Question

which of the following regressions represents the strongest positive linear relationship between x and y?
regression 1
$y = ax + b$
$a = 19.3$
$b = 0$
$r = 1.0882$
regression 2
$y = ax + b$
$a = - 16.9$
$b = 1.6$
$r = - 0.8343$
regression 3
$y = ax + b$
$a = 13.2$
$b = - 15.5$
$r = 0.0694$
regression 4
$y = ax + b$
$a = 7$
$b = 12.1$
$r = 0.7823$
answer
regression 1
regression 2
regression 3
regression 4

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 positive \(r\) indicates a positive linear relationship, and the closer \(|r|\) is to \(1\), the stronger the linear relationship.

Step2: Analyze each regression's \(r\) value

• For Regression 1: \(r = 1.0882\). But \(|r|\leq1\), so this value is invalid (likely a calculation error).
• For Regression 2: \(r=-0.8343\). This is a negative correlation.
• For Regression 3: \(r = 0.0694\). The value is close to \(0\), indicating a very weak linear relationship.
• For Regression 4: \(r = 0.7823\). It is positive and among the valid positive \(r\) values (since Regression 1's \(r\) is invalid), and its \(|r|\) is relatively large.

Answer:

Regression 4