Question

which of the following regressions represents the strongest linear relationship between x and y?
regression 1
$y = ax + b$
$a = 4.5$
$b = 2.2$
$r = 0.8673$
regression 2
$y = ax + b$
$a = 14$
$b = -1.2$
$r = 0.5898$
regression 3
$y = ax + b$
$a = 3.5$
$b = 18.9$
$r = 0.865$
regression 4
$y = ax + b$
$a = 9.7$
$b = 12.5$
$r = 1.0724$
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 \(1\). The closer \(|r|\) is to \(1\), the stronger the linear relationship.

Step2: Analyze the given \(r\) values

For Regression 1: \(r = 0.8673\), so \(|r|=0.8673\)
For Regression 2: \(r = 0.5898\), so \(|r| = 0.5898\)
For Regression 3: \(r=0.865\), so \(|r| = 0.865\)
For Regression 4: \(r = 1.0724\). But since \(|r|\leq1\), this value is invalid (likely a calculation error).

Comparing the valid \(|r|\) values: \(0.8673>0.865 > 0.5898\)

Answer:

Regression 1