To find the least-squares regression line equation, we need to compute the slope and the y-intercept. The regression line equation is given by:

$y = mx + b$

Where:

• $y$ is the price of milk,
• $x$ is the price of eggs,
• $m$ is the slope,
• $b$ is the y-intercept.

The slope $m$ and y-intercept $b$ can be calculated using the following formulas:

1. Slope (m): $m = \frac{n \sum (xy) - \sum x \sum y}{n \sum x^2 - (\sum x)^2}$

2. Y-intercept (b): $b = \frac{\sum y - m \sum x}{n}$

Where:

• $n$ is the number of data points,
• $\sum x$ is the sum of the prices of eggs,
• $\sum y$ is the sum of the prices of milk,
• $\sum xy$ is the sum of the product of the corresponding prices of eggs and milk,
• $\sum x^2$ is the sum of the squares of the prices of eggs.

Now, let's compute the necessary sums and then find the slope and intercept.

Given Data:

\hline \text{Price of Eggs} (x) & \text{Price of Milk} (y) \\ \hline 1.33 & 3.12 \\ 1.77 & 3.08 \\ 0.89 & 3.07 \\ 1.37 & 3.43 \\ 1.50 & 3.57 \\ 1.63 & 3.89 \\ 1.83 & 3.77 \\ 0.93 & 2.89 \\ 1.26 & 2.73 \\ 1.46 & 3.11 \\ \hline \end{array}$$ I'll calculate the sums and the regression coefficients. numpy as np # Given data: price of eggs (x) and price of milk (y) eggs = np.array([1.33, 1.77, 0.89, 1.37, 1.50, 1.63, 1.83, 0.93, 1.26, 1.46]) milk = np.array([3.12, 3.08, 3.07, 3.43, 3.57, 3.89, 3.77, 2.89, 2.73, 3.11]) # Number of data points n = len(eggs) # Sums of x, y, xy, x^2 sum_x = np.sum(eggs) sum_y = np.sum(milk) sum_xy = np.sum(eggs * milk) sum_x_squared = np.sum(eggs ** 2) # Calculate slope (m) and intercept (b) using the formulas m = (n * sum_xy - sum_x * sum_y) / (n * sum_x_squared - sum_x ** 2) b = (sum_y - m * sum_x) / n m, bThe least-squares regression line for predicting the price of milk from the price of eggs is: $$y = 0.7483x + 2.2206$$ Where: - $$y$$ is the price of milk, - $$x$$ is the price of eggs, - The slope $$m = 0.7483$$, - The y-intercept $$b = 2.2206$$. Would you like more details or have any follow-up questions? Here are some related questions you might find interesting: 1. How do we interpret the slope and y-intercept in this context? 2. How would we use this regression line to predict the price of milk if the price of eggs is $2.00? 3. What does the correlation coefficient tell us about the relationship between the price of eggs and milk? 4. How can we check the accuracy of this regression model using residuals? 5. How does the least-squares method minimize the error in this case? **Tip**: When working with regression lines, always check the residuals (the differences between the observed and predicted values) to assess how well the model fits the data.