To solve this, we need to calculate the correlation coefficient $r$ for the given data points. The correlation coefficient formula 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]}} ]

Step 1: Organize the data

Here’s the provided data:

$x = \{2, 3, 4, 5, 6, 7, 8\}$ $y = \{18.76, 18.24, 19.62, 20.6, 21.78, 25.66, 23.34\}$

We'll compute:

1. $\sum x$, $\sum y$
2. $\sum x^2$, $\sum y^2$
3. $\sum (xy)$

Let me calculate these values. Define the data x = [2, 3, 4, 5, 6, 7, 8] y = [18.76, 18.24, 19.62, 20.6, 21.78, 25.66, 23.34]

Calculate the required sums

n = len(x) # Number of data points sum_x = sum(x) sum_y = sum(y) sum_x2 = sum([xi2 for xi in x]) sum_y2 = sum([yi2 for yi in y]) sum_xy = sum([xi * yi for xi, yi in zip(x, y)])

Store the results for display

sum_x, sum_y, sum_x2, sum_y2, sum_xyHere are the computed sums:

• $\sum x = 35$
• $\sum y = 148.00$
• $\sum x^2 = 203$
• $\sum y^2 = 3171.4992$
• $\sum (xy) = 770.74$

Step 2: Plug into the formula for $r$

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

Let me compute $r$. math

Compute the numerator and denominator

numerator = n * sum_xy - sum_x * sum_y denominator = math.sqrt((n * sum_x2 - sum_x2) * (n * sum_y2 - sum_y2))

Calculate the correlation coefficient

r = numerator / denominator rThe correlation coefficient $r$ is approximately:

$r = 0.893$

This indicates a strong positive linear relationship between the year ($x$) and the high temperature ($y$).

Would you like a step-by-step explanation of how the formula was applied or further clarifications?

Here are 5 related questions to explore:

1. What does the correlation coefficient signify in practical terms?
2. How can you compute the coefficient of determination $r^2$ from $r$?
3. How do you interpret a negative correlation coefficient?
4. What are other methods to assess the relationship between two variables besides correlation?
5. Can we use this correlation to predict future temperatures?

Tip: When calculating correlation coefficients, always double-check your data input and ensure no outliers significantly skew the results.