Question

which of the following regressions represents the strongest negative linear relationship between x and y?
regression 1
$y = ax + b$
$a = - 15$
$b = - 18.2$
$r = - 0.1439$
regression 2
$y = ax + b$
$a = - 6.5$
$b = - 0.3$
$r = - 0.7514$
regression 3
$y = ax + b$
$a = 16.9$
$b = - 12.4$
$r = 0.7737$
regression 4
$y = ax + b$
$a = - 5.3$
$b = - 13.6$
$r = - 1.0863$

Explanation:

Step1: Recall the property of the correlation coefficient \(r\)

The correlation coefficient \(r\) measures the strength and direction of a linear relationship. The value of \(r\) ranges from \(- 1\) to \(1\). A negative \(r\) indicates a negative linear relationship. The closer \(|r|\) is to \(1\), the stronger the linear relationship.

Step2: Analyze each regression

• For Regression 1: \(r=-0.1439\), \(|r| = 0.1439\)
• For Regression 2: \(r=-0.7514\), \(|r|=0.7514\)
• For Regression 3: \(r = 0.7737\) (positive, so not a negative linear relationship)
• For Regression 4: \(r=-1.0863\) (invalid since \(|r|\leq1\))

Answer:

Regression 2