To calculate the correlation coefficient (r) for the given data, 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 data from the image:

   • $x = [2, 3, 4, 5, 6, 7]$
   • $y = [23, 21, 10, 11, 15, 14]$

2. Compute intermediate values:

   • $\sum x, \sum y, \sum x^2, \sum y^2, \sum xy$
   • $n = 6$ (number of data points)

Let me calculate these values step-by-step.import pandas as pd

Data from the image

data = { 'x': [2, 3, 4, 5, 6, 7], 'y': [23, 21, 10, 11, 15, 14] }

Convert to DataFrame

df = pd.DataFrame(data)

Calculate the necessary sums

df['x^2'] = df['x'] ** 2 df['y^2'] = df['y'] ** 2 df['xy'] = df['x'] * df['y']

sums = df.sum() # Summing up all necessary values n = len(df) # Number of data points

Compute correlation coefficient components

numerator = n * sums['xy'] - sums['x'] * sums['y'] denominator = ((n * sums['x^2'] - sums['x'] ** 2) * (n * sums['y^2'] - sums['y'] ** 2)) ** 0.5 correlation_coefficient = numerator / denominator

correlation_coefficient_rounded = round(correlation_coefficient, 2)

correlation_coefficient_rounded-0.63-0.63successThe correlation coefficient (rounded to two decimal places) for the given data is -0.63.

This indicates a moderate negative relationship between the variables $x$ and $y$.

Would you like a detailed breakdown of the calculations or further explanation?

Expanding Questions:

1. What does a negative correlation coefficient signify?
2. How does the strength of the correlation coefficient affect data interpretation?
3. What are potential applications of calculating the correlation coefficient?
4. Could this data represent any real-world phenomenon? If so, what could it be?
5. How does this correlation coefficient compare to a perfect positive or negative correlation?

Tip:

Always visualize your data with a scatter plot when interpreting correlation—it provides valuable insights into the relationship.