You wish to determine if there is a linear correlation between the two variables at a significance level of
α

To determine the correlation coefficient $r$ for this data set, we can use the Pearson correlation formula, defined as:

$r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \cdot \sum (y_i - \bar{y})^2}}$

where:

• $x_i$ and $y_i$ are individual data points for variables $x$ and $y$,
• $\bar{x}$ and $\bar{y}$ are the mean values of $x$ and $y$,
• $\sum$ denotes summation across all data points.

I'll calculate this correlation coefficient now.The correlation coefficient $r$ for this data set is approximately:

$r = 0.651$

This indicates a moderate positive linear relationship between $x$ and $y$.

Would you like to proceed with further analysis to test the significance of this correlation at $\alpha = 0.10$?

Here are five additional questions related to this problem:

1. What is the interpretation of this correlation coefficient in the context of the data?
2. How can we test if this correlation is statistically significant?
3. What would a correlation coefficient close to 1 or -1 indicate?
4. How do outliers affect the correlation coefficient?
5. What other methods can be used to measure correlation besides Pearson’s $r$?

Tip: Always visualize your data with a scatter plot to gain insights into the pattern before calculating correlation.