To calculate the correlation coefficient $r$ between the year $X$ and the high temperature $Y$, you can use the Pearson correlation formula:

[ 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:

• $X$ and $Y$ are the values from the table.
• $n$ is the number of data points (in this case, 15).

Steps:

1. Calculate $\sum X$, $\sum Y$, $\sum X^2$, $\sum Y^2$, and $\sum XY$.
2. Plug these values into the correlation formula.
3. Compute $r$.

I'll help you calculate this in detail:

Data from the table:

• $X = [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]$
• $Y = [25.76, 23.02, 23.68, 23.64, 24.4, 21.06, 21.02, 19.98, 20.04, 19.8, 18.26, 16.22, 16.78, 14.14, 13.00]$

I'll now perform the necessary calculations to find the correlation coefficient $r$.The correlation coefficient $r$ between the year $X$ and the high temperature $Y$ is approximately $-0.963$ (to three decimal places). This indicates a strong negative correlation, meaning that as the year increases, the high temperature tends to decrease.

Do you need further details or explanations?

Here are five related questions you might find useful:

1. What does a negative correlation coefficient imply in this context?
2. How do you interpret the strength of the correlation coefficient?
3. How could we visualize this data using a scatter plot?
4. What other statistical measures can we compute to analyze this data further?
5. How might external factors influence the temperature trend over the years?

Tip: A correlation coefficient close to -1 or 1 indicates a very strong relationship, while values near 0 suggest a weak relationship.