Question

7.4 what is a correlation coefficient? 1. what information does a correlation coefficient tell us about the data in a scatter plot? 2. which value best estimates the value for the correlation coefficient of the scatter plot: -1, -0.8, -0.2, 0.2, 0.8, or 1? explain your reasoning.

Explanation:

Question 1

A correlation coefficient (usually denoted as \( r \)) tells us two key things about the data in a scatter plot:

1. Direction of the relationship: The sign of \( r \) (positive or negative) indicates if there is a positive linear relationship (as one variable increases, the other tends to increase) or a negative linear relationship (as one variable increases, the other tends to decrease).
2. Strength of the relationship: The magnitude of \( r \) (between -1 and 1, where \( r = 1 \) or \( r = -1 \) is a perfect linear relationship, and \( r = 0 \) means no linear relationship) tells us how closely the data points follow a straight - line pattern. Values closer to -1 or 1 indicate a strong linear relationship, while values closer to 0 indicate a weak linear relationship.

1. Direction: First, observe the scatter plot. The points seem to trend downward from left to right, which means as \( x \) increases, \( y \) decreases. So the correlation coefficient should be negative (eliminating 0.2, 0.8, and 1).
2. Strength: Now, consider the strength. A correlation coefficient of -1 would mean a perfect straight - line (all points on the line), and -0.2 would mean a very weak negative relationship. The points in the scatter plot do not lie perfectly on a line (so not -1), but they do show a relatively strong downward trend (not as weak as -0.2). A value of -0.8 indicates a strong negative linear relationship, which matches the pattern of the points (they are somewhat close to the line but not perfectly on it).

Answer:

A correlation coefficient tells us the direction (positive/negative) and strength (weak/strong) of the linear relationship between the two variables in the scatter plot.

Question 2