Question

1. if the correlation coefficient between x and y is 0.85, which of the following statements are true?

as x increases, y decreases.
an increase in x causes an increase in y.
x and y are positively correlated.
x and y are negatively correlated.
x and y are not correlated.
as x increases, y increases.

Explanation:

Step1: Recall correlation coefficient rules

The correlation coefficient \( r \) ranges from -1 to 1. A positive \( r \) (like 0.85) means positive correlation, where as one variable increases, the other tends to increase. A negative \( r \) means negative correlation (one increases, the other decreases). Correlation does not imply causation.

Step2: Analyze each statement

• "As \( x \) increases, \( y \) decreases": False (positive \( r \) means same direction).
• "An increase in \( x \) causes an increase in \( y \)": False (correlation ≠ causation).
• " \( x \) and \( y \) are positively correlated": True ( \( r = 0.85>0 \) ).
• " \( x \) and \( y \) are negatively correlated": False ( \( r>0 \) ).
• " \( x \) and \( y \) are not correlated": False ( \( r = 0.85 \) is a strong correlation).
• "As \( x \) increases, \( y \) increases": True (positive correlation implies this trend).

Answer:

The true statements are:

• \( x \) and \( y \) are positively correlated.
• As \( x \) increases, \( y \) increases.