To find the correlation coefficient $r$ for the given data, we use the formula:

$r = \frac{n \sum xy - \sum x \sum y}{\sqrt{\left(n \sum x^2 - (\sum x)^2\right)\left(n \sum y^2 - (\sum y)^2\right)}}$

Where:

• $n$ is the number of data points
• $x$ and $y$ are the values from the table
• $\sum x$, $\sum y$, $\sum xy$, $\sum x^2$, and $\sum y^2$ are the sums needed for the formula.

Step 1: Organize the data

Let’s calculate each necessary term:

Step 2: Calculate the sums

Now add up all the columns:

$\sum x = 1.5 + (-1.8) + (-2.5) + (-2.1) + (-13.2) + 2.4 = -15.7$

$\sum y = -5.4 + 5.5 + 4 + 1 + (-6.5) + (-4.9) = -6.3$

$\sum xy = -8.1 + (-9.9) + (-10) + (-2.1) + 85.8 + (-11.76) = 43.94$

$\sum x^2 = 2.25 + 3.24 + 6.25 + 4.41 + 174.24 + 5.76 = 196.15$

$\sum y^2 = 29.16 + 30.25 + 16 + 1 + 42.25 + 24.01 = 142.67$

The number of data points $n = 6$.

Step 3: Substitute into the formula

$r = \frac{n \sum xy - \sum x \sum y}{\sqrt{\left(n \sum x^2 - (\sum x)^2\right)\left(n \sum y^2 - (\sum y)^2\right)}}$

Substitute the known values:

• $n = 6$
• $\sum xy = 43.94$
• $\sum x = -15.7$
• $\sum y = -6.3$
• $\sum x^2 = 196.15$
• $\sum y^2 = 142.67$

$r = \frac{6(43.94) - (-15.7)(-6.3)}{\sqrt{\left(6(196.15) - (-15.7)^2\right)\left(6(142.67) - (-6.3)^2\right)}}$

1. Calculate each term:

   • $6(43.94) = 263.64$
   • $(-15.7)(-6.3) = 98.91$
   • $6(196.15) = 1176.9$
   • $(-15.7)^2 = 246.49$
   • $6(142.67) = 856.02$
   • $(-6.3)^2 = 39.69$

2. Substitute back:

$r = \frac{263.64 - 98.91}{\sqrt{\left(1176.9 - 246.49\right)\left(856.02 - 39.69\right)}}$

Simplify the terms:

• $263.64 - 98.91 = 164.73$
• $1176.9 - 246.49 = 930.41$
• $856.02 - 39.69 = 816.33$

$r = \frac{164.73}{\sqrt{930.41 \cdot 816.33}}$

3. Multiply and take the square root:

• $930.41 \cdot 816.33 = 759876.5$
• $\sqrt{759876.5} \approx 871.68$

4. Final division:

$r = \frac{164.73}{871.68} \approx 0.19$

Final Answer:

The correlation coefficient $r$ is approximately 0.19.

Do you want further clarification or breakdown of any steps?

Related Questions:

1. What does the value of $r = 0.19$ indicate about the relationship between $x$ and $y$?
2. How do you interpret a weak positive correlation in real-world data?
3. Can you explain the process for calculating $r$ in a spreadsheet (e.g., Excel or Google Sheets)?
4. What are the properties and limitations of the correlation coefficient $r$?
5. How would the correlation coefficient change if one outlier was removed from the data set?

Tip:

To check your work efficiently, use a graphing calculator, spreadsheet software, or statistical tools like Python or R. These tools can calculate $r$ automatically!