Question

which of the following regressions represents the strongest negative linear relationship between x and y?
regression 1
$y = ax + b$
$a = - 10$
$b = 16.7$
$r = - 0.4592$
regression 2
$y = ax + b$
$a = - 2.7$
$b = - 7.3$
$r = - 0.7702$
regression 3
$y = ax + b$
$a = 16.7$
$b = - 2.8$
$r = 0.8222$
regression 4
$y = ax + b$
$a = - 16.5$
$b = 9.1$
$r = - 1.0024$
answer
regression 1
regression 3
regression 2
regression 4

Explanation:

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

The correlation coefficient \(r\) measures the strength and direction of a linear relationship. For a negative linear relationship, \(r<0\). The closer \(|r|\) is to \(1\), the stronger the linear relationship.

Step2: Analyze each regression

• Regression 1: \(r = - 0.4592\), \(|r|=0.4592\)
• Regression 2: \(r=-0.7702\), \(|r| = 0.7702\)
• Regression 3: \(r = 0.8222>0\), represents a positive linear relationship (so we can rule it out for a negative relationship)
• Regression 4: \(r=-1.0024\). But \(|r|\leq1\), so this value is likely due to calculation error (in a real - world context, we consider the magnitude of \(r\) values within the valid range \(- 1\leq r\leq1\)). Among the valid \(r\) values (\(r=-0.4592\), \(r = - 0.7702\)), since \(| - 0.7702|>| - 0.4592|\)

Answer:

Regression 2