To calculate the correlation coefficient $r$, we will use the 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]}} ]

Steps:

1. Extract the values of $x$ and $y$ from the table.
2. Calculate:
   • $\sum{x}$
   • $\sum{y}$
   • $\sum{xy}$
   • $\sum{x^2}$
   • $\sum{y^2}$
3. Substitute the results into the formula.

Let me perform these calculations for you. numpy as np

Data from the table

x = np.array([4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]) y = np.array([39.21, 33.1, 30.79, 30.68, 27.07, 24.86, 22.55, 18.04, 17.73, 13.22, 12.21, 7.4])

Calculate required sums

n = len(x) sum_x = np.sum(x) sum_y = np.sum(y) sum_xy = np.sum(x * y) sum_x2 = np.sum(x2) sum_y2 = np.sum(y2)

Correlation coefficient formula

numerator = n * sum_xy - sum_x * sum_y denominator = np.sqrt((n * sum_x2 - sum_x2) * (n * sum_y2 - sum_y2)) r = numerator / denominator

rThe correlation coefficient $r$ is approximately $-0.992$ (to three decimal places).

This indicates a very strong negative linear relationship between the year ($X$) and the high temperature ($Y$).

Do you want me to break down the calculations further or assist with anything else?

Here are some related questions to explore:

1. What does a negative correlation coefficient signify in practical terms?
2. How would you interpret this trend in terms of temperature changes over the years?
3. Can you use the correlation coefficient to predict temperatures? Why or why not?
4. How is the correlation coefficient related to causation?
5. What would happen to the correlation if there were an outlier in the data?

Tip: Always visualize the data (e.g., scatter plot) to better understand the correlation!