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

Steps to Solve

1. Understand the Data
   The data consists of two sets of values: Heights of rectangles and their corresponding Lengths. We will analyze the correlation between these two sets.

2. Calculate the Correlation Coefficient
   Use the formula for the correlation coefficient $r$: $$ r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}} $$
   Here, $x$ represents heights, $y$ represents lengths, and $n$ is the number of pairs.

3. Collect Necessary Values
   For calculations, compute the following:

   • $n = 10$ (number of pairs)
   • $\sum x$ (sum of heights)
   • $\sum y$ (sum of lengths)
   • $\sum xy$ (sum of products of heights and lengths)
   • $\sum x^2$ (sum of squared heights)
   • $\sum y^2$ (sum of squared lengths)

4. Substitute Values
   Substitute the computed values into the formula to calculate $r$.

5. Interpret the Correlation Coefficient
   Based on the result of $r$:

   • If $r \approx 1$, there is a strong positive correlation.
   • If $r \approx -1$, there is a strong negative correlation.
   • If $r \approx 0$, there’s little to no correlation.
   • Values close to 1 or -1 indicate strong correlations.

6. Select the Correct Answer
   Determine which option (strong negative, strong positive, weak positive, weak negative correlation) best describes the calculated $r$.