To find the value of $b$ in the equation $y = a + bx$, we use the formula for the slope $b$ in a linear regression equation:

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

Where:

• $n$ is the number of data points,
• $x$ and $y$ are the given data points.

Step 1: Compute the required sums

The data points are:

$x: \{15, 17, 19, 20, 20, 21, 22, 22, 23\}$ $y: \{40, 40, 50, 52, 50, 65, 65, 66, 70\}$

Compute the following:

1. $\sum x$,
2. $\sum y$,
3. $\sum xy$ (product of corresponding $x$ and $y$),
4. $\sum x^2$ (square of each $x$).

Let me calculate these values.import pandas as pd

Data

x = [15, 17, 19, 20, 20, 21, 22, 22, 23] y = [40, 40, 50, 52, 50, 65, 65, 66, 70]

Create DataFrame

data = pd.DataFrame({'x': x, 'y': y})

Calculate sums

n = len(x) sum_x = data['x'].sum() sum_y = data['y'].sum() sum_xy = (data['x'] * data['y']).sum() sum_x2 = (data['x'] ** 2).sum()

Results

n, sum_x, sum_y, sum_xy, sum_x2(9, 179, 498, 10127, 3613)(9, 179, 498, 10127, 3613)successThe computed sums are:

$n = 9, \quad \sum x = 179, \quad \sum y = 498, \quad \sum (xy) = 10127, \quad \sum (x^2) = 3613$

Step 2: Substitute values into the formula for $b$

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

Now, let me calculate $b$.# Calculate b b = (n * sum_xy - sum_x * sum_y) / (n * sum_x2 - sum_x**2) round(b, 1)4.24.2successThe value of $b$ in the equation $y = a + bx$ is:

$b = 4.2$

Would you like a detailed breakdown of the calculation?

Related Questions:

1. How is the intercept $a$ calculated in $y = a + bx$?
2. What is the predicted $y$-value when $x = 25$?
3. Can we verify the goodness-of-fit (e.g., calculate $R^2$) for this data?
4. How does the slope $b$ influence the relationship between $x$ and $y$?
5. What are some examples of real-world applications for regression equations like $y = a + bx$?

Tip:

Understanding the slope $b$ helps you quantify how much $y$ changes for every unit increase in $x$.