To compute the least-squares regression line $\hat{y} = b_0 + b_1x$, we need the following:

1. Calculate the slope ($b_1$): $b_1 = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}$

2. Calculate the intercept ($b_0$): $b_0 = \bar{y} - b_1\bar{x}$

Let me compute these values step by step using the data from the table:

$x = [3.8, 6.1, 4.4, 4.3, 1.6, 3.4, 3.3]$ $y = [4.2, 4.5, 5.6, 4.8, 8.8, 5.5, 4.6]$

Step 1: Find means of $x$ and $y$

We need to calculate $\bar{x}$ and $\bar{y}$.

$\bar{x} = \frac{\sum x}{n}, \quad \bar{y} = \frac{\sum y}{n}$

Step 2: Compute $\sum (x_i - \bar{x})(y_i - \bar{y})$ and $\sum (x_i - \bar{x})^2$

Step 3: Compute $b_1$ and $b_0$

I’ll calculate these values now. numpy as np

Data

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])

Step 1: Calculate means of x and y

x_mean = np.mean(x) y_mean = np.mean(y)

Step 2: Calculate the terms for slope (b1) and intercept (b0)

numerator = np.sum((x - x_mean) * (y - y_mean)) # Sum of (x_i - x_mean) * (y_i - y_mean) denominator = np.sum((x - x_mean) ** 2) # Sum of (x_i - x_mean)^2

b1 = numerator / denominator # Slope b0 = y_mean - b1 * x_mean # Intercept

b0, b1The least-squares regression line is:

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

Let me know if you'd like a detailed explanation of each step or assistance with additional parts of the problem!

Here are 5 related questions you might explore:

1. How do we interpret the slope ($b_1$) in a regression line?
2. How do outliers affect the regression line?
3. What is the coefficient of determination ($R^2$) and how can we calculate it?
4. Can we predict $y$ for a specific $x$ value using this equation?
5. How do we check if this regression line is statistically significant?

Tip: Always check the assumptions of linear regression (linearity, homoscedasticity, and independence) when interpreting results!