https://file.aiquickdraw.com/mathbot/file/1727387248453jrzk2cfi.jpg

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.

Given z-scores for three data pairs, calculate the sum of the products of each pair and then calculate the correlation coefficient.

Learn how to calculate the correlation coefficient by summing the products of z-scores and applying the Pearson correlation formula. This problem involves three pairs of z-scores, demonstrating statistical correlation calculations. Suitable for students in Grades 10-12.

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