Question

functions and lines
linear relationship and the correlation coefficient

1. for which data set is the correlation coefficient r closest to 1?
2. which data set shows the least evidence of a linear relationship?
3. for which data set does the correlation coefficient r appear to be equal to - 1?

Explanation:

Step1: Recall correlation coefficient concept

The correlation coefficient $r$ close to 1 indicates a strong positive - linear relationship, close to - 1 indicates a strong negative - linear relationship, and close to 0 indicates a weak or no linear relationship.

Step2: Analyze Figure 1

The points in Figure 1 seem to have a positive - linear trend, but not a perfect one.

Step3: Analyze Figure 2

The points in Figure 2 seem to have a negative - linear trend, but not a perfect one.

Step4: Analyze Figure 3

The points in Figure 3 seem to follow a positive - linear pattern quite well. The points are closely clustered around an imaginary straight - line with a positive slope, so the correlation coefficient $r$ is close to 1.

Step5: Analyze Figure 4

The points in Figure 4 are scattered randomly, showing little to no linear relationship. So it shows the least evidence of a linear relationship.

Step6: Analyze for $r=-1$

None of the figures show a perfect negative - linear relationship (where all points lie exactly on a straight - line with a negative slope), so there is no data set with $r = - 1$.

Answer:

1. The data set in Figure 3
2. The data set in Figure 4
3. None of the above