https://file.aiquickdraw.com/mathbot/file/1729747759226ptafum5u.jpg

To solve this problem, we analyze the two scatterplots and the given answer choices.

- **Plot A** shows a clear nonlinear relationship; the points curve, indicating a parabolic shape, which is typical of a nonlinear relationship.
- **Plot B** shows a negative linear relationship, as the points appear to follow a downward sloping straight line.

Now, let's evaluate each option:

1. **Option A**: Plot A indicates a nonlinear relationship between the variables.
   - This is true since Plot A shows a curved (parabolic) pattern.

2. **Option B**: Plot B indicates a linear relationship between the variables.
   - This is true since Plot B shows points that follow a linear trend.

3. **Option C**: The correlation coefficient for the data in Plot B is close to 1.
   - This cannot be true. A correlation coefficient of 1 indicates a perfect positive linear relationship. However, Plot B shows a **negative** linear relationship, so the correlation should be close to -1, not 1.

4. **Option D**: The correlation coefficient for the data in Plot A is close to 0.
   - This is true. Since Plot A shows a nonlinear pattern, the correlation coefficient would not capture the relationship well and would likely be close to 0.

5. **Option E**: The \( r^2 \) value of the least-squares regression line for Plot B is greater than the \( r^2 \) value of the least-squares regression line for Plot A.
   - This is true. Plot B follows a strong linear trend, so the \( r^2 \) value would be high, while Plot A's \( r^2 \) value would be lower due to its nonlinear pattern.

Thus, the statement that **cannot** be true is **Option C**: The correlation coefficient for the data in Plot B is close to 1.

Would you like more details on the concepts of correlation and \( r^2 \) values?

Here are 5 related questions for deeper understanding:
1. What is the difference between correlation and \( r^2 \) in data analysis?
2. How is the correlation coefficient calculated, and what are its typical ranges?
3. Can a nonlinear relationship ever have a high \( r^2 \) value? If so, under what conditions?
4. What are some methods for detecting nonlinear relationships in scatterplots?
5. How does the least-squares regression line differ between linear and nonlinear models?

**Tip:** In scatterplot analysis, the correlation coefficient reflects the strength and direction of a **linear** relationship, while the \( r^2 \) value measures the proportion of variance explained by a model, which can apply to both linear and nonlinear fits.

Based on the scatterplots, which statement CANNOT be true?

Learn how to analyze scatterplots to determine which statement cannot be true. Understand key concepts like correlation coefficient, linear vs. nonlinear relationships, and the r^2 value. Ideal for high school students studying statistics and data analysis.

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