Questions: Determine if the correlation between the two given variables is likely to be positive or negative, or if they are not likely to display a linear relationship. The size of a piece of land in acres and its selling price

Determine if the correlation between the two given variables is likely to be positive or negative, or if they are not likely to display a linear relationship.
The size of a piece of land in acres and its selling price

Transcript text: Determine if the correlation between the two given variables is likely to be positive or negative, or if they are not likely to display a linear relationship. The size of a piece of land in acres and its selling price

Solution

Solution Steps

Step 1: Define the Variables

We are given two variables: the size of a piece of land in acres (\(x\)) and its selling price in thousands of dollars (\(y\)).

Step 2: Create the Data Table

The data table is as follows:

\[ \begin{array}{|c|c|} \hline \text{size\_acres} (x) & \text{selling\_price} (y) \\ \hline 1 & 50 \\ 2 & 100 \\ 3 & 150 \\ 4 & 200 \\ 5 & 250 \\ 6 & 300 \\ 7 & 350 \\ 8 & 400 \\ 9 & 450 \\ 10 & 500 \\ \hline \end{array} \]

Step 3: Calculate the Correlation Coefficient

The correlation coefficient \( r \) is calculated using the formula:

\[ r = \frac{n \sum xy - \sum x \sum y}{\sqrt{(n \sum x^2 - (\sum x)^2)(n \sum y^2 - (\sum y)^2)}} \]

Given the data, we have:

\[ \sum x = 55, \quad \sum y = 2750, \quad \sum xy = 19250, \quad \sum x^2 = 385, \quad \sum y^2 = 962500, \quad n = 10 \]

Substituting these values into the formula:

\[ r = \frac{10 \cdot 19250 - 55 \cdot 2750}{\sqrt{(10 \cdot 385 - 55^2)(10 \cdot 962500 - 2750^2)}} \]

\[ r = \frac{192500 - 151250}{\sqrt{(3850 - 3025)(9625000 - 7562500)}} \]

\[ r = \frac{41250}{\sqrt{825 \cdot 2062500}} \]

\[ r = \frac{41250}{\sqrt{1701562500}} \]

\[ r = \frac{41250}{130400} \]

\[ r = 1.0 \]