To compute the least-squares regression line, we need to determine the slope ($m$) and the y-intercept ($b$) using the formula for the regression line:

$\hat{y} = mx + b$

The formulas for $m$ and $b$ are: $m = \frac{n\sum(xy) - \sum x \sum y}{n\sum(x^2) - (\sum x)^2}$ $b = \frac{\sum y - m\sum x}{n}$

Let’s extract the data: $x = [3.8, 6.1, 4.4, 4.3, 1.6, 3.4, 3.3]$ $y = [4.2, 4.5, 5.6, 4.8, 8.8, 5.5, 4.6]$

We will:

1. Calculate the necessary sums ($\sum x$, $\sum y$, $\sum xy$, and $\sum x^2$).
2. Substitute the sums into the formulas for $m$ and $b$.
3. Round the results to 4 decimal places.

Let me calculate this for you. numpy as np

Data points

x = np.array([3.8, 6.1, 4.4, 4.3, 1.6, 3.4, 3.3]) y = np.array([4.2, 4.5, 5.6, 4.8, 8.8, 5.5, 4.6])

Number of data points

n = len(x)

Calculations for the least-squares regression

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

Slope (m)

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

Intercept (b)

b = (sum_y - m * sum_x) / n

Rounding results to 4 decimal places

m_rounded = round(m, 4) b_rounded = round(b, 4)

m_rounded, b_roundedThe equation of the least-squares regression line is:

$\hat{y} = -0.8421x + 8.6646$

Would you like a detailed explanation of the calculations, or do you have any questions?

Here are five related questions to explore:

1. How can we interpret the slope ($-0.8421$) in the context of this data?
2. What does the y-intercept ($8.6646$) represent?
3. How can we calculate the coefficient of determination ($R^2$) to measure the fit of the line?
4. How do we use this regression equation to predict values for new $x$ inputs?
5. What assumptions underlie the least-squares regression model?

Tip: Always visualize your data with a scatter plot and the regression line to understand the relationship visually!