Question

which of the following regressions represents the strongest negative linear relationship between x and y?
regression 1
$y = ax + b$
$a = - 9$
$b = - 6$
$r = - 0.4563$
regression 2
$y = ax + b$
$a = 13.8$
$b = - 8.5$
$r = 0.3374$
regression 3
$y = ax + b$
$a = 15.2$
$b = - 12.8$
$r = 0.8255$
regression 4
$y = ax + b$
$a = - 5.2$
$b = - 15.2$
$r = - 0.7049$

Explanation:

Step1: Recall Correlation Coefficient

The correlation coefficient \( r \) measures the strength and direction of a linear relationship. For negative relationships, \( r < 0 \). The strength is determined by the absolute value of \( r \); closer to \( -1 \) (or \( 1 \) for positive) means stronger.

Step2: Identify Negative \( r \) Values

Regression 1: \( r = -0.4563 \), Regression 4: \( r = -0.7049 \).

Step3: Compare Absolute Values

\( |-0.4563| = 0.4563 \), \( |-0.7049| = 0.7049 \). Since \( 0.7049 > 0.4563 \), Regression 4 has a stronger negative linear relationship. Also, check slopes (but \( r \) is key for strength). Regression 4's \( r \) is more negative (in absolute value) than Regression 1.

Answer:

Regression 4 (with \( y = ax + b \), \( a = -5.2 \), \( b = -15.2 \), \( r = -0.7049 \)) represents the strongest negative linear relationship between \( x \) and \( y \).