Question

functions and lines
linear relationship and the correlation coefficient

1. which data set indicates the strongest linear relationship between its two variables?
2. which data set has an apparent negative, but not perfect, linear relationship between its two variables?
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 or - 1 indicates a strong linear relationship. A perfect positive linear relationship has $r = 1$, and points lie exactly on an upward - sloping line.

Step2: Analyze Figure 1

Points in Figure 1 are somewhat scattered and do not form a clear linear pattern.

Step3: Analyze Figure 2

It shows a negative, but not perfect, linear relationship as points generally trend downwards but are not exactly on a line.

Step4: Analyze Figure 3

Points in Figure 3 form a tight upward - sloping pattern, indicating a strong positive linear relationship.

Step5: Analyze Figure 4

Points in Figure 4 are scattered and do not show a clear linear trend.

Step6: Find $r = 1$ case

None of the given figures have points that lie exactly on a straight line. So for the third question, there is no correct answer among the given figures.

Answer:

1. The data set in Figure 3
2. The data set in Figure 2
3. None of the above (since no figure has points exactly on a straight - line for $r = 1$)