Which of the following regressions represents the weakest linear relationship between x and y?

Regression 1
y, equals, a, x, plus, by=ax+b
a, equals, minus, 19, point, 4a=−19.4
b, equals, 17, point, 7b=17.7
r, equals, minus, 0, point, 7, 0, 3, 7r=−0.7037 Regression 2
y, equals, a, x, plus, by=ax+b
a, equals, 5, point, 8a=5.8
b, equals, 16, point, 7b=16.7
r, equals, 0, point, 3, 9, 8, 4r=0.3984 Regression 3
y, equals, a, x, plus, by=ax+b
a, equals, minus, 19, point, 5a=−19.5
b, equals, minus, 0, point, 6b=−0.6
r, equals, minus, 0, point, 2, 9, 6r=−0.296 Regression 4
y, equals, a, x, plus, by=ax+b
a, equals, 6, point, 9a=6.9
b, equals, 13, point, 4b=13.4
r, equals, 0, point, 2, 4, 1, 2r=0.2412
Answer
Multiple Choice Answers
Regression , 1Regression 1
Regression , 2Regression 2
Regression , 3Regression 3
Regression , 4Regression 4

1 answer

To determine which regression represents the weakest linear relationship between \( x \) and \( y \), we need to focus on the correlation coefficient \( r \). The strength of a linear relationship is indicated by the absolute value of \( r \), where:

Values close to 1 (or -1) indicate a strong linear relationship.
Values close to 0 indicate a weak linear relationship.

Here are the correlation coefficients provided for each regression:

Regression 1: \( r = -0.7037 \) (strong negative relationship)
Regression 2: \( r = 0.3984 \) (moderate positive relationship)
Regression 3: \( r = -0.296 \) (weak negative relationship)
Regression 4: \( r = 0.2412 \) (weak positive relationship)

To find the weakest linear relationship, we will look for the \( r \) value with the smallest absolute value:

\( |r| \) for Regression 1 = 0.7037
\( |r| \) for Regression 2 = 0.3984
\( |r| \) for Regression 3 = 0.296
\( |r| \) for Regression 4 = 0.2412

Comparing these absolute values, we see that:

Regression 4 has the smallest absolute value of \( r = 0.2412 \).

Thus, Regression 4 represents the weakest linear relationship between \( x \) and \( y \).

The correct answer is Regression 4.