Question

question 4
1 pts
which one of the following statements is true?
○ if r is negative, this means that as one variable goes up, the other variable always goes down.
○ we compute r when the relationship between quantitative variables is linear or nonlinear.
○ a correlation of r = -0.52 is stronger than a correlation of r = 0.20.
○ if r is negative, r² will also be negative.
○ when r is negative, this means that either the response variable or the explanatory variable must have a
distribution that is skewed to the left.

Explanation:

Step1: Analyze each option

• Option 1: A negative correlation (r negative) indicates a general trend of one variable increasing as the other decreases, but not "always" (there can be variability). So this is false.
• Option 2: The correlation coefficient \( r \) is used to measure the strength and direction of a linear relationship between quantitative variables. It is not appropriate for nonlinear relationships. So this is false.
• Option 3: The strength of a correlation is determined by the absolute value of \( r \). \( | - 0.52|=0.52 \) and \( |0.20| = 0.20 \). Since \( 0.52>0.20 \), the correlation \( r=- 0.52 \) is stronger (in terms of the magnitude of the linear relationship) than \( r = 0.20 \). This statement is true.
• Option 4: \( r^{2}\) is the square of \( r \), and squaring a negative number gives a positive result (e.g., if \( r=-0.5 \), \( r^{2}=0.25 \)). So \( r^{2}\) is always non - negative, regardless of the sign of \( r \). This is false.
• Option 5: The sign of \( r \) (negative or positive) is related to the direction of the linear relationship between two variables, not the skewness of the distribution of individual variables. Skewness of a variable's distribution and the sign of the correlation between two variables are different concepts. So this is false.

Answer:

A correlation of \( r = - 0.52 \) is stronger than a correlation of \( r=0.20 \). (The option corresponding to this statement, assuming the options are labeled as per the order in the question, e.g., if it's the third option, then "The third option: A correlation of \( r = - 0.52 \) is stronger than a correlation of \( r = 0.20 \)")