Questions: The following table presents prices, in thousands of dollars, of single-family homes in various metro areas of the United States for last year and this year. Metro Area, Last year, This year, Metro Area, Last year, This year Atlanta, GA, 87.8, 115.1, Philadelphia, PA, 193.5, 197.7 Baltimore, MD, 218.1, 226.5, Phoenix, AZ, 129.9, 169.0 Boston, MA, 311.5, 332.2, Portland, OR, 208.6, 246.5 Chicago, IL, 157.2, 159.4, Riverside, CA, 174.3, 216.7 Cincinnati, OH, 112.5, 121.0, St. Louis, MO, 103.7, 111.0 Cleveland, OH, 84.9, 101.0, San Diego, CA, 359.5, 412.3 Dallas, TX, 148.2, 160.4, San Francisco, CA, 448.0, 593.9 Denver, CO, 226.4, 261.2, Seattle, WA, 265.4, 312.6 Houston, TX, 152.1, 163.7, Tampa, FL, 131.9, 141.8 Los Angeles, CA, 122.1, 155.1, Washington, DC, 311.6, 348.7

Solution Steps

The mean (\(\mu\)) of last year prices is calculated using the formula: \[ \mu = \frac{\sum_{i=1}^N x_i}{N} \] Substituting the given values: \[ \mu = \frac{3947.2}{20} = 197.36 \]

The mean (\(\mu\)) of this year prices is calculated using the same formula: \[ \mu = \frac{\sum_{i=1}^N x_i}{N} \] Substituting the given values: \[ \mu = \frac{4545.8}{20} = 227.29 \]

The variance (\(\sigma^2\)) is calculated using: \[ \sigma^2 = \frac{\sum (x_i - \mu)^2}{n-1} \] Substituting the given values: \[ \sigma^2 = 9629.81 \] The standard deviation (\(\sigma\)) is: \[ \sigma = \sqrt{9629.81} = 98.13 \]

The variance (\(\sigma^2\)) is calculated using the same formula: \[ \sigma^2 = \frac{\sum (x_i - \mu)^2}{n-1} \] Substituting the given values: \[ \sigma^2 = 15082.34 \] The standard deviation (\(\sigma\)) is: \[ \sigma = \sqrt{15082.34} = 122.81 \]

The covariance (\(\text{Cov}(X,Y)\)) is given as: \[ \text{Cov}(X,Y) = 11859.79 \] The correlation coefficient (\(r\)) is calculated using: \[ r = \frac{\text{Cov}(X,Y)}{\sigma_X \sigma_Y} \] Substituting the given values: \[ r = \frac{11859.79}{98.13 \times 122.81} = 0.98 \]

Final Answer