To compute the correlation coefficient $r$ by hand, we will use the formula for the Pearson correlation coefficient:

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

Given Data:

$x = [2, 4, 6, 6, 7]$
$y = [4, 8, 12, 13, 19]$
$n = 5$ (since there are 5 pairs of data)

We need to calculate the following values:

• $\sum x$
• $\sum y$
• $\sum xy$
• $\sum x^2$
• $\sum y^2$

I will calculate these step by step for you.import numpy as np

Given data

x = np.array([2, 4, 6, 6, 7]) y = np.array([4, 8, 12, 13, 19]) n = len(x)

Calculations

sum_x = np.sum(x) sum_y = np.sum(y) sum_xy = np.sum(x * y) sum_x2 = np.sum(x ** 2) sum_y2 = np.sum(y ** 2)

Correlation coefficient formula

r = (n * sum_xy - sum_x * sum_y) / np.sqrt((n * sum_x2 - sum_x ** 2) * (n * sum_y2 - sum_y ** 2)) rThe correlation coefficient $r$ is approximately $0.955$ (rounded to three decimal places).

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:

1. How is the correlation coefficient interpreted in terms of strength and direction?
2. What are the limitations of using the Pearson correlation coefficient?
3. How does the value of $r$ relate to positive and negative association?
4. What is the difference between correlation and causation?
5. How do you compute the correlation coefficient when given larger datasets?

Tip: A correlation coefficient close to 1 indicates a strong positive linear relationship between variables.