To find the y-intercept ($b_0$) of the regression line $\hat{y} = b_0 + b_1x$, we need the following steps.

1. Calculate the means of both $x$ (price) and $y$ (number of bids).

Let’s define the variables:

• $x$ is the price in dollars.
• $y$ is the number of bids.

The data is:

• $x = \{21, 23, 29, 43, 50\}$
• $y = \{1, 2, 4, 5, 7\}$

We need to compute the mean of both $x$ and $y$: $\bar{x} = \frac{\sum x_i}{n}, \quad \bar{y} = \frac{\sum y_i}{n}$ where $n = 5$ (the number of data points).

2. Compute the slope ($b_1$).

The formula for $b_1$ (slope) is: $b_1 = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}$

3. Find the y-intercept ($b_0$).

Once we know $b_1$, the y-intercept is found using: $b_0 = \bar{y} - b_1 \bar{x}$

Let’s calculate the necessary values to find the y-intercept.The estimated y-intercept ($b_0$) of the regression line is approximately $-2.197$ (rounded to three decimal places).

Would you like further details or have any questions about this calculation?

Here are five related questions for further exploration:

1. How is the slope ($b_1$) of the regression line interpreted in this context?
2. How do we determine if the correlation coefficient is statistically significant?
3. What assumptions must be met to use linear regression?
4. How would the regression line change if one of the data points were an outlier?
5. How can you use the regression line to make predictions?

Tip: Always check the correlation coefficient before using a regression model to make predictions. A weak correlation suggests the model may not be reliable.