Consider the data set {(2, 2), (3, 4), (5, 4)}, 

1. Calculate the correlation coefficient r for this set of data.
2. Explain what a correlation coefficient of r=1 would suggest about the data.

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:  
\( (2, 2), (3, 4), (5, 4) \)  

\[
\begin{aligned}
\sum x &= 2 + 3 + 5 = 10 \\
\sum y &= 2 + 4 + 4 = 10 \\
\sum xy &= (2)(2) + (3)(4) + (5)(4) = 4 + 12 + 20 = 36 \\
\sum x^2 &= 2^2 + 3^2 + 5^2 = 4 + 9 + 25 = 38 \\
\sum y^2 &= 2^2 + 4^2 + 4^2 = 4 + 16 + 16 = 36 \\
n &= 3
\end{aligned}
\]

---

### 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!

Consider the data set {(2, 2), (3, 4), (5, 4)}, 1. Calculate the correlation coefficient r for this set of data. 2. Explain what a correlation coefficient of r=1 would suggest about the data.

Discover how to calculate the correlation coefficient for the data set {(2, 2), (3, 4), (5, 4)}. Follow a step-by-step solution, using the Pearson correlation formula, to understand the linear relationship between variables x and y. Suitable for high school students in Grades 10-12.

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