Let's work through the steps from the image you provided:

Step 4: Sum the products of each pair of z-scores

We have the following pairs of z-scores:

• $z_{x1} \cdot z_{y1} = -1.94$
• $z_{x2} \cdot z_{y2} = 0.8475$
• $z_{x3} \cdot z_{y3} = -0.132$

To find the sum, we calculate: $\sum(z_{xi} \cdot z_{yi}) = (-1.94) + 0.8475 + (-0.132)$

Let me compute that.

The sum of the products of the z-scores is:

$\sum(z_{xi} \cdot z_{yi}) = -1.2245$

Step 5: Calculate the correlation coefficient

The formula for the correlation coefficient is:

$r = \frac{\sum(z_{xi} \cdot z_{yi})}{n - 1}$

Given that $n = 3$ (since there are three pairs of z-scores), we can calculate:

$r = \frac{-1.2245}{3 - 1} = \frac{-1.2245}{2}$

Let me calculate that.

The correlation coefficient $r$ is:

$r = -0.61225$

This indicates a moderate negative correlation between the variables.

Do you want more details or have any questions?

Here are 5 related questions to expand your understanding:

1. How is the correlation coefficient interpreted in terms of strength and direction of a relationship?
2. What are the potential causes of negative correlations?
3. How would you calculate the correlation if there were more data points?
4. What is the significance of $r$ values close to 1, 0, and -1?
5. How can outliers affect the correlation coefficient?

Tip: Always check for linearity in data before relying on the correlation coefficient for meaningful interpretation.