Question

which of the following regressions represents the strongest linear relationship between

regression 1
$y = ax + b$
$a = 12.5$
$b = - 19.7$
$r = 0.1839$

regression 2
$y = ax + b$
$a = - 12$
$b = - 17.7$
$r = - 1.125$

regression 3
$y = ax + b$
$a = 11.3$
$b = - 0.2$
$r = 0.6063$

regression 4
$y = ax + b$
$a = - 18.5$
$b = 19.6$
$r = - 0.6607$

answer
regression 1
regression 3
regression 2
regression 4

Explanation:

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

The strength of a linear relationship is determined by the absolute value of the correlation coefficient \(r\), \(|r|\). The closer \(|r|\) is to \(1\), the stronger the linear relationship.

Step2: Calculate the absolute values of \(r\) for each regression

• For Regression 1: \(|r| = |0.1839|=0.1839\)
• For Regression 2: \(|r| = | - 1.125| = 1.125\) (but \(r\) values range from \(-1\) to \(1\), so this value is invalid as \(|r|\leq1\))
• For Regression 3: \(|r|=|0.6063| = 0.6063\)
• For Regression 4: \(|r|=|-0.6607|=0.6607\)

Since Regression 2 has an invalid \(r\) value (because \(|r|>1\)), among the valid ones (\(r\) values for Regression 1, Regression 3 and Regression 4), we compare \(|r|\) values.

Answer:

Regression 4