Question

jackson countys parks and recreation department is considering establishing a new park. as part of the decision - making process, the department asked an intern to conduct a park usage study. the study considered the area (in square kilometers), x, and the number of visitors last year, y, of each county park. the least squares regression line of this data set is: y = 7341.372x - 30,495.811. complete the following sentence: if a parks area was one square kilometer larger, the least squares regression line predicts that it would have had more visitors last year.

Explanation:

Step1: Identify the regression - line formula

The least - squares regression line is $y = 7341.372x-30495.811$, where $x$ is the area of the park in square kilometers and $y$ is the number of visitors.

Step2: Analyze the slope

The slope of the regression line is $m = 7341.372$. A positive slope indicates a positive linear relationship. That is, as $x$ (park area) increases, $y$ (number of visitors) increases.

Step3: Calculate the predicted increase

If the park's area is one square kilometer larger ($\Delta x=1$), we substitute $\Delta x = 1$ into the regression equation. Since $y = 7341.372x-30495.811$, when $x$ changes to $x + 1$, $y_{new}=7341.372(x + 1)-30495.811=7341.372x+7341.372 - 30495.811$. The change in $y$ is $y_{new}-y=(7341.372x + 7341.372-30495.811)-(7341.372x-30495.811)=7341.372$.

Answer:

The least - squares regression line predicts that if a park's area was one square kilometer larger, it would have 7341.372 more visitors last year.