The image you uploaded provides data for calculating the correlation coefficient $r$, the slope $b$, and the intercept $a$ for a linear regression model. Let's break it down and calculate each:

Given Data:

• $X = [30, 45, 32, 55, 50, 45, 61, 44, 50, 32]$
• $Y = [9, 30, 12, 24, 15, 54, 22, 17, 5, 17]$

Step 1: Calculating the correlation coefficient $r$

The formula for the correlation coefficient $r$ 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]}} ]

Where:

• $n$ is the number of data points
• $\sum x$ is the sum of $X$
• $\sum y$ is the sum of $Y$
• $\sum xy$ is the sum of the products of corresponding $X$ and $Y$ values
• $\sum x^2$ is the sum of the squares of $X$
• $\sum y^2$ is the sum of the squares of $Y$

Step 2: Calculating the slope $b$ and intercept $a$

The formulas for $b$ (slope) and $a$ (intercept) are:

$b = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}$ $a = \frac{\sum y - b(\sum x)}{n}$

Let me calculate these values for you now. numpy as np

Given data

X = np.array([30, 45, 32, 55, 50, 45, 61, 44, 50, 32]) Y = np.array([9, 30, 12, 24, 15, 54, 22, 17, 5, 17])

Number of data points

n = len(X)

Summing the required terms

sum_x = np.sum(X) sum_y = np.sum(Y) sum_x2 = np.sum(X2) sum_y2 = np.sum(Y2) sum_xy = np.sum(X * Y)

Calculating the correlation coefficient (r)

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

Calculating the slope (b) and intercept (a)

b = (n * sum_xy - sum_x * sum_y) / (n * sum_x2 - sum_x**2) a = (sum_y - b * sum_x) / n

r, b, aHere are the calculated values:

• The correlation coefficient $r$ is approximately 0.2214.
• The slope $b$ is approximately 0.2959.
• The intercept $a$ is approximately 7.3601.

Would you like further clarification or details on any of these calculations?

Here are some related questions you might find useful:

1. How does the correlation coefficient $r$ influence the relationship between variables?
2. What is the interpretation of the slope $b$ in a linear regression model?
3. How can you use the equation of the line $Y = a + bX$ to predict $Y$ from $X$?
4. How can you visualize the data and regression line?
5. What are some possible reasons for a low correlation coefficient?

Tip: A positive correlation coefficient indicates that as $X$ increases, $Y$ tends to increase as well, though the value here suggests only a weak relationship.