To find the estimated y-intercept $b_0$ of the regression line $\hat{y} = b_0 + b_1x$, we will use the formula:

$b_0 = \bar{y} - b_1 \bar{x}$

Where:

• $\bar{y}$ is the mean of the number of bids (the dependent variable),
• $\bar{x}$ is the mean of the list prices (the independent variable),
• $b_1$ is the slope of the regression line, which we will calculate first.

Step 1: Calculate the means of $x$ (list price) and $y$ (number of bids).

The values for $x$ (price in dollars) are: $23, 34, 44, 46, 50$

The values for $y$ (number of bids) are: $1, 2, 4, 9, 10$

First, let's compute the means:

$\bar{x} = \frac{\sum x}{n} = \frac{23 + 34 + 44 + 46 + 50}{5} = \frac{197}{5} = 39.4$

$\bar{y} = \frac{\sum y}{n} = \frac{1 + 2 + 4 + 9 + 10}{5} = \frac{26}{5} = 5.2$

Step 2: Calculate the slope $b_1$

The formula for the slope $b_1$ is:

$b_1 = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}$

We need to compute the following:

1. $(x_i - \bar{x})$ for each $x$,
2. $(y_i - \bar{y})$ for each $y$,
3. The product $(x_i - \bar{x})(y_i - \bar{y})$ for each pair,
4. The squared differences $(x_i - \bar{x})^2$.

Let me now calculate $b_1$ and then $b_0$. numpy as np

Given data

x = np.array([23, 34, 44, 46, 50]) y = np.array([1, 2, 4, 9, 10])

Calculate means

mean_x = np.mean(x) mean_y = np.mean(y)

Calculate slope (b1)

numerator = np.sum((x - mean_x) * (y - mean_y)) denominator = np.sum((x - mean_x)**2) b1 = numerator / denominator

Calculate y-intercept (b0)

b0 = mean_y - b1 * mean_x

Round the y-intercept to three decimal places

round(b0, 3)The estimated y-intercept $b_0$ of the regression line is approximately -7.784.

Would you like to proceed with calculating the slope or any other details? Here are a few related questions:

1. How would you calculate the slope $b_1$ for this data?
2. How can you assess whether the correlation coefficient is statistically significant?
3. What would be the interpretation of the y-intercept value in this context?
4. How do you use the regression line to make predictions based on the list price?
5. What are the assumptions of linear regression that we need to consider in this analysis?

Tip: Always check the residuals (the differences between actual and predicted values) to assess the goodness of fit and whether the linear model is appropriate.