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To determine the correlation coefficient between the heights and lengths of the rectangles, we need to compute the Pearson correlation coefficient \(r\), which measures the strength and direction of a linear relationship between two variables.

The formula for the Pearson correlation coefficient is:

\[
r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n \sum x^2 - (\sum x)^2][n \sum y^2 - (\sum y)^2]}}
\]

Where:
- \(x\) represents the height values,
- \(y\) represents the length values,
- \(n\) is the number of data points.

From the table:
- Heights (in.): \( 41, 70, 21, 34, 10, 92, 54, 24, 35, 42, 66 \)
- Lengths (in.): \( 21, 25, 32, 12, 16, 45, 40, 23, 35, 21, 14 \)

### Step 1: Calculating the necessary sums

Let me compute these for you.The Pearson correlation coefficient \(r\) for the given data is approximately \(0.424\).

This indicates a **weak positive correlation** between the height and length of the rectangles. A weak positive correlation means that as the height increases, the length tends to increase as well, but the relationship is not very strong.

Would you like more details or have any questions?

Here are some related questions to explore further:

1. How would the interpretation change if the correlation coefficient were negative?
2. What is considered a strong correlation versus a weak correlation?
3. How does the correlation coefficient relate to linear regression?
4. What factors could affect the strength of the correlation in this dataset?
5. Can correlation imply causation, and if not, why?

**Tip**: Correlation coefficients close to 1 or -1 indicate stronger relationships, while those closer to 0 indicate weaker ones.

What does the correlation coefficient for the data indicate about the strength of the linear association between the height and length of these rectangles?

Calculate and interpret the Pearson correlation coefficient to determine the strength of the linear relationship between the height and length of rectangles. This problem involves understanding the correlation between two variables and using statistical formulas. Suitable for high school students.

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