Question

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 $r$ ranges from - 1 to 1. A value of $r = 1$ indicates a perfect positive - linear relationship, $r=-1$ indicates a perfect negative - linear relationship, and $r = 0$ indicates no linear relationship.

Step2: Analyze Figure 1

The points in Figure 1 are somewhat scattered and do not show a strong linear pattern.

Step3: Analyze Figure 2

The points in Figure 2 also do not form a very tight linear cluster.

Step4: Analyze Figure 3

The points in Figure 3 follow an upward - sloping linear trend quite closely. So, the correlation coefficient $r$ for Figure 3 is closest to 1.

Step5: Analyze Figure 4

The points in Figure 4 are randomly scattered with no discernible linear pattern, indicating the least evidence of a linear relationship.

Step6: Find $r=-1$ case

None of the given figures show a perfect negative - linear relationship (where all points lie on a straight - line with a negative slope).

Answer:

1. The data set in Figure 3
2. The data set in Figure 4
3. None of the above figures show a correlation coefficient $r=-1$.