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Solve for the sample correlation coefficient

To solve for the sample correlation coefficient \( r \), we'll use the formula for Pearson's correlation coefficient:

\[
r = \frac{n \sum xy - \sum x \sum y}{\sqrt{n \sum x^2 - (\sum x)^2} \sqrt{n \sum y^2 - (\sum y)^2}}
\]

Where:
- \( n \) is the number of data points,
- \( x \) and \( y \) are the given per capita income values for 1980 and 1999,
- \( xy \) is the product of each pair of \( x \) and \( y \),
- \( \sum x \), \( \sum y \), \( \sum xy \), \( \sum x^2 \), and \( \sum y^2 \) are the necessary sums to calculate.

### Steps:
1. Extract the values of \( x \), \( y \), and \( xy \) from the table.
2. Compute the necessary sums: \( \sum x \), \( \sum y \), \( \sum xy \), \( \sum x^2 \), and \( \sum y^2 \).
3. Plug the sums into the formula to find the correlation coefficient \( r \).

Let me extract the data and calculate these sums now.The sample correlation coefficient \( r \) is approximately \( 0.829 \), indicating a strong positive correlation between the 1980 and 1999 per capita incomes in this dataset.

Would you like further details or explanations about this calculation? 

Here are some additional questions to expand on this topic:
1. What does a correlation coefficient of 0.829 imply about the relationship between two variables?
2. How would you calculate the regression line based on this data?
3. What are the limitations of Pearson's correlation coefficient?
4. Could the correlation coefficient be affected by outliers in the data?
5. How can we interpret this correlation in economic terms?

**Tip:** The closer the correlation coefficient is to 1, the stronger the linear relationship between the two variables.

Learn how to calculate Pearson's correlation coefficient using per capita income data from 1980 and 1999. This solution walks you through the step-by-step process, covering key statistical concepts, including correlation and linear relationships. Suitable for students in Grades 10-12.

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