Below are four bevariate data sets and their scatter plots. (Note that all of the scatter plots are displayed with the same scale.) Each data set is made up of sample values drawn from a population

xy

10

54

1032

20 68

2045

30 77

30 38

40 63

4.0 5.1

5073

50 48

6045

6073

70 50

70 61

5036

80 79

9046

90 70

100 25

Figure 1

100 81

Figure 2

wt

1071

20 91

30 72

40 59

5.0 7.9

6052

7041

073

10 78

20 47

30 99

4.0 5.7

50 20

6049

7090

8039

90.93

100 6.8

Flaure

90 61

10041

Figure 3

Answer the following questions about the relationships between pairs of variables and the values of r, the sample correlation coefficient. The same response may be the correct answer for more than one question.

(a) For which data set is the sample correlation coefficient closest to 0?

X

(Choose one) ▼

(b) Which data set indicates the strangest positive Inear relationship between its two variables?

(Choose one)

▼

(c) For which data set is the sample correlation coefficient closest to -17

(Choose one) ▼

(4) For which data set is the sample correlation coefficient equal to 17

(Choose one) Y

To answer these questions regarding the sample correlation coefficient \( r \) for the provided bivariate data sets and their scatter plots, let's first understand what \( r \) represents:

- The sample correlation coefficient \( r \) measures the strength and direction of a linear relationship between two variables.
  - \( r = 1 \): Perfect positive linear relationship.
  - \( r = -1 \): Perfect negative linear relationship.
  - \( r = 0 \): No linear relationship.

**Assumptions Based on Your Descriptions:**

1. **(a) For which data set is the sample correlation coefficient closest to 0?**

   - A correlation coefficient close to 0 suggests that there is little to no linear relationship between the variables. This would be indicated by a scatter plot with points that are scattered randomly without forming any visible line.

   **Answer**: Likely Figure 4.

2. **(b) Which data set indicates the strongest positive linear relationship between its two variables?**

   - The strongest positive linear relationship will have a scatter plot where all points closely align along a positively sloped straight line.

   **Answer**: Likely Figure 2.

3. **(c) For which data set is the sample correlation coefficient closest to -1?**

   - A correlation coefficient close to -1 suggests a strong negative linear relationship, where the scatter plot points closely align along a negatively sloped straight line.

   **Answer**: Likely Figure 3.

4. **(d) For which data set is the sample correlation coefficient equal to 1?**

   - A correlation coefficient of 1 indicates a perfect positive linear relationship. The scatter plot points should form a perfect straight line with a positive slope.

   **Answer**: Likely Figure 1.

Please confirm if the scatter plots match these descriptions, or if you would like further clarification based on the actual plots provided.

**Do you have any questions or need further explanation?**

Here are 5 related questions:
1. How would the value of \( r \) change if the data in one of the sets were shifted upward?
2. What does a correlation coefficient of -0.5 indicate about a data set?
3. How could you visually estimate the correlation coefficient from a scatter plot?
4. What does it mean if a data set has a correlation coefficient of 0.8?
5. What are the limitations of using the correlation coefficient to describe relationships between variables?

**Tip:** Correlation does not imply causation—two variables can have a strong correlation without one necessarily causing the other.

Explore how to interpret the sample correlation coefficient (r) for bivariate data sets. Answer questions about correlations ranging from zero to perfect positive and negative correlations. Includes explanations and visual interpretations.

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