Question

name date period
cool down
lesson 7: the correlation coefficient
cool down: what is a correlation coefficient?

1. what information does a correlation coefficient tell you 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:

For question 1:
A correlation coefficient \(r\) measures the strength and direction of a linear relationship between two variables in a scatter - plot. If \(r>0\), the relationship is positive (as one variable increases, the other tends to increase). If \(r < 0\), the relationship is negative (as one variable increases, the other tends to decrease). The closer \(|r|\) is to \(1\), the stronger the linear relationship. If \(r = 0\), there is no linear relationship.

For question 2:

1. Direction:

• The slope of the line of best - fit in the scatter - plot is negative. So, the correlation coefficient \(r<0\).

1. Strength:

• The value of \(r=-1\) would mean all the data points lie exactly on a straight line with a negative slope. The value of \(r = - 0.2\) would mean a very weak negative linear relationship.
• The data points in the given scatter - plot are somewhat close to the line of best - fit, indicating a moderately strong negative linear relationship. The value of \(r=-0.8\) is a better estimate than \(r=-1\) (because the points are not perfectly on the line) and \(r=-0.2\) (because the relationship is stronger than a weak negative relationship).

Answer:

1. A correlation coefficient tells you the direction (positive or negative) and the strength (how close the data points are to a line of best - fit) of a linear relationship between two variables in a scatter - plot.
2. The value \(-0.8\) best estimates the correlation coefficient. The negative sign is due to the negative slope of the line of best - fit, and the magnitude of \(0.8\) (closer to \(1\) than to \(0\)) indicates a moderately strong linear relationship (since the points are not perfectly on the line but are not too scattered for a weak relationship).