https://file.aiquickdraw.com/mathbot/file/1733424791746kbz4z7zc.jpg

The question asks for an estimate of the correlation coefficient \( r \) for the scatter plot provided. The options for \( r \) are:

- \( a. -2.50 \)
- \( b. -1.00 \)
- \( c. 2.50 \)
- \( d. 1.00 \)

### Observations:
The scatter plot shows a **negative linear trend**, as \( x \) increases, \( y \) decreases. This indicates that \( r \) must be a negative value. The points align relatively well, suggesting a strong correlation.

### Evaluation:
1. The correlation coefficient \( r \) always falls between \(-1\) and \( 1\). Hence, options \( -2.50 \) and \( 2.50 \) are invalid because they are out of this range.
2. A perfect negative correlation (\( r = -1 \)) occurs when all points lie exactly on a straight descending line. Since the points in this scatter plot are close to a line but not perfectly aligned, \( r \) is likely close to \(-1\), but not exactly \(-1\).

### Conclusion:
The best estimate for \( r \) is **\( b. -1.00 \)**. However, if exact values aren't required, the correlation is close to \(-1\), indicating a strong negative relationship.

Would you like a more detailed explanation about correlation or scatter plots?

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### Related Questions:
1. What is the mathematical formula to calculate the correlation coefficient \( r \)?
2. How does \( r \) differ between strong and weak correlations?
3. Why can't \( r \) exceed the range of \([-1, 1]\)?
4. What real-world scenarios might produce strong negative correlations?
5. How does the scatter plot's density affect the estimated value of \( r \)?

### Tip:
When estimating \( r \) visually, focus on both the direction and how tightly the points cluster around an imaginary line.

Estimate the value of r for the scatter plot provided.

Learn how to estimate the value of the correlation coefficient r for a scatter plot. This problem demonstrates key statistical concepts, including identifying negative trends and the properties of r values. Suitable for high school students in Grades 9-12.

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