Let's walk through the process of finding the correlation coefficient $r$ for the given data set $\{(2, 2), (3, 4), (5, 4)\}$.

The formula for the Pearson correlation coefficient $r$ is:

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

where:

• $n$ = number of points
• $\sum xy$ = sum of the product of corresponding $x$ and $y$ values
• $\sum x$ = sum of $x$-values
• $\sum y$ = sum of $y$-values
• $\sum x^2$ = sum of squares of $x$-values
• $\sum y^2$ = sum of squares of $y$-values

---

Step 1: Calculate the necessary sums

Given data points:
$(2, 2), (3, 4), (5, 4)$

\sum x &= 2 + 3 + 5 = 10 \\ \sum y &= 2 + 4 + 4 = 10 \\ \sum xy &= (2)(2) + (3)(4) + (5)(4) = 4 + 12 + 20 = 36 \\ \sum x^2 &= 2^2 + 3^2 + 5^2 = 4 + 9 + 25 = 38 \\ \sum y^2 &= 2^2 + 4^2 + 4^2 = 4 + 16 + 16 = 36 \\ n &= 3 \end{aligned}$$ --- ### Step 2: Plug values into the correlation formula \[ r = \frac{3(36) - (10)(10)}{\sqrt{\left[3(38) - 10^2\right] \left[3(36) - 10^2\right]}} \] Simplify each part: $$3(36) - 100 = 108 - 100 = 8$$ $$3(38) - 100 = 114 - 100 = 14 \quad \text{and} \quad 3(36) - 100 = 108 - 100 = 8$$ $$r = \frac{8}{\sqrt{14 \times 8}} = \frac{8}{\sqrt{112}}$$ Since $$\sqrt{112} \approx 10.583$$: $$r = \frac{8}{10.583} \approx 0.756$$ --- ### Step 3: Interpret the result The correlation coefficient $$r \approx 0.756$$ indicates a **moderate to strong positive linear relationship** between the variables $$x$$ and $$y$$. --- ### Explanation of $$r = 1$$ A **correlation coefficient of $$r = 1$$** suggests that all the data points lie exactly on a **positive sloping line**. In other words, there is a **perfect positive linear relationship** between the two variables—when one variable increases, the other increases proportionally. --- Do you have any questions or want further details? Here are five related questions to deepen your understanding: 1. What is the difference between positive and negative correlation? 2. How would the value of $$r$$ change if the dataset had more variability? 3. How is the value of $$r$$ affected if we reverse the roles of $$x$$ and $$y$$? 4. What are some real-life applications where correlation coefficients are useful? 5. How is a correlation coefficient different from covariance? **Tip:** A correlation coefficient close to $$0$$ indicates little to no linear relationship, but it does not imply that the variables are unrelated in all ways!