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To solve for the correlation coefficient \( r \) between the two variables \( X \) (year) and \( Y \) (temperature) based on the data provided, I will first extract the \( X \) and \( Y \) values from the table. After that, I'll use the formula for the Pearson correlation coefficient, which is:

\[
r = \frac{n(\sum XY) - (\sum X)(\sum Y)}{\sqrt{[n \sum X^2 - (\sum X)^2][n \sum Y^2 - (\sum Y)^2]}}
\]

Where:
- \( X \) represents the year,
- \( Y \) represents the high temperature,
- \( n \) is the number of data points (13 in this case).

Let me calculate this for you.The correlation coefficient \( r \) between \( X \) (the year) and \( Y \) (the high temperature) is approximately:

\[
r = 0.974
\]

This indicates a strong positive linear relationship between the two variables.

Would you like any further details or have questions about this calculation?

Here are 5 questions that might help expand your understanding:
1. What does a correlation coefficient close to 1 imply about the relationship between two variables?
2. How would the interpretation of the correlation coefficient change if the result were negative?
3. Can correlation coefficients ever exceed 1 or fall below -1?
4. What assumptions are involved in calculating Pearson's correlation coefficient?
5. How does the correlation coefficient differ from covariance?

**Tip:** While correlation shows the strength of a relationship between two variables, it doesn't imply causation! Always consider additional factors when interpreting results.

Annual high temperatures in a certain location have been tracked for several years. Let X represent the year and Y the high temperature. Based on the data shown below, calculate the correlation coefficient (to three decimal places) between X and Y. Use your calculator!

This problem involves calculating the Pearson correlation coefficient between the year and high temperature data. Using the provided formula, we find the correlation coefficient to three decimal places, reflecting the strength and direction of the linear relationship.

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