To find the estimated y-intercept ($b_0$) of the regression line equation $\hat{y} = b_0 + b_1 x$, we need to first compute the slope ($b_1$) and then use the formula to find $b_0$.

The formula for the slope ($b_1$) is:

$b_1 = \frac{n \sum{x y} - \sum{x} \sum{y}}{n \sum{x^2} - (\sum{x})^2}$

And once we have $b_1$, the y-intercept $b_0$ can be calculated using the following formula:

$b_0 = \frac{\sum{y} - b_1 \sum{x}}{n}$

Where:

• $x$ represents the list price,
• $y$ represents the number of bids,
• $n$ is the number of data points (in this case, $n = 5$).

Step 1: Calculate the necessary sums

We are given the following data:

Price (x)	Number of Bids (y)
21	1
26	3
28	5
35	6
43	9

Now, let's compute the necessary sums:

• $\sum{x} = 21 + 26 + 28 + 35 + 43 = 153$
• $\sum{y} = 1 + 3 + 5 + 6 + 9 = 24$
• $\sum{xy} = (21 \times 1) + (26 \times 3) + (28 \times 5) + (35 \times 6) + (43 \times 9)$
• $\sum{x^2} = (21^2) + (26^2) + (28^2) + (35^2) + (43^2)$

Let's calculate each of these:

$\sum{xy} = (21 \times 1) + (26 \times 3) + (28 \times 5) + (35 \times 6) + (43 \times 9) = 21 + 78 + 140 + 210 + 387 = 836$

$\sum{x^2} = (21^2) + (26^2) + (28^2) + (35^2) + (43^2) = 441 + 676 + 784 + 1225 + 1849 = 4975$

Step 2: Compute the slope $b_1$

Now, substitute these sums into the formula for $b_1$:

$b_1 = \frac{5 \times 836 - 153 \times 24}{5 \times 4975 - (153)^2}$

$b_1 = \frac{4180 - 3672}{24875 - 23409}$

$b_1 = \frac{508}{466}$

$b_1 \approx 1.09$

Step 3: Compute the y-intercept $b_0$

Now, substitute the values of $b_1$, $\sum{x}$, and $\sum{y}$ into the formula for $b_0$:

$b_0 = \frac{24 - 1.09 \times 153}{5}$

$b_0 = \frac{24 - 166.77}{5}$

$b_0 = \frac{-142.77}{5}$

$b_0 \approx -28.554$

Final Answer:

The estimated y-intercept $b_0$ is approximately -28.554.

Would you like to proceed with further steps to calculate the slope and check for statistical significance? Let me know if you need more details!

Related Questions:

1. How can you calculate the slope of the regression line using the least squares method?
2. What does the correlation coefficient tell you about the strength of the relationship between the price and number of bids?
3. Why might the regression line not be a good predictor if the correlation coefficient is not statistically significant?
4. How do you interpret the value of the y-intercept in this context?
5. What would the next steps be to calculate the regression line’s equation fully?

Tip:

Remember, while the y-intercept and slope are important for finding the equation of the regression line, the significance of these values depends on the statistical analysis, such as calculating the p-value of the correlation coefficient!