Question

there are four scatter plots with correlation coefficients r = 0, r = -0.9, r = 0.5, r = 1 above them. the first scatter plot (top - left) has points showing a positive trend. the second scatter plot (top - right) has points with no obvious trend. the third scatter plot (bottom - left) has points scattered randomly. the fourth scatter plot (bottom - right) has points showing a negative trend.

Explanation:

Step1: Recall Correlation Coefficient

The correlation coefficient \( r \) measures the strength and direction of a linear relationship between two variables. \( r = 1 \) means a perfect positive linear relationship, \( r = - 1 \) perfect negative, \( r = 0 \) no linear relationship, \( r>0 \) positive, \( r < 0 \) negative. Magnitude near 1 is strong, near 0 is weak.

Step2: Analyze Each Graph

• Top - Left Graph: Points form a clear upward - sloping linear pattern, so \( r \) should be close to 1 (positive strong).
• Top - Right Graph: Points are scattered with no clear linear trend, maybe \( r\approx0 \) or weak, but not matching \( r=-0.9,0.5,1 \) well.
• Bottom - Left Graph: Points are scattered randomly, no linear trend, \( r\approx0 \).
• Bottom - Right Graph: Points form a clear downward - sloping linear pattern, so \( r \) should be negative (close to - 0.9 as it's a strong negative linear relationship).

Step3: Match \( r = - 0.9 \)

The bottom - right graph shows a strong negative linear relationship, which matches \( r=-0.9 \) (since \( r = - 0.9 \) indicates a strong negative linear correlation).

Answer:

The graph with \( r=-0.9 \) is the bottom - right scatter plot (the fourth graph in the grid, considering top - left, top - right, bottom - left, bottom - right order).