To determine the value of $a$ in the equation $y = a + bx$, we need to perform a linear regression. The general formula for the slope ($b$) and intercept ($a$) in a linear regression is:

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

$a = \frac{\sum y - b \sum x}{n}$

Where:

• $n$ is the number of data points
• $\sum x$ is the sum of the $x$-values
• $\sum y$ is the sum of the $y$-values
• $\sum(xy)$ is the sum of the product of $x$ and $y$
• $\sum(x^2)$ is the sum of the squares of $x$-values

Let's compute these values.import numpy as np import pandas as pd

Data

x = np.array([15, 17, 19, 20, 20, 21, 22, 22, 23]) y = np.array([40, 40, 50, 52, 50, 65, 65, 66, 70])

Number of data points

n = len(x)

Calculations for linear regression

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

Calculating slope (b) and intercept (a)

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

a_rounded = round(a, 1)

a_rounded-28.3-28.3successThe value of $a$ in the equation $y = a + bx$ is $-28.3$ (rounded to the nearest tenth).

Would you like details on the full calculations or have any questions?

Here are 5 related questions:

1. How is the slope ($b$) calculated, and what is its significance in the equation?
2. What is the full equation of the line of best fit for this dataset?
3. Can we calculate the correlation coefficient ($r$) for this data?
4. How would the result change if the dataset was modified?
5. What are the practical applications of regression analysis in real-world problems?

Tip: Always check for outliers in the data, as they can significantly influence the regression line.