To calculate the correlation coefficient $r$ between $x$ and $y$, we use the formula:

[ 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:

• $n$ is the number of data points,
• $\sum x$, $\sum y$, $\sum xy$, $\sum x^2$, and $\sum y^2$ are the sums of the respective variables.

Let's calculate this step by step:

$x$	$y$	$xy$	$x^2$	$y^2$
3	80.8	242.4	9	6528.64
4	81.2	324.8	16	6593.44
5	73.4	367.0	25	5388.36
6	70	420.0	36	4900
7	67.4	471.8	49	4542.76
8	69.2	553.6	64	4788.64
9	61	549.0	81	3721
10	60.4	604.0	100	3648.16

Now, let's compute the sums:

$\sum x = 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 52$ $\sum y = 80.8 + 81.2 + 73.4 + 70 + 67.4 + 69.2 + 61 + 60.4 = 563.4$ $\sum xy = 242.4 + 324.8 + 367 + 420 + 471.8 + 553.6 + 549 + 604 = 3532.6$ $\sum x^2 = 9 + 16 + 25 + 36 + 49 + 64 + 81 + 100 = 380$ $\sum y^2 = 6528.64 + 6593.44 + 5388.36 + 4900 + 4542.76 + 4788.64 + 3721 + 3648.16 = 40111$

Finally, we'll substitute these values into the formula for $r$.

I'll calculate that for you.The correlation coefficient $r$ is $-0.960$, rounded to three decimal places.

This indicates a strong negative correlation between $x$ and $y$, meaning as $x$ increases, $y$ tends to decrease.

Would you like more details or have any questions?

Here are 5 related questions:

1. How does the correlation coefficient affect data interpretation?
2. What is the significance of a negative correlation?
3. How is a correlation coefficient interpreted in real-world scenarios?
4. Can correlation imply causation between two variables?
5. What is the difference between correlation and regression analysis?

Tip: Correlation coefficients range from -1 to 1, where values closer to -1 or 1 indicate stronger relationships.