-Score Calculation
The z-score is calculated using the formula:

\[
z = \frac{x - \mu}{\sigma}
\]

Where:
- $x$ is the value for which the z-score is being calculated.
- $\mu$ is the mean.
- $\sigma$ is the standard deviation.

Given:
$x = 312,000$
$\mu = 275,000$
$\sigma = 59,000$

\[
z = \frac{312,000 - 275,000}{59,000} \approx \frac{37,000}{59,000} \approx 0.63
\]

### Part 3: Empirical Rule (68% Range)
The Empirical Rule states that for a normal distribution:
- Approximately 68% of the data falls within 1 standard deviation from the mean.

The range can be calculated as:
\[
\text{Lower bound} = \mu - \sigma = 275,000 - 59,000 = 216,000
\]
\[
\text{Upper bound} = \mu + \sigma = 275,000 + 59,000 = 334,000
\]

### Part 4: Chebyshev's Theorem (87% Range)
Chebyshev's Theorem states that for any distribution, the proportion of values that lie within $k$ standard deviations of the mean is at least $ \left(1 - \frac{1}{k^2}\right) $.

For 87%, set $ \left(1 - \frac{1}{k^2}\right) = 0.87 $, solving for $k$:
\[
\frac{1}{k^2} = 0.13 \quad \Rightarrow \quad k^2 = \frac{1}{0.13} \approx 7.6923 \quad \Rightarrow \quad k \approx \sqrt{7.6923} \approx 2.77
\]

Now, calculate the range:
\[
\text{Lower bound} = \mu - k\sigma = 275,000 - 2.77 \times 59,000 \approx 275,000 - 163,430 = 111,570
\]
\[
\text{Upper bound} = \mu + k\sigma = 275,000 + 2.77 \times 59,000 \approx 275,000 + 163,430 = 438,430
\]

### Summary:
- **Coefficient of Variation:** 21.5%
- **Z-Score for $312,000:** 0.63
- **Empirical Rule Range (68%):** $216,000 to $334,000
- **Chebyshev's Theorem Range (87%):** $111,570 to $438,430

Would you like more details or have any questions?

---

**Related Questions:**
1. How would the CV change if the standard deviation were higher?
2. What does a higher z-score indicate about the price of a house?
3. How does the Empirical Rule compare to Chebyshev's Theorem in terms of range width?
4. Can the z-score be negative, and what would it indicate if it were?
5. How would the range of prices change if we used 95% in the Empirical Rule?

**Tip:** The z-score helps determine how far a data point is from the mean, in terms of standard deviations.

Assume the average selling price for houses in a certain county is $275,000 with a standard deviation of $59,000. a) Determine the coefficient of variation. b) Calculate the z-score for a house that sells for $312,000. c) Using the Empirical Rule, determine the range of prices that includes 68% of the homes around the mean. d) Using Chebyshev's Theorem, determine the range of prices that includes at least 87% of the homes around the mean.

Explore how to analyze house prices in a county using statistical methods. Learn to calculate the coefficient of variation, z-score for a $312,000 house, and determine the price ranges using the Empirical Rule and Chebyshev's Theorem. Suitable for Grades 10-12 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation